"Never forget for a moment that as soon as one part of the body moves the whole body moves. Do not move just one part independently." T'ai-chi Touchstones: Yang Family Secret Transmissions. Compiled and trans. by Douglas Wile, p. 113.
Everything is connected. This is the only way to move effortlessly in a relaxed fashion. It is also the only way to move with real power and speed. So often you see people demonstrate kata without the waist or the legs when they are attacking or blocking with the arms or the hands. It's most notable when people turn in kata to face another direction. They appear to ground themselves firmly in some rigid stance before they attack or block with the hands or arms. This sometimes happens in the first turn-around in Seisan kata. The students demonstrating kata turn and sink firmly into basic stance before the arms move. Or students stepping forward in the second move of Seipai move in such a way that you can see the hands moving independently of the lower part of the body, instead of being connected and really "powered" by the movement of the waist and lower half of the body. Or students stepping up from zenkutsu-dachi to a feet parallel stance with an overhead hammer fist attack in the middle of Saifa will stand up and then attack with the hammer fist. But even this swinging or arcing hammer fist should be connected to the waist and powered by the rest of the body. Or when blocking in the last sequence in Saifa (turning before the final mawashi and cat-stance), the forearm block should be executed by engaging the waist--in other words, when the arm moves to block, the whole body moves. This is rather hard to convey in words, but it is easy to see.
"The circle of retreat is easy; the circle of advance is difficult." T'ai-chi Touchstones: Yang Family Secret Transmissions. Compiled and trans. by Douglas Wile, p. 89.
When I read this, I think of the angles of counter-attack shown in the Goju-ryu classical katas. We see retreat directly back in Seiunchin. We see "blocking" or receiving movement stepping back along a diagonal in Kururunfa. We see stepping off-line to the sides--at a 90 degree angle to the opponent's attack--in Seisan and Sanseiru. We even see defense from a position with no movement in Seipai, as if one is caught with one's back to the wall, unable to move off-line in any direction. But with a very real attack it has always seemed to me that the most difficult movement is to move along the forward diagonal, along the tangent of the circle of advance. We see this in the diagonal steps before the kick in Saifa or before the first kick in Seipai or the corner techniques in Seipai. This movement is more difficult, but it is certainly made easier if one keeps in mind the admonition of the first quote above: "...as soon as one part of the body moves the whole body moves."
Pictures: Seipai, Saifa, Saifa, Seipai.