Navigation shows how vessels are directed in their course upon the great waters. In proportion as the “paths of the seas” have become open, safe and free for all, they are found paths of knowledge and civilization. The science, small at its beginning, has grown to its present advanced state by slow degrees, helped by contributions from the most opposite sources. Practical but uneducated seamen have doubtless done much, as their ingenuity is often, in emergencies, taxed to supply means of safety and success that are wanting. More has been contributed by scholars, secluded philosophic men whose lives are spent “in communion with the skies,” in observing the motions of the heavenly bodies and studying the laws by which they are regulated. But perhaps the most valuable service has been rendered by another class who combine an experience of the sea with much knowledge of astronomical science, men acquainted with the needs of seamen and qualified to meet them. The introduction of the mariner’s compass early in the fifteenth century was an epoch in the history of navigation, as it made seamen in a measure independent of the sun and stars. This was an incalculable advantage, as soon became apparent to those who adopted the compass as their guide. Of the many improvements and helps in the science of navigation we can only name, as conspicuous, the invention of Mercator’s chart[3] in 1569, Davis’s quadrant[4] about 1600, and Hadley’s quadrant a century later. The character of the instruments and a glance at the Nautical Almanac will show how largely both mathematics and astronomy enter into the science of navigation. Nor is it quite safe to take passage with a shipmaster who has but limited knowledge of either. He should at least thoroughly understand his instruments and be a ready, accurate computer. Geometry grew out of the practice of surveying, and now embodies many of the laws and principles of the science. There are several distinct systems of surveying, classed according to the purposes contemplated. It is astronomically employed in determining the figure of the earth by the actual measurement of arcs. A fair knowledge of mathematics and trigonometry is required in what are known as coast surveys. Land surveying is of the plainest kind, and employed in finding the contents of areas, or in dividing large tracts into lots of smaller dimensions. The chief difficulty is in getting the exact bearing of the lines and the measure of the angles when the plot is an irregular polygon.