## 3 Results
### Principal Component Analysis and Summary  
The PCA conducted on all the racket variables reported that PCA1 accounted for 35% of the variance of the data, PCA2 for 17% and PCA3 for 12%. These three principal components were investigated further to see if they varied with material or brand. Material significantly affected PCA1 (Between-ANOVA: F (6525) = 17.813, p \(<\) 0.001), with a large effect size (\(\eta_{\rm p}\)\({}^{2}\) = 0.181); brand also affected PCA1 with a medium effect size (Between-ANOVA: F (13,525) = 3.919, p \(<\) 0.001, \(\eta_{\rm p}\)\({}^{2}\) = 0.095). Material did not significantly affect PCA2, but brand did, with a medium effect size (Between-ANOVA: F (13,525) = 4.266, p \(<\) 0.001, \(\eta_{\rm p}\)\({}^{2}\) = 0.103). Neither material nor brand had a significant effect on PCA3. Figure 5: Change in geometry of tennis rackets from 1870s; (**a**) estimated head area; (**b**) string bed density and (**c**) the ratio between frame depth and thickness. Figure 6: Change in inertial properties of tennis rackets from 1870s: (**a**) Mass; (**b**) estimated transverse MOI; and (**c**) polar MOI. Figure 8a shows PCA1 to increase with year, particularly with the introduction of aluminium and composite rackets with more varied PCA1 values. When a bivariate Pearson's correlation was conducted on all the racket variables with PCA1, head width had the best correlation (r = 0.897, p \(<\) 0.001), followed by head length (r = 0.841, p \(<\) 0.001). Indeed, Figure 8a for PCA 1 vs. year has a similar shape to Figure 5a for head area vs. year. Figure 8b shows head area vs. PCA1, with large headed composite rackets having higher and more varied PCA1 values than their wooden predecessors. Therefore, the largest variation in the rackets, captured by PCA1, could be explained by differences in head area. Natural frequency was also well-correlated to PCA1 (r = 0.813, p \(<\) 0.001), as the composite rackets had higher natural frequencies (Figure 7a). Figure 8a for PCA1 vs. year also has a similar shape to Figure 7a for natural frequency vs. year. Figure 8c shows PCA2 to be larger in older and newer rackets (1870s and 1990s), and smaller from 1930 to 1960. This trend of PCA2 vs. year is also similar to that for CoM location (Figure 3b) and grip length (Figure 3b). When a bivariate Pearson's correlation was conducted on all the racket variables with PCA2, CoM location had the largest correlation coefficient (r = 0.854, p \(<\) 0.001), followed by grip length (r = 0.745, p \(<\) 0.001). PCA2 could, therefore, be explained by differences in CoM location, as well as grip length. Figure 8d shows rackets with a CoM closer to tip tended to have higher PCA2 values. PCA2 values for brands that were well represented, with many rackets spanning a range of dates (e.g., Spalding, Wilson, HEAD, Prince, Dunlop Slazenger and Donnay), tended to change over time with the trend of the racket population. The medium effect of brand on PCA2 and PCA1 may, therefore, have been due to differences in the ages of the rackets between brands (e.g., Feltham and Babolat), rather than differences in racket parameters between brands at a given time. Figure 7: Dynamic properties of tennis rackets from the 1870s: (**a**) Natural frequencies; (**b**) damping coefficients; (**c**) natural frequencies vs.damping and (**d**) vibration signals in time domain (legend indicates natural frequency and damping coefficient). By comparing the photographs in Figure 9 (and those in Video S1 in the Supplementary Material) it is clear that the tennis racket has developed considerably since the 1870s. The wooden racket from the 1870s (Figure 9a) is characterised by the lopsided head typical of early designs. The racket from 1886 (Figure 9b) is symmetrical and highlights the move away from the lopsided design. The racket from 1922 (Figure 9c) has a wooden handle and steel head and throat that were intended to combat issues with wooden frames, such as breaking or warping.