Thomas Edison is credited with creating the first device for recording and playing back sounds in 1877. His approach used a very simple mechanism to store an analog wave mechanically. In Edison's original phonograph, a diaphragm directly controlled a needle, and the needle scratched an analog signal onto a tinfoil cylinder:
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You spoke into Edison's device while rotating the cylinder, and the needle "recorded" what you said onto the tin. That is, as the diaphragm vibrated, so did the needle, and those vibrations impressed themselves onto the tin. To play the sound back, the needle moved over the groove scratched during recording. During playback, the vibrations pressed into the tin caused the needle to vibrate, causing the diaphragm to vibrate and play the sound.
This system was improved by Emil Berliner in 1887 to produce the gramophone, which is also a purely mechanical device using a needle and diaphragm. The gramophone's major improvement was the use of flat records with a spiral groove, making mass production of the records easy. The modern phonograph works the same way, but the signals read by the needle are amplified electronically rather than directly vibrating a mechanical diaphragm.