By the Regency, hydrogen balloons were more typically used than hot air. The problem with hot air balloons at that time was they did not have a good fuel source, as we do now. So they could stay up only a short time, whereas hydrogen balloons could be used for long flights, depending on conditions. It was a hydrogen balloon which made the first channel crossing, although just barely.
Nova tells us, “The year was 1783, a milestone year for aviation—the dream of flying had finally been realized. On October 15 of that year, a few months after the duck’s historic flight, a balloon called Aerostat Reveillon, launched in France and carrying scientist Jean-François Pilâtre De Rozier, rose to the end of its 250-foot tether. It stayed aloft for 15 minutes and then landed safely nearby. A month later, De Rozier and a companion, the Marquis d’Arlandes, flew untethered to 500 feet and traveled about five and a half miles in a 20-minute flight—the first “free flight” made by man.
“Designed by the Montgolfier brothers, Joseph-Michel and Jacques-ítienne, the balloon was heated by a straw fire that later caused the balloon to catch on fire. But the Montgolfier brothers, undaunted, went on to design other balloons, including the first successful balloon that was unmanned (and unducked, for that matter).”
The TV show used drawings from 1783 of the crowds surrounding the ascent. King Louis was interested in ballooning but did not want to risk a human, so the French first sent up a chicken, a duck, and a sheep. The animals survived. Later Louis allowed the balloonists to send up men in a balloon, as long as they did so away from his palace.
In the Nova program, reportedly, the French air command refused to allow the reenactors to cut the tethers.
Amazing to think that these balloons were the beginning of aviation. The various ideas of DaVinci and tales of Icarus kept the idea in men’s minds, but most experts in the field say aviation started with those first balloon flights.
In the early days of ballooning, crossing the English Channel was considered the first step to long-distance ballooning. Two years after his historic first balloon ride, De Rozier attempted the crossing. De Rozier’s experimental system consists of a hydrogen balloon and a hot air balloon tied together. Tragically, the craft explodes half an hour after takeoff, and de Rozier and his copilot are killed. This double balloon helium/hot air system, however, remains among the most successful designs for long-distance ballooning. This same year, 1785, French balloonist Jean-Pierre Blanchard and American John Jeffries become the first to fly across the English Channel.
On December 1, 1783, just ten days after the first hot air balloon ride, the first gas balloon was launched by physicist Jacques Alexander Charles and Nicholas Louis Robert. This flight too started in Paris, France. The flight lasted 2½ hours and covered a distance of 25 miles. The gas used in the balloon was hydrogen, a lighter than air gas that had been developed by an Englishman, Henry Cavendish in 1776, by using a combination of sulphuric acid and iron filings.
Gas balloons soon became the preferred mode of air travel. The balloon shown at left is the Royal Vauxhall Balloon typical of gas balloons which were flown in the 1830’s and 1840’s. Unlike hot air balloons, gas balloons did not depend upon fire to get them aloft and stay up and therefore they were able to stay up longer and their altitude could be controlled somewhat easier with the use of ballasts. Gas balloons continued to be the primary mode of air travel until the invention of the fixed wing aircraft by the Wright brothers in America in 1903. However, it was expensive to and time consuming to inflate a gas balloon so flying was not something just anyone could afford. Hot air balloons, however, had no dependable heat source, so hot air ballooning was not very practical.
The first parachute was used by a person in a balloon.
An animal was the first one sent out of the balloon with a parachute. A woman died when she fell from a balloon and other ballooners died when their balloons tore or exploded.
They (parachutists) would cut loose from the balloon, not actually jump from it. The 5th illustration at this link gives you an idea: http://en.m.wikipedia.org/wiki/André-Jacques_Garnerin
Sources:
Every Woman Dreams... 
Great post and nice to see this history since I’ve included a small bit about a balloon in my new book. Thanks as always.
Glad you found it useful.