The magician, with hands obviously empty, takes up a small plate or coffee cup saucer and extends it toward a spectator, asking the spectator to place a coin on the plate.
The magician then rubs the coin on the surface of the plate. As his fingers come away from the plate, the coin is seen to have multiplied to two coins.
The magician rubs the two coins around the surface of the plate and the spectators see that the two coins have become three.

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T. Nelson Downs was known for his stage presentation of The Miser's Dream. However in his book on coin manipulations he discusses a variety of close up and impromptu coin effects. Among them is this dodge of multiplying coins by rubbing them on the surface of a plate.
Downs described multiplying one borrowed coin to four, although considering the method it seems more advisable to shorten the presentation by progressing from one to three.
Props
A small plate or coffee cup saucer. Two coins. A bar of soap or, alternately, magicians' wax.
Setup
The two coins are stuck to the underside of the plate using a bit of soft soap to cause them to adhere temporarily. The coins should be under the rim of the plate, about a half inch / 12 mm apart.
Once prepared the plate can be on the table, positioned so the coins are directly away from the audience.

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Top left, the two coins stuck to the bottom rim of the plate.
Handling sequence showing the hands shifting position during the course of the routine to hide the coins from view and to steal each coin prior to its production on the top of the plate. Follow the routine with plate in hand to understand the repositioning of the hands as the presentation progresses.

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Back view with coin finger palmed as hand starts process of multiplying one coin to two. Front view the as visible coin is moved around the plate by the finger tips. Finally the palmed coin is snapped against the plate, fingers moving back to reveal the multiplication.
Presentation
Casually show the hands empty without advertising the fact and request the loan of a coin of the same value as those attached to the plate.
As a spectator is getting a coin to fulfill your request, pick up the plate with the left hand. The left fingers go under the plate to cover the two coins, the left thumb rests on the top edge of the plate. Move the plate forward, gesturing with the right hand for the spectator to place the coin on the plate.
After the coin is on the plate, the left fingers under the plate pull slightly to the left to expose the right-most coin. The right hand moves under the plate to cover the exposed coin, taking hold of the plate. The left hand releases its hold and moves away.
Raise the plate up a couple of inches as if to move closely examine the coin. Lower the plate the plate, at the same time bring the left hand over to hold the plate on the left. As the left hand takes hold of the plate the right hand moves back taking the right coin with it and then moves forward to pick up the coin on the plate.
With the right finger tips just touching the surface of the plate, the right thumb works the two coins so they are on top of each other and pressed against the inside of the right fingers.
Rub the edges of the coins around the center of the plate in a tight circle, pushing the coins down with the right thumb and raising the right fingers a bit.
When about half the front surface of the front coin is visibly extended from the tips of the right fingers, snap the two coins down on the plate and continue the circular rubbing motion. At the same time the fingers spread apart to show that there are now two coins on the plate.
Move the right hand away from the plate, then down to take hold of the plate at the back, the right fingers covering the remaining coin under the rim of the plate. The left fingers release their hold on the plate. The right hand raises the plate a couple of inches while the left fingers move over the center of the plate and push the two coins with the left finger tips.
Lower the plate again, in doing so the left hand takes hold of the rim of the plate on the left, the right hand steals the second hidden coin and then moves forward to pick up the two coins on the plate.
The handling is repeated at this point with the edges of the coins being rubbed in a circular pattern around the center of the plate, slowly worked out at the finger tips and then snapped on the plate to show that there are now three coins on the plate.
Performance Notes
The proper pacing is crucial to this illusion. You must not hesitate as you multiply the coin from one to two to three, but you should not rush any of the moves. The repositioning of the hands holding the rim of the plate has to be done without thinking so practice is required.
The process of rubbing the coins on the surface of the plate should appear to be a delicate gesture and you should keep your eyes firmly fixed on the plate surface during the entire routine.
Done properly, the illusion is quite convincing and pretty to watch. To finish, the coins could be tipped onto the table surface or into a container for use in a subsequent routine.