Visible crew/equipment: After Young Indy locks the two treasure hunters in the reptile train car he pulls a snake out of his shirt, and just as the third young treasure hunter tackles Indy we can see a crewmember's arm and equipment at the left side of the screen, before it cuts to the long shot. (00:07:00)
Visible crew/equipment: When Young Indy is trapped in the train car with the lion, he tosses up the whip to Fedora, and as he's being pulled up in the interior shot, he's wearing a stunt apparatus around his waist above his beaded belt. [The stunt double is also wearing a glove on his right hand holding what looks like skewered meat, to presumably draw the trained lion, but that's a lot harder to see without pausing]. (00:08:35)
Visible crew/equipment: In the catacombs after Indy passes the picture of the Ark Of The Covenant and approaches the next wall with the lighter, (when the screen goes upwards a bit) the shadow of a boom mic can be seen dropping into view at the top of the lighted wall. (00:31:40)
Visible crew/equipment: When Indiana and the Protector of the Grail are fighting in the boat that is about to be chewed up by the giant screw/propeller, when the angle changes to the top point of view, you can see the rope pulling the front of the boat around to position the rear for destruction... (00:39:10)
Visible crew/equipment: After the Venice boat chase scene, in the shot which takes place on the boat while it is being chopped up by the propeller, look at the top of the boat (opposite where Indy is holding Kazim.) You can see a series of round-shaped set lights reflecting in the boat. (00:39:30)
Visible crew/equipment: After Indy and Henry have escaped from Castle Brunwald, Indy jumps into one of the boats, pulls the motor starter cord and jumps back out, then just as he bends over to release the boat from the piling, right between Indy's legs the black covered arm of a hidden crewmember appears from under the tarp taking hold of the throttle, steering the boat away from the pier. (01:02:40)
Visible crew/equipment: After Elsa gives Donavan the false Grail, he turns and you can see the reflection of the camera lens in the cup. (01:51:45)
Visible crew/equipment: When Indy tries to reach the Grail his father convinces him to "Let it go" and Henry pulls his son to safety, then in the next wideshot of Indy, Henry, Sallah, and Marcus a crewmember wearing a short sleeved, white shirt and blue pants can be seen at the right side of the screen, just before it cuts to Henry's closeup. This crewmember can be seen again after the closeup of the knight. (01:57:45)
Visible crew/equipment: When the lead bandit opens the magic cart door to find young Indy running away on the track, a crew member wearing a white shirt can be seen reflected on the door.
Visible crew/equipment: After Indy's father is cured and the Nazis drop the weapons, he stands up and set lights are reflected on his glasses.
Visible crew/equipment: By the cliff, Indy's hat comes rolling towards him. Watch closely and you'll see a long shadow a meter above the ground, parallel to that of the crew member throwing the hat.
Visible crew/equipment: When Indy Sr. is shot, two big stage lights are reflected on his glasses.
Visible crew/equipment: At the hotel where Indy knocks on Elsa's bathroom door, she opens it and shuts the music off. The camera pulls backward and its shadow is visible on the left side of the door frame. It's fairly large.
Visible crew/equipment: At the beach, right before Dr. Jones scares the birds there's a zoom-in close-up of his face and a crew member approaching is reflected on his glasses.
Visible crew/equipment: After Dr. Jones is healed, Indy tells him to stand up. Stage lights are reflected in Dr. Jones's glasses.
Answer: Henry says, in response to Indy asking if he remembered the details of the trials: "I wrote them down in my diary so that I wouldn't have to remember." So, obviously he did NOT know them by heart. Also, as the other answer says, they didn't want the diary to either be in the Nazis' possession or be burned.