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To use it, you align the arrow on the blue ring for the first key event, Translunar Insertion, with the actual day and time that it occurred, as shown on the outer ring. Then, by rotating the inner ring so that its arrow points to a day and time of interest, you can read in the cutout window the events scheduled to transpire at that time.
The back of the analyzer deals with launch, lunar module activities, and Earth reentry activities. When I scanned in the image above, I set the window to show the activities during EVA 2 at the Surveyor 3 site.
I would have killed to have something like this back during the missions! Circular slide rules still have their uses today, primarily by pilots.
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