As long as it has been known that the shortest distance between two points on the earth’s surface is not a straight line but an arc of a great circle containing both of these points, the problem of great circle sailing has been perplexing to navigators. Determining what courses will carry one along this great circle is the problem, for the geographical direction of the course constantly changes as one proceeds along toward the destination. Take, for example, the familiar route from San Francisco to Tokyo. If traversed by the great circle method, it passes just south of the Aleutian Islands. The initial heading out of San Francisco is north-west, but the final leg into Tokyo is south-west. To follow this shortest, quickest, fuel-saving route exactly requires the navigator to compute new courses continuously by one of the several methods available. Needless to say, shipboard navigators find this task time-consuming, and aerial navigators know that an exact traversal is well-nigh impossible at aircraft speeds.
The decision whether or not to use a great circle course depends on several considerations. At the lower latitudes, the ordinary rhumb line and the great circle course nearly coincide, and the rhumb line between places near the same meridian is almost a great circle. For short voyages the gain is not worth the extra work or trouble, but on a long voyage, such as the one from Valparaiso to Sydney, the saving is considerable. In this instance, the great circle route is about 748 miles shorter than the rhumb line.
Many methods are now available for finding great circle courses. They may be plotted as straight lines on gnomonic and Lambert conformal projections. These charts, however, are not convenient for ordinary navigation. Great circles appear as curves on the Mercator charts used, and thus may not be plotted directly. The usual solution is to plot the course on the gnomonic or Lambert conformal projection, and then transfer it by a series of rhumb line approximations to the useful Mercator chart. A series of rhumb line courses that comes close to a great circle may also be found by involved computation, by conversion angles found in Bowditch, or by reference to H. O. 214, etc. Each of these methods involves the solution of a spherical triangle on the surface of the earth. These systems have one disadvantage in common: they give instantaneously correct values, but since the course angle is constantly changing, a course laid out by these methods must be a series of approximations.
This paper offers a proposal to solve this problem by mechanical means. A simple mechanical computer can give continuous, accurate values of the course to steer, and, in addition, estimates the remaining distance to the destination. The computer’s output can be shown on a dial at the helm on board ship, or in aircraft it can be driven directly into the automatic pilot.
This computer is similar in principle to those used to solve the fire control problem. The only inputs required are present latitude and longitude, obtained from the dead reckoning tracer by direct drive, and the destination’s latitude and longitude, manually set. This computer is designed to solve the spherical triangle shown in Figure 1. A spherical triangle is determined by three points on the earth’s surface; in this case, they are the North Pole, the present position, and the destination. The sides of the triangle are the distance to the destination expressed in angular units, the complement of the present latitude, and the complement of the latitude of the destination. These values are related by two simple equations, of which the unknowns are the desired quantities. The computer is designed to solve these equations simultaneously, when supplied with the above inputs.
If “d” is the distance to destination, and “C” is the course, then, cos d = cos (90 — L1) cos (90 — L2) + sin (90 — L1) sin (90 —L2) cos (λ1 — λ2), and
sin C/sin (90 — L2) = sin (λ1 + λ2)/sin d
These may be easily reduced to the simpler form of
(A) cos d = sin (L1) sin (L2)
+ cos (L1) cos (L2) cos (λ1 + λ2)
(B) sin C = cos (L1) sin (λ1 + λ2) cos d.
These equations are true for any two positions on the surface of the earth.
The next question is the method by which the computer solves these equations. Figure 2 shows the process schematically. Remember that initial values of L2 and λ2 are set in manually, and that L1 and λ1 are continuously supplied by the dead reckoning tracer. The computer simply proceeds through the above equations. Modified component solvers resolve L1 and L2 into their respective sines and cosines. λ1 and λ2 are added by a differential, and resolved by another modified component solver, as seen on the right side of Figure 2. Now, to solve equation (A), sin (L1) and sin (L2) are taken to a multiplier and their product is obtained. Likewise, the product of cos (L1) cos (L2) cos (λ1 + λ2)results from multipliers. The outputs of these multipliers are added in a differential, giving the value of cos d, as shown. Distance itself is found by- applying cos d to an inverse cosine cam, and may be shown on a dial as the distance remaining.
The second step is the solution for the course. A cosecant cam is used to convert “d” to cosecant d. Cos (L2) and sin (λ1 + λ2) are obtained from their respective component solvers and multiplied with csc d, as seen in Figure 2. From (B), this product equals the sine of the course angle. This sine of the course angle is converted to the course angle itself by means of an inverse sine cam. Thus, course and distance are continuously and accurately produced as one proceeds.
The component parts of which the computer is composed are similar to the mechanisms which make up an ordinary range- keeper used in fire control. It requires three modified component solvers, five multipliers, two differentials, and three cams, with their associated connections and gears, and indicators.
The fact that this computer weighs little, takes up little space, and requires virtually no power to operate makes it practicable for airborne use. In fact, it is its use in aircraft that is the most promising. High speeds of flight intensify the need for a continuous great circle course without computation by flight personnel, and a combination of this computer and the automatic pilot should greatly simplify long distance flying. Since the output appears in the mechanical rotation of a dial, it could be led, with no alteration, directly to the automatic pilot.
It should be realized, of course, that the output of the computer can be no more accurate than the values from the dead reckoning tracer which are fed to it. As is now done, the DRT must be corrected every time a fix is obtained, whether it be a celestial fix or a check-point seen from an airplane.
Even if the navigator takes the time to lay out a great circle course by existing methods, he can never be as accurate as this computer, for his course must be a series of approximations. In addition, the computer saves him the time he would ordinarily spend laying out such a course. It would make truly automatic the “automatic” pilot in large aircraft. It would effect savings in fuel and in time. Not only naval vessels and aircraft could benefit from this computer: merchant vessels and passenger liners, whose schedules are determined to the last minute, would find the great circle course computer to be a valuable aid. Its development should give simplicity, accuracy, and economy to the practice of navigation.