The 40 times figure comes from a review of USCTA Accident Statistics for 1990 and 1991 (American Medical Equestrian Associaton
"The place where most accidents occurred was on cross country. Cross country involves jumping fixed obstacles at speed. If a hors hits one of these obstacles, either the rider or horse and rider will fall. The second most common area was either stadium or other unspecified. Warmup areas for the jumping phases were the next most likely place for an injury. It comes as no surprise the jumping phases accounted for 86% of the injuries. Dressage accounted for only 1% and the stable area and other accounted for 12%, again indicating the surprisingly large number of unmounted injuries."
If one assumes twice as many events involving jumping, then the 86:1 ration of injuries would give about a 40 fold increase in injuries during jumping. The last time I looked for more current statistics, the report basically divided the injuries into during the jump and just before & after the jump. It had dropped the dressage phase, apparently from lack of injuries.
These are crude measures. There was another study that had:
A Cambridge University study of 1000 riding accident hospital admissions has shown:5
* One injury for 100 h of leisure riding
* One injury for 5 h for amateur racing over jumps
* One injury for 1 h of cross-country eventing
That would be a 20-fold increase, but those figures sound very high to me. You can find it referred here: Spinal injuries resulting from horse riding accidents
I do think it is reasonably established that jumping involves significantly greater risk than riding the flat, and thus warrants safety steps like good instruction, use of helmet, etc.