In this paper two new approaches are developed to calculate the astronomical vessel position (AVP). Basically, determining the AVP is originated from the spherical equal altitude circles (EACs) concept; therefore, based on the Sumner line's idea, which implies the trial-and-error procedure in assumption, the AVP is determined by using the two proposed approaches. One consists in taking the great circle of spherical geometry to replace the EAC to fix the AVP and the other implements the straight line of the plane geometry to replace the EAC to yield the AVP. To ensure the real AVP, both approaches choose the iteration scheme running in the assumed latitude interval to determine the final AVP. Several benchmark examples are demonstrated to show that the proposed approaches are more accurate and universal as compared with those conventional approaches used in the maritime education or practical operations.