The Gaussian method of reducing circummeridian alti tudes of the sun by referring them to the instant of the sun's maximum altitude, is in this work rigorously investi gated, and a small term, overlooked or disregarded by gauss, has been added to the formula. A new and brief approximative method of finding the latitude by two altitudes near the meridian when the time is not known, is given in Vol. 1. Arts. 195 and 204, and another by three altitudes near the meridian, in Art. 205, which will probably be found useful as nautical methods.