In the maneuvering board problem of using a prescribed speed to intercept a ship that will change course or speed in a specified manner, it is well known that we must first determine what *leg* it will be on when the interception takes place. The course and speed on that leg must be used to find the starting point of the fictitious ship employed in working out the solution. If the wrong leg is used, the reference ship will intercept the *fictitious* ship—not the actual one.

In Dutton it is stated that some experimentation might be necessary and that an estimate or approximate solution based on the course and speed made good by the maneuvering ship might be made to determine the approximate time of interception.

There is, however, a simple rule of thumb for taking practically all guesswork out of the problem, and it can be used without additional plotting—generally by inspection and with only mental calculations.

The normal solution, as we know, requires that we first plot the *actual movements *of the maneuvering ship until we reach the leg on which it will be intercepted. If, at the same time, we had some way of following the *movements of the reference ship*, we would merely have to note where their tracks cross to determine the time and place of their meeting.

This article will show how this can be done within close enough limits to serve our purpose.

While the maneuvering ship is moving along on various courses and speeds, the reference ship will be *moving outward, from the center of the board* at a constant (specified) speed. Its course is unknown until the final solution has been made, but that is immaterial. Whatever the course, the reference ship will *always* be located somewhere on a circle of radius equal to its speed times the elapsed time. The idea is similar to that of a circle of equal altitude in celestial navigation or a range circle in piloting. This “circle of position” is constantly expanding in the same manner as the waves in a pond when a stone is dropped into it.

In the case at hand, the moving circle is concentric with those of the maneuvering board, itself. It can thus be located for any instant by simply multiplying the speed of the reference ship by the time since the start of the interception run.

Up to this point we are dealing solely with *actual* movement of both the maneuvering and reference ships, and we must continue to do so until the expanding circle of the reference ship *passes over the maneuvering ship*. The time and place where this happens is the time and place of the interception we are seeking. If there were some easy method of finding *exactly* where this point is, our entire problem would be solved and there would be no need to make a relative movement plot at all. But at isn’t quite as easy as that. The exact point could only be determined by tedious trial and error. But there is a way around the difficulty. We can usually tell by mere inspection at any selected time whether the interception has already taken place. It follows, then, that if we take two points on the track of the meanuvering ship, it is a simple matter to determine whether it has happened between them. For this purpose, the two most logical points to select are successive ones at which the maneuvering ship changes course or speed—for we must plot them anyhow as a prerequisite to a relative movement solution. Furthermore, these are the points that determine the legs of the zigzag course, and it is the proper leg that we are trying to find.

(There is a comparatively rare case where the maneuvering ship is moving faster than the reference ship that would require intermediate test points for a strict application of the method, but even there the imminence of contact will ordinarily be apparent and there is a definite warning signal as will be explained further on.)

Once the leg is found, our circles have served their purpose, and we proceed with the conventional relative movement solution. This is done by plotting back to the point where the maneuvering ship would have been had it used only one course and speed from the beginning of the interception run. This gives us the point Mi.

So much for the theory. Let’s see how the circles work out in practice.

*Dutton* (Tenth Edition: 1951) gives the following problem on Page 705:

*Scale*: 3:1 in knots—-10:1 in miles.

*Situation*: A battleship is maneuvering some distance from your destroyer. At 1000 it bears 070° from you, distant 80 miles. It is on course 030°, speed 15 knots, but at 1100 it will change course to 340° and at 1230 it will change course to 290° and increase speed to 18 knots.

*Order received*: At 1000 proceed at 30 knots to deliver important mail to the battleship.

*Required*: (1) Your course. (2) Estimated time of interception.

Now for the solution:

Let the battleship be the maneuvering ship and plot its 1000 position from your own ship at the center (Fig. 1).

From this point plot the true course of the battleship for a distance equal to its ruil from 1000 until the change of course at 1100.

Here is where we stop for a moment and make our first use of the circle method!

Your ship will have moved 30 miles during the hour between 1000 and 1100; that is to say, during the time the battleship is on its first leg. Therefore, your circle of position will be the No. 3 circle on the maneuvering board (for we are using a scale of 10:1 in miles). Glance at this circle and you will see that it is *nearer the center of the board* than the battleship’s 1100 position, and so the interception* will not take place on the first leg*. Notice that we use the 1100 positions of both the battleship and your own ship, for that is the time at the *end* of the leg under consideration. You should also note that the 1100 circle does not intersect the leg at any point, for that is a good habit to form as will be shown later.

Now we return to our drawing and plot the true course of the battleship until 1230 (which is the time of the next change of course and speed) and mentally determine the position circle of your ship at 1230. This will have a radius of 75 miles because 2½ hours have elapsed and you have been moving outward (in a still-unknown direction) at a constant speed of 30 knots. Again this circle is inside of (has not reached) the 1230 position of the battleship and does not cross the leg—and again no interception.

Now we know the interception must take place (if at all) on the *final* course and speed given because the battleship will hold that course and speed until you reach it.

