A surcingle is a strap made of leather or leather-like synthetic materials such as nylon or neoprene, sometimes with elastic, that fastens around the horse's girth.
A show Friesian horse wearing a surcingle
Detail of a training surcingle or "roller"
Longeing or lungeing is a technique for training and exercising horses where the horse travels in a circle around the handler. It is also a critical component of the sport of equestrian vaulting.
Surcingle
…trainers first teach a young horse to accept girth pressure by strapping on a surcingle before a saddle and girth. The surcingle is commonly used for longeing (a horse training technique), often as a base from which to attach training equipment such as side reins, overcheck, lauffer reins (sliding side reins…
A horse in training for equestrian vaulting at the halt on a longe line
17th century copperplate engraving of a horse and rider being worked on a longe line
A horse and vaulter on a longe line with handler on the ground. (In vaulting, the handler is called a longeur.) The longe is also used to develop a person's equestrian skills.
Longe cavesson fitted with a bridle. Leg wraps on the front legs and stirrups run up the leathers are protective measures.