The purpose of astigmatic corneal surgery is to flatten the steeper meridian of the preoperative cylinder, to steepen the flatter meridian, or both. Therefore, it may be useful to quantitate the surgical effect by calculating the equivalent dioptric value of the postoperative cylinder in these principal meridians. In this study, the dioptric value projected on the preoperatively steeper meridian is termed the with-the-power (WTP) component, the portion projected on the flatter meridian, the against-the-power (ATP) component. Consider a preoperative net astigmatism of the power N in the meridian a. After astigmatic corneal surgery, the postoperative corneal cylinder is M in the meridian b. For the postoperative cylinder, the WTP component = M x sin2([b + 90]-a). The ATP component = M x cos2([b + 90]-a). The astigmatic polar value is defined as the difference between these magnitudes: AKP = M x (sin2[(b + 90)-a] - cos2[(b + 90)-a]). By calculating the astigmatic polar value, the surgeon immediately knows the outcome of the surgical procedure (i.e., whether the preoperative astigmatism has been undercorrected, overcorrected, or perfectly corrected). We describe the theory behind this new formula and discuss its applications and limitations.