Bear in mind that the only marks we have made on our maneuvering board up to this point are those we would have had to plot anyhow in the conventional solution of the problem. The circles shown in Fig. 1 need not be drawn for we can locate them by eye from the design of the maneuvering board itself. Even in close or doubtful cases we can locate a circle with a mere sweep of the dividers.

We proceed from here on in the regular way by laying off the reciprocal of course 290° from the 1230 position and thus locate point Mi of a fictitious ship which has been on course 290°, speed 18 knots since 1000 (Fig. 2). That enables us to complete the solution.

That’s all there is to the problem given by *Dutton*, but for the sake of illustrating our circle method, let us go back for a moment and suppose that we had been given still another course and speed for the battleship at 1330. We could determine quickly by inspection that the interception would take place prior to 1330 because the course line beginning at 1230 (290°) heads back into the maneuvering board while the 1330 circle of position for the reference ship (105 miles for 3½ hours at 30 knots) would be beyond the limits of the circular diagram and therefore *would have passed ove*r the 1330 position. In other words, it would have passed over the *battleship itself* sometime prior to 1330 and, consequently, the two ships would have met at some time between 1230 and 1330 and somewhere on the 1230-1330 leg. If this had been a doubtful case, we could still have plotted the 1330 position of the maneuvering ship and noted that it would lie *within the area* of the 1330 position circle of the reference ship.

As we said above, the point of interception on the *actual movement plot* is always one where the constantly expanding circle of position passes over the maneuvering ship itself. It must, moreover, lie on the speed vector of the reference ship (or its extension) for that delineates its actual track. Quite obviously, it must also be on the track of the maneuvering ship. This means that it must be the point of intersection of the er vector (or its extension) with the plotted track of the maneuvering ship on the leg for which the relative movement plot has been worked out. It can therefore be used to check the accuracy of our work after the solution is complete.

Such a check can be made as follows:

Measure the position of the reference ship along the *er* vector from the center of the board (Fig. 2) using the number of miles it traverses between the start of the interception run and the time of the interception. Then measure the position of the maneuvering ship along the final course line using the number of miles it traverses between the last change of course or speed and the time of the interception. These points should coincide!

Let’s go back to our problem from *Dutton* and see how this works out. *Dutton* gives as answers that the course of your ship is 045° and the estimated time of interception (ETI) is 1303. You would therefore have travelled 91.5 miles in the 3 hours and 3 minutes between 1000 and 1303. We can plot this as a point 91.5 miles from the center on the bearing 045°. The battleship, on the other hand, would have covered 9.8 miles on course 290°, measuring from its 1230 position, during the 33 minutes elapsed between 1230 and 1303. These positions coincide (as shown in Fig. 2 at the spot marked “Point of interception”), and we have the satisfaction of knowing that our work has been accurate.

In the above example we have been discussing a case where the speed of the reference ship was at all times greater than that of the maneuvering ship. In such cases the system will work without fail. There remains, however, a sometimes deceptive instance when the maneuvering ship is moving at greater speed than the reference ship on a leg being tested. Here there is a possibility that the maneuvering ship may *run out of the circle* (or rather outrun the circle) after a contact has been made so that the point at the end of the leg may not lie within the circle. Fortunately there is a clear warning signal in all such cases: The circle of position as of the time at the end of the leg *will intersect the leg itself* at one or two points. The mere fact of such intersection does not necessarily mean that interception has taken place on the leg—but it is a red flag. It tells us that interception *may* have taken place, and it will always be present if interception *has* taken place. Inspection will generally reveal whether the maneuvering ship is clearly outdistancing the reference ship. In case of doubt we could resort to the trial and error method of testing intermediate points on the leg within the area of the circle to see how closely the ships are approaching, but even that isn’t really necessary. If the evidence indicates a possible contact, why not make a relative movement plot for the leg at once and see whether the *er* vector (or its extension) crosses the leg itself! It if does, we already have the correct solution from our relative movement plot. If not, we pass on to consideration of the next leg, for we have intercepted the fictitious ship rather than the actual one.

The ease and convenience of the whole circle method lies in making only one test for the *end* of each leg—one test per leg. Let us bear in mind that it is not a substitute for a relative movement plot. While it gives us a general idea of the course of the reference ship and the point of interception, it is merely a short cut and approximation—an easy way to find the key to the final solution of the problem. It involves no extra plotting and usually requires nothing more than a glance at a circle of the maneuvering board at each point where the maneuvering ship changes course and/or speed. If the circle of the reference ship has expanded to or beyond the final point of any successive leg, we can be *certain* the interception will occur on that leg. This holds true regardless of which ship is moving the faster.

It is only when three special factors coincide that we need consider even the possibility of a “hidden” interception. These are (1) the circle must *intersect* the leg, (2) the final point of the leg must lie *outside* of the circle, and (3) the *maneuvering* ship must be moving faster than the reference ship.

The circle method is equally as useful when the maneuvering ship will make only one change of course as when it will follow a complicated zig-zag plan—for we must know whether the interception will occur *before* or *after* the turn.

In any event, it’s just a case of moving a circle mentally out toward the maneuvering ship as we plot its intended track.