In using these logarithmic scales for the solution or arithmetical problems, in trigonometry or navigation, it follows, from the nature of logarithms, that the fourth term of four proportional magnitudes being to the third, as the second to the first, if these magnitudes be A, II, C, and ll, we have That is, the extent from the log. A to the log. B, these two quantities being of the same kind, will reach from the log. C to the log. D, always carefully observing- that the extent in the second case must be taken in the same direction as in the first. This circumstance, which must never be lost sight of, requires to be particularly ob served in solving problems where the scale of logarith mic tangents is employed, and the thil d term is the tan gent of 45° on the radius. In such cases, when the extent is from the left to the right, in respect of the two first terms, the extent ought to be taken in the same direction on the scale of tangents ; but the manner of laying down the scale not admitting this, the extent is taken from right to left, only the complement of the number of de grees and minutes corresponding to it is taken, in place of the numbers actually marked on the scale. A similar precaution is to be observed when either the first or second term of an analogy is greater than 45°, the extent in the second case being always reckoned in the sante direction of what it would have been, if the line of loga rithmic tangents had been continued to the right hand of 45°.
Methods of Solving the various Cases in Geografibical .A,"avigation.
THE earth being of a spherical form, (see As-rno NOMY, ancl GEoGuArriv,) the lines conceived to be traced on its surface, to represent the distance from one place to another, must, on the principles of spheri cal geometry, be great circles of the sphere ; and the angles which these great circles make with each other, must also, in conformity with the assumed pleasure of a spherical angle, be the satne as the angles formed by their respective planes. The solution of the various problems in navigation, either in reference to distance or angular position, would seem, therefore, at first sight, to require the application or sphelical trigonometty, a branch of geotnetry not very accessible to the ordinary mariner, and which would have been attended with very great difficulties in reducing its deductions to the actual circumstances of the case. Happily, however, this is
rendered wholly unnecessary, by means of a very simplo, but ingenious contrivance, according to which the circles on the surface of the sphere are projected into straight lines; and the calculations connected with their magni tudes and positions brought within the ranRe of plane trigonometry. For this reason, WC shall first explain what is called plane sailing, in which it is assumed that the surface of the earth is an extended plane, as the re sults to which this supposition leads, though not true, may afterwards be modified by the contrivance we have alluded to, so as to accord with the actual figure of the earth.
In plane sailing, the meridians are conceived to be all parallel to one another, as well as the parallels of lati tude ; and the ship's course is estimated by the angle which the line of direction, in which she has sailed, makes with the meridian. In delineating the ship's position with respect to the point from which she has set out, the straight line on which she has sailed is called the dis tance ; the number of miles she has advanced northward or southward, is denominated the difference of latitude ; and the term departure is applied to the perpendicular distance she has receded from the meridian. In exhibit ing. a geometrical representation of these lines, the me ridians are drawn upwards and downwards, towards the top being reckoned northward, and towards the bottom southward. This being agreed upon, towards the right hand must be east, and towards the left west. The lines of rhumbs or chords, according as the course is ex pressed in points or degrees, are employed for laying down the angular position of the distance with the me ridian ; and the distance and difference of latitude, and departure, are all measured by some assumed scale of equal parts.
The difference of latitude and departure being always at right angles to each other, the lines by which they are represented will form with the distance a right-angled triangle. Hence any two of these being given, or the course, and any one of them, the other parts of the tri angle will be determinable. The cases in plane sailing are therefore six in number.