If the limits of your knowledge of the story of the Man in the Iron Mask end with Leonardo DiCaprio and Gérard Depardieu, you might be surprised to learn that Alexandre Dumas's fictional story is based on the true legend of a mysterious French prisoner whose identity remained unknown even after his death. Historians have several theories as to who the man may have been, ranging from a traitorous general to an Italian diplomat to the illegitimate half-brother to King Louis XIV.

But those historians can have all the theories they want. Dr. Carter Nichols has something better:

Batman, Superman, Robin, and a time machine!

Really, given this possibility, why are there any mysteries left in the universe? What became of the lost colony of Roanoke? What happened to Amelia Earhart? Was the universe birthed of a big bang or did God say, "Let there be light" and will it into being? What prompted single-celled organisms to evolve into multi-celled ones? Who better to figure out such things than the world's greatest detective, his acrobatic teenage sidekick, and an invincible, superpowered alien with jouralism training?

Technically, Clark is right that Dr. Nichols has sent Batman and Robin into the past before--in fact, just three issues before this, he sent them to Baghdad in 955 A.D. where they met Ali Baba and Aladdin. However, Nichols had no idea that's what he was doing.

As far as he was concerned, Dr. Nichols sent Bruce Wayne and Dick Grayson a thousand years into the past. Now, as brilliant as the plan of sending Batman and Robin hurtling backward in time might be, the idea of sending a millionaire playboy and the teenager who lives with him is equally insipid. That would be like ignoring a team of astronauts and sending Paris Hilton and Justin Beiber to explore Mars.

Dr. Nichols adds Clark Kent to the mission--because "Paris Hilton, Justin Beiber, and Anderson Cooper: Mars Explorers" makes so much more sense--and sends the trio back to 1696, where they immediately cross paths with the Three Musketeers and D'Artagnan.
What's most interesting about Batman and Superman's insistence that they be sent back in time as their alter egos, is that they shed those alter egos almost immediately upon arrival. The purpose Batman's cowl is twofold. First, it protects his identity, and, second, it strikes fear into the hearts of the cowardly and superstitious criminals of Gotham City who wonder whether he might be some kind of supernatural half-man, half-bat. But by putting it on in 17th century France, it undermines both purposes.

First, by venturing into the past as Bruce Wayne, then interacting with historical figures such as Charles de Batz-Castelmore d'Artagnan and King Louis XIV as Batman, a suspicious coincidence arises. It's difficult enough to explain why it is that Bruce Wayne always disappears whenever Batman shows up in 1955 Gotham City, but at least they both live there and have a reason to be there. How are they supposed to explain that Bruce Wayne, Dick Grayson, and Clark Kent all just happened to disappear around the same time that Batman, Robin, and Superman broke through a temporal barrier in the exact same time and place?
Second, as quickly as he dons his trusty tights and cowl, Batman then takes on the persona of Armand d'Athos, putting a wide-brimmed musketeer hat that covers his bat ears and kills any chance of anyone mistaking him for a human-bat hybrid.

Meanwhile, Superman poses as Porthos and the three storm the castle of Pignerol where the Man in the Iron Mask is being held.
But when the rescue operation goes sour, Superman reminds us, it wasn't a rescue operation in the first place.
The prisoner is taken to the Bastille and Superman follows to protect him while Batman and Robin run to get the king, who... is supposed to care for some reason or other...

Unfortunately, Batman chooses the event of meeting the King of France as an opportune moment to lose the musketeer outfit, and King Louis reacts as one might when seeing two masked men leap onto your tenth story balcony--one of whom might be an ungodly bat-man genetic amalgam--by charging at them, sword drawn.
So while Batman undresses, shaves, and impersonates Le Roi Soleil, who he, in turn, dresses up in his Batman costume, word gets to the Bastille that the king is coming. This prompts an execution order since if the king finds out about the Man in the Iron Mask... he'll... care for some reason or other...

Superman, not content to let Batman have all the stripping and impersonating fun, protects the prisoner the best way he can.
When the king arrives, it's revealed that the Man in the Iron Mask was innocent and was framed for crimes committed by the jailer who was trying to kill him... though no evidence is really presented and no explanation for why he was convicted in the first place is given. Instead, King Louis accepts the word of Batman, who he initially thought was an assassin until he was knocked unconscious then woke up wearing tights and a satin bat mask inside a prison.
But that's to be expected. He had Superman there to back him up, and they are the two most respected superheroes in the world... or at least will be 250 years later!