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How did the parts work together?
The longest screw moved the mount, and thus the specimen, in two directions: up and down and back and forth.
This shorter screw moved the mount, and thus the specimen, in and out, that is, closer to or farther from the lens. It did this by pressing against the body plate, pushing it away from the specimen pin.
With most microscope designs, as with most telescope designs, the specimen stays still and the lens moves closer or farther. If the specimen is dry, flat, and fixed on an immovable stage, as with a modern glass slide, it can move in only two dimensions, controlled by hand when placing the slide or shifted later. The other dimension is controlled by moving the lens closer or farther. You can't see the back side of the object on the slide unless you turn the slide over.
Leeuwenhoek, however, especially early in his career, was often observing three-dimensional objects that either were moving or could be rotated on an axis. It made more sense to keep the lens fixed and to move the specimen. His microscope had to accomplish three tasks: magnification, resolution (separating the details), and visibility to the human eye. In 1699, he wrote:
But to mount such small glasses well, requireth a far greater judgment, than to make them.
The lower screw, in addition to acting as a handle, adjusts the vertical position of the pin. The screw going into the stage adjusts how close the pin is to the lens. The little knob or handle lets the specimen be rotated.
The diagram in the middle shows the same four screws as well as the double plates that the lens is mounted between. It does not show the concavity that was punched into each plate for the lens to fit between.