A scholar and explorer, Dr. Samuel Ferguson, accompanied by his manservant Joe and his friend professional hunter Richard "Dick" Kennedy, sets out to travel across the African continent — still not fully explored — with the help of a balloon filled with hydrogen. He has invented a mechanism that, by eliminating the need to release gas or throw ballast overboard to control his altitude, allows very long trips to be taken. This voyage is meant to link together the voyages of Sir Richard Burton and John Hanning Speke in East Africa with those of Heinrich Barth in the regions of the Sahara and Chad. The trip begins in Zanzibar on the east coast, and passes across Lake Victoria, Lake Chad, Agadez, Timbuktu, Djenné and Ségou to St Louis in modern-day Senegal on the west coast. The book describes the unknown interior of Africa near modern-day Central African Republic as a desert, when it is actually savanna.
A good deal of the initial exploration is to focus on the finding of the source of the Nile, an event that occurs in chapter 18 (out of 43). The second leg is to link up the other explorers. There are numerous scenes of adventure, composed of either a conflict with a native or a conflict with the environment. Some examples include:
Rescuing of a missionary from a tribe that was preparing to sacrifice him.
Running out of water while stranded, windless, over the Sahara.
An attack on the balloon by condors, leading to a dramatic action as Joe leaps out of the balloon.
The actions taken to rescue Joe later.
Narrowly escaping the remnants of a militant army as the balloon dwindles to nothingness with the loss of hydrogen.
An anachronistic killing of a Bluebuck antelope, a species which was already extinct.*
In all these adventures, the protagonists overcome by continued perseverance more than anything else. The novel is filled with coincidental moments where trouble is avoided because wind catches up at just the right time, or the characters look in just the right direction. There are frequent references to a higher power watching out for them.
The balloon itself ultimately fails before the end, but makes it far enough across to get the protagonists to friendly lands, and eventually back to England, therefore succeeding in the expedition. The story abruptly ends after the African trip, with only a brief synopsis of what follows.

Question: Where does the explorers trip in Africa begin?
Answer: Zanzibar/on the east coast
Alternative answer: The east coast of Zanzibar.