# Determination of Seed Volume Based on Selected Seed Dimensions
## 2 Materials and Methods
### Physical Parameters of Seeds  
The basic seed parameters were measured in three perpendicular directions with the use of the MWM 2325 (PZO, Warszawa, Poland) laboratory microscope and a dial indicator thickness gauge designed by the authors. Seed parameters were determined to the nearest 0.01 mm with the use of both devices.  
crosshair in the reticle was aligned with one end of the seed contour. The position of the seed contour was recorded. The stage was then moved until the crosshair in the reticle was aligned with the other end of the seed contour. The position of the seed contour was recorded. Seed length \(L\) was defined as the absolute difference between the two measurements. Seed width \(W\) was determined without changing the seed's position on the slide, and the above procedure was repeated by moving the microscope stage with the X knob. Seed thickness \(T\) was determined with the dial indicator gauge. The gauge was calibrated, the seed was placed between flat contact points parallel to the contact surface, and seed thickness was read from the dial indicator. The same procedure was applied to measure the seeds of all analyzed plant species.  
The geometric mean diameter \(D\), the aspect ratio \(R\), and the sphericity index \(\Phi\) were calculated based on the measured seed dimensions [28]:  
\[D=\sqrt[3]{T\cdot W\cdot L} \tag{4}\]  
\[R=\frac{W}{L} \tag{5}\]  
\[\Phi=\frac{D}{L} \tag{6}\]  
The sphericity index \(\Phi\) equals 1 in spherical particles (where all dimensions are equal), and it is smaller than 1 in seeds modeled by other geometric figures.  
The volume \(V_{p}\) of all seeds in a given sample was determined with a 25 cm\({}^{3}\) glass pycnometer with a capillary tube and a thermometer. The pycnometer was filled with distilled water. The following parameters were determined: seed mass (\(m_{1}\)), the mass of the pycnometer filled with distilled water (\(m_{2}\)), and the mass of the pycnometer containing the seeds of each plant species after excess water had been removed (\(m_{3}\)). Seeds were weighed on the WAA 100/C/2 laboratory scale (RADWAG Radom, Poland) to the nearest 0.1 mg. The total seed volume \(V_{p}\) of every plant species was determined with the use of Equation (7) by summing up all three mass measurements and dividing them by the density \(\rho\) of distilled water:  
\[V_{p}=\frac{m_{1}+m_{2}-m_{3}}{\rho} \tag{7}\]  
Seed volume coefficients were calculated with the use of the below equations based on seed volume \(V_{p}\) and seed dimensions (\(T\), \(W\), and \(L\)) in each plant species (with \(n\) seeds per sample):  
\[k_{0}=\frac{V_{p}}{\sum(L\cdot W\cdot T)} \tag{8}\]  
\[k_{1}=\frac{V_{p}}{\sum(L^{2}\cdot W)} \tag{9}\]  
\[k_{2}=\frac{V_{p}}{\sum(L^{2}\cdot T)} \tag{10}\]  
\[k_{3}=\frac{V_{p}}{\sum(W^{2}\cdot L)} \tag{11}\]  
\[k_{4}=\frac{V_{p}}{\sum(W^{2}\cdot T)} \tag{12}\]  
\[k_{5}=\frac{V_{p}}{\sum(T^{2}\cdot L)} \tag{13}\]  
\[k_{6}=\frac{V_{p}}{\sum(T^{2}\cdot W)} \tag{14}\]  
\[k_{7}=\frac{V_{p}}{\sum L^{3}} \tag{15}\]\[k_{8}=\frac{V_{p}}{\sum W^{3}} \tag{16}\] \[k_{9}=\frac{V_{p}}{\sum T^{3}} \tag{17}\]  
Seed volume \(V\) of each plant species was calculated with the following equation:  
\[V=k_{0}\cdot L\cdot W\cdot T \tag{18}\]  
Since not all seed dimensions are always known, seed volume can also be calculated with one of the following formulas:  
\[V_{1}=k_{1}\cdot L^{2}\cdot W \tag{19}\] \[V_{2}=k_{2}\cdot L^{2}\cdot T\] (20) \[V_{3}=k_{3}\cdot W^{2}\cdot L\] (21) \[V_{4}=k_{4}\cdot W^{2}\cdot T\] (22) \[V_{5}=k_{5}\cdot T^{2}\cdot L\] (23) \[V_{6}=k_{6}\cdot T^{2}\cdot W\] (24) \[V_{7}=k_{7}\cdot L^{3}\] (25) \[V_{8}=k_{8}\cdot W^{3}\] (26) \[V_{9}=k_{9}\cdot T^{3} \tag{27}\]  
The error in seed volume calculations involving Equations (19)-(27) was determined relative to volume \(V\) (Equation (18)) as the most accurate value, and the following formula was applied:  
\[d_{i}=\frac{|V_{i}-V|}{V}\cdot 100 \tag{28}\]  
where \(i\) ranges from 1 to 9.