It was the invention of the 'leaping head' around 1820-30 - a specific addition/adaptation to the pommels of the sidesaddle - that enabled ladies (respectable ones, anyway) several decades later to jump safely, gallop across country and ride actively and energetically. They could even hunt on something approaching an equal basis, once actually mounted of course, with men.
The sidesaddle seat is so very secure that the security itself can cause danger, albeit not commonly.
The heyday of the sidesaddle was a short one, though, as riding astride started to become acceptable for women on the approach of WW1, and after it had ended, so, largely, did the teaching and development of side-saddle riding and sidesaddle riding soon became to very large extent the preserve of the elderly, the disabled and those who might be described by some as 'eccentric' ( but who I would describe as 'interesting').
It would not be an exaggeration to claim that the invention of the leaping head was one of the early steps towards women's suffrage and the supposed equality women are said to enjoy in the present day.