The American flag displayed near the beginning of the race has 48 stars. In 1908, when this movie supposedly takes place, there were only 46 states (New Mexico and Arizona were admitted in 1912, and Alaska and Hawaii were admitted in 1959).
In the scene on the beach in the tent when Leslie and Maggie DuBois drink champagne, Leslie puts on a phonograph record of "The Desert Song." The film is set in about 1908; "The Desert Song" did not premiere until 1926.
Just before Maggie Dubois sings "The Sweetheart Tree," Hezekiah is playing the guitar left handed. After he puts it down, Maggie picks it up and immediately begins playing right handed without re-stringing it.
When Leslie is putting on the parachute in the hot air balloon, we see a shot of the crowd below, in which the spectators are clearly visible, not much more than a hundred feet. But when he jumps, the pull back shot is much higher, possibly thousands of feet.
The balloon was supposed to be losing altitude due to the hole from Professor Fate's arrow. Additionally, the first shot is likely not high enough for a parachute to work.
During the scene towards the beginning of the race where Professor Fate and Max put out their engine fire with the foamy fire extinguisher, the car in the background, which in a previous shot lost its wheels, now has the wheels reattached.
When the Leslie Special car is being pushed out of the garage its on red carpet immediately in front of the garage. When viewed from above the red carpet is now just surrounding the car. There's no red carpet in front of the garage's opening.
Shortly before the starting gun, a very tall police officer (who strikingly resembles Max Baer Jr.) picks up two children on his shoulders. A couple minutes before that, while Peter Falk is cranking up the Hannibal 8 automobile, you can see the cop in the background with the kids already on his shoulders.
Towards the beginning of the race where Professor Fate and Max put out their engine fire with the foamy fire extinguisher, Max gets his long scarf snagged on the front of the car. The scene cuts to the approaching car of Maggie Dubois momentarily, then quickly cuts back to Max, whose scarf is now completely untangled and arranged neatly around his neck.
When Professor Fate and Max are driving along the rail tracks they run into a train coming in the opposite direction. This happens when they are out of sight, having just disappeared round a bend. They come into view again, being pushed by the train, but their car has mysteriously turned around: instead of rolling backwards on the track it is pointing in the same direction as the train.
At the and of the pie-throwing scene (done on an interior set), Natalie Wood is thoroughly covered with pie filling, topping, and crust. When she is seen entering the car (an exterior shot), though this ostensibly takes place just moments after the end of the pie fight, she is now only picturesquely dabbed with pie debris.
When Prof Fate is showing Max the completed Hannibal Twin 8's features, they enter the car and Fate has Max raise the vehicle. They then produce the cannon which Max fires when told to lower the car. Once the cannon is fired, the view goes outside and we see Fate's garage falling apart, the doors falling down, and the Hannibal inside in a lowered position.
Natalie Wood (Maggie Dubois) sends photo negatives to the newspaper by using homing pigeons. The negatives from the camera she is using would be too heavy for a pigeon to carry. Also, since there is no protective covering, the negative would have been ruined long before reaching its destination,
When Professor Fate is lifted by an airplane flown by Max, the plane resembles a Curtiss Headless Pusher. This plane was not designed and built until 1912, four years after the year in which this movie was set. In July of 1908 Glenn Curtiss flew his June Bug design, which was the first public exhibition of flight in the US (the Wright Brothers had not yet flown publicly). The 1908 June Bug and the 1912 Headless Pusher bear little resemblance to each other. In reality, the June Bug was not as nimble and stable as the Pusher, so it is understandable why the 1912 design was used in the movie.
As the Baron prepares to jump out the window after the sword-fight with Leslie, he attributes his parting quip to "a famous English gentleman" that starts out "He who fights and runs away . . . ". That passage, however, is from a poem by Oliver Goldsmith, a doctor from Ireland.
When Leslie is attempting to rescue the Prince, Hezekiah and Maggie from the Baron's castle he confronts two guards. One swings his rifle and hits his partner. The rifle can clearly be seen to bend indicating that it is rubber.
When Frisbee loses his footing when retrieving the pigeon at the newspaper office, his right leg "disappears" as he hangs from the ledge. This is due to his leg swinging into the matte shot that was created to "place" him into the shot to complete the building's appearance - the initial setup was done on a soundstage.
When Fate first reveals the Hannibal Twin 8 to Max, the camera pulls back to a wide shot. To change from the dramatic lighting of the close up, you can see many more off-camera lights being switched on to light up the entire garage for the reveal of the car.
Professor Fate's car, the Hannibal Twin 8, has the ability to elevate the chassis above the ground. However, whenever this function is used, both the engine and drive train of the vehicle also rise, leaving the wheels on the ground with no method of propulsion nor steering. So the car is driving around on wheels apparently not powered by anything. Note that while the car is being driven around the castle courtyard, small boxes are visible next to the rear wheels: these are almost certainly batteries for an electric motor. And in this example, with there being no ability to steer, the wheels were almost certainly fixed to drive in a circle.
The goof item below may give away important plot points.
In a scene towards the end of the movie, the communication antenna on the Eiffel Tower is shown but when Professor Fate blows up the Eiffel Tower in the final scene, the communication antenna is not present.