From my experience, the circular method works very well for putting an expedient edge on a blade and for doing initial, coarse bevel work. The method of moving in circles breaks up the burr as you go, making it a little more difficult to tell if you've full reground an edge, and also makes it tougher to remove the burr (depending on stone type) as the trailing and lateral portions of the circle don't remove the burr as well as edge leading. On the upside the burr doesn't have quite the same lopsided characteristics of one done with a linear movement and for general use, they hold up well (longer) than a quickly ground edge using edge leading or edge trailing strokes (an edge that likely has some burring left over such as might be the case if sharpening on the go). It also seems that the burr created is a bit smaller than might be expected for the same amount of grinding done laterally. The grind scratches are erratic, leading to an edge that cuts a little erratic, but for field use very fast and effective (when talking about grinding with fairly rough grades of abrasive). Also makes the most of your grind time as the stone is always working on the steel. This might seem like a small deal, but when sharpening with stones found on the trail or waterway, a 20% increase in grinding speed could equal many minutes of sharpening, and if doing heavy repair work or sharpening a larger blade is worth considering. I've done some experimenting and for field sharpening machetes, hatchets, and axes, now use a circular method with the coarse side and then give it several fast scrubbing/back and forth passes with the fine side, finishing with some edge leading to completely remove the burr. At that point I apply some compound to a stick or branch and strop.