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"My father was a master mechanic; I grew up with a screwdriver in one hand and a pair of pliers in the other."
- Frank Herbert

Telepomp (Matter Transmission)  
  A device that transmitted matter from one place to another.  

This is the earliest reference (as far as I know) to the idea of matter transmission or teleportation.

The narrator of the story is visiting the Arsenal Museum in Central Park, and encounters a bizarre talking head. It turns out that this is all that remains of a scientist who invented a new way to travel - the Telepomp.

"Listen," he said. "In the course of my experiments with the telephone I became convinced that the same principle was capable of indefinite expansion. Matter is made up of molecules and molecules, in their turn, are made up of atoms. The atom, you know, is the unit of being... Their dissolution may be accomplished by chemical affinity or by a sufficiently strong electric current..."

"There was no reason why matter could not be telegraphed, or to be etymologically accurate, 'telepomped.' It was only necessary to effect at one end of the line the disintegration of the molecules into atoms and to convey the vibrations of the chemical dissolution by electricity to the other pole, whhere a corresponding reconstruction could be effected from other atoms. As all atoms are alike, their arrangement into molecules of the same order, and the arrangement of those molecules into an organization similar to the original organization, would be practically a reproduction of the original. It would be a materialization - not in the sense of the spiritualist's cant, but in all the truth and logic of stern science."

From The Man Without a Body, by Edward Page Mitchell.
Published by The Sun in 1877
Additional resources -

Once implemented, the Telepomp worked perfectly.

In my rooms on Joy Street, in Boston, I had about five miles of wire. I had no difficulty in sending simple compounds, such as quartz, starch, and water, from one room to another over this five-mile coil. I shall never forget the joy with which I disintegrated a three-cent postage stamp in one room and found it immediately reproduced at the receiving instrument in the other. This success with inorganic matter emboldened me to attempt the same thing with a living organism. I caught a cat - a black and yellow cat - and I submitted him to a terrible current from my two-hundred-cup battery. The cat disappeared in a twinkling. I hastened to the next room and, to my immense satisfaction, found Thomas there, alive and purring, although somewhat astonished.

He also draws power from the following analogy: "I constructed an instrument by which I could pull down matter, so to speak, at the anode and build it up again on the same plan at the cathode."

Compare to the displacement booth from Flash Crowd (1972) by Larry Niven, the stepping discs from Ringworld (1970) by Larry Niven and the trip box from Eye of Cat (1982) by Roger Zelazny.

Also, see the libra-transmitter from Into the Meteorite Orbit by Frank R. Kelly, the cosmic express from The Cosmic Express by Jack Williamson, Jaunte from The Stars My Destination, the Transo from Time is the Simplest Thing by Clifford Simak and the geofractor (1939) from One Against the Legion by Jack Williamson.

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Additional resources:
  More Ideas and Technology from The Man Without a Body
  More Ideas and Technology by Edward Page Mitchell
  Tech news articles related to The Man Without a Body
  Tech news articles related to works by Edward Page Mitchell

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