## 7 Computation of Magnetization in Body Fragments from Vases 1-6  
Measurements by the three-dimensional orthogonal system of sensors/magnet apparatus are taken in the same position, with the \(x\)-sensor oriented perpendicularly to the horizontal and vertical component of the geomagnetic field. The measurement positions are situated at the edges of a mesh formed from lines vertical to the traces of the grooves from the formation of the six vases (Figure 4b) on a pottery wheel. To exploit the rotational symmetry of vases (Figure 14a), the measurements are taken with respect to a right-hand coordinate system, with the \(x\)-axis along the groove direction, the \(y\)-axis perpendicular to the surface and pointing towards the inside of the fragments, and the \(z\)-axis pointing towards the base of the vases. The components Mi (i = x, y, z) are computed from the measurements B\({}_{i}\) of the field, in the rotating position where the grooves are oriented in the direction of the x-sensor. The components of the measured field, B, and of the magnetization, M, in the considered coordinate system (Figure 14a) are computed by the relations:  
B\({}_{x}\) = B. sin\(\theta_{B}\). cos\(\varphi\), B\({}_{y}\) = B.sin\(\theta_{B}\).sin\(\varphi\), B\({}_{z}\) = B.cos\(\theta_{B}\), M\({}_{x}\) = M. sin\(\theta\). cos\(\varphi\), M\({}_{y}\) = M.sin\(\theta\). sin\(\varphi\), M\({}_{z}\) = M.cos\(\theta\)  
To determine the deviation, \(\theta\), of the magnetization from the vertical (Figure 14b), measurements of the B\({}_{xz}\), B\({}_{y}\) are taken, by rotating the fragments at an angle \(\varphi_{B}\) = \(\varphi\) and orienting B\({}_{xz}\) in the x-direction, in the position where B\({}_{z}\) = 0 and B\({}_{xz}\) > 0. The distance, D, between the measurement position and the fragment edge in the common direction of the **M\({}_{xz}\)**,**B\({}_{xz}\)** (Figure 14c) is measured in the negative or positive \(x\)-semi-axis, when B\({}_{y}\) > 0 or B\({}_{y}\) < 0. The distance D (Figure 14d) may be sufficient (D \(>\) d = L.|tan\(\theta\)|) or insufficient (D \(<\) d), for the measured fragment thickness, L, at the measurement position. In any case, the magnetization angles \(\theta=\theta_{\rm D}\) or \(\theta=\theta_{\rm L}\) are approximated [41] by the solution of the 3 rd degree equations (using the Newton-Raphson method):  
\[ \begin{split}\text{D}>\text{d}:&\frac{\text{B}_{ xx}}{\text{B}_{y}}=\tan\theta_{\rm B}=\tan\theta_{\rm L}\cdot\sqrt{\frac{1+(\frac{ \text{B}_{xx}\text{B}_{\rm L}}{\text{L}})^{2}}{1+(\frac{\text{B}_{xx}\text{B} _{\rm L}}{\text{L}})^{2}}}\\ \text{D}<\text{d}:&\frac{\text{B}_{xx}}{\text{B}_{y}}= \tan\theta_{\rm B}=\tan\theta_{\rm D}\cdot\sqrt{\frac{1+(\frac{\text{B}_{yy} \text{B}_{\rm L}}{\text{D}})^{2}}{1+(\frac{\text{B}_{yy}\text{B}_{\rm L}}{ \text{D}})^{2}}}\end{split}  \tag{7}\]  
The constants \(\alpha_{xx}\)/\(\alpha_{y}\)+, \(\lambda\) are computed (Table 2) from the field measurements in co-belonging base fragments of the vases:  
* If \(\theta_{\rm L}>\theta_{\rm D}\), then \(\text{D}>\text{d}=\text{L.tan}\theta_{\rm L}\) and \(\theta=\theta_{\rm L}\). For the computation of the components of the magnetization, M, the sufficient length Equations (4) are used. * -If \(\theta_{\rm L}<\theta_{\rm D}\), then \(\text{D}<\text{d}=\text{L.tan}\theta_{\rm D}\) and \(\theta=\theta_{\rm D}\). For the computation of the components of the magnetization, M, the insufficient length Equations (5) are used. Experimental results (analytical results at each calculation stage of magnetization are presented in \"Supplementary Files\" for the body of vase 6. For the reproducibility of the results, the magnetic field measurements for the remaining vases in each position of their body are listed therein) show that the computed magnitude, M, displays similar values in the base and body fragments, which is a criterion for finding ceramic fragments of the same vase from archaeological excavations.