A painting is two-dimensional, where depth is captured/reproduced through the position and size of objects and people in the picture.
Now, do you remember Back To the Future 2. Think of the usual poster, two dimensional, that you see advertising a film, and compare that to where the shark springs out of the advertising display for JAWS 19, like 3-D.
The writer is saying, a painting is 2 dimensional, yet they are trying to pretend it's real, as 3-D as real life; and the hand is painted as breaking the flat surface of this 2-D representation - it doesn't 'jump out', but it does protrude out of the rest of the picture. Another way of seeing it is looking down into water and seeing fish and seaweed under the surface, and some log sticking out of the water, breaking the surface of the water.
It's not the hand itself that shouldn't break the surface - t's the flat surface of the picture itself which shouldn't have been broken.
He says the illusionist baroque art uses every trick to make a picture seem real. How far will they go in trying to give this 'realistic' impression? -as far as right down to the trick of making the hand break out of the canvas and give a 3-D effect. So, right down to = as far as you can go.
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