Instead of starting out with the outside construction and working our way in, here we start with the inside (the ellipse) and work towards the square around it.
To explain some of the ideas below, take a look at the circle on the left first. Since this is a circle in simple front view, drawing a square around it is easy. The sides of the square should be equal, the lines are perpendicular, the middle of each side should connect with the circle and the middle of both coincide. Easy. To translate this to a perspective view though is a bit harder. For one, the lines of the cube will no longer appear perpendicular, and the middle of the ellipse (the mathematical ellipse mind you) isn't in the middle of the square surface anymore. An important aspect that remains unchanged is the fact that the ribs of the square are still tangent to the circle (or ellipse). This is what we will use in the following steps to find a square around the ellipse.
Step 1: Start out with a horizontal ellipse. Make sure it follows the rules as explained in assignment 5 for drawing ellipses.
Step 2: Choose a direction. This can be anything, but remember to let it pass through the real middle of the circle in perspective and not through the crossing of the major and minor axis. Notice that the line you just drew is actually 1-3 or the line 2-4 in the example on the left. The choice you make here defines the perspective of your square around the ellipse.
Step 3: There are actually two ways to go about this, I'll explain the second in the next step-by-step example. Draw the line perspectively parallel to your first direction, but now on the spot where it will be tangent to the ellipse. Basically, keep the direction of the first line in mind and try to find where it'll touch the ellipse on the edge.
Step 4: Now do the same on the other side. Keep in mind that there should be some perspective in those lines, they should go to the same vanishing point somewhere off the page.
Step 5: Mark the spots where both the lines of the last two steps touch the ellipse and connect them. If you find one of the to be off (because your line doesn't pass through the middle) correct this.
Step 6: No do the same as in step 3 and 4 but now for the other sides of the square. Note that the direction you choose in step 2 determines the whole of the rest of the square you draw. You can draw any square around it you like by choosing a different direction.