AN INTERPOLATING INSTRUMENT TO EXPEDITE THE USE OF AQUINO'S TABLES IN NAVIGATION
By Lieutenant (J. G.) Alexander Forbes, U. S. N. R. F.
The following is a description of a simple instrument made to facilitate interpolation from mathematical tables. Although the particular instrument which has been made as a sample was designed primarily for the purpose of saving time and calculations in determining line of position by Aquino's method, the principle applies just as well to interpolations in the use of other tables, such as logarithms, etc.; and indeed the sample instrument can, by the addition of one or two spare parts, be adapted to any tables in which interpolation is needed, provided the function be virtually linear.
The particular reason for developing this instrument in its present form and for describing it as an adjunct to navigation is as follows:
Aquino's method has been recommended by Commander G. R. Marvell, U. S. N. (U. S. Naval Institute Proceedings, March, 1910), who found in it a saving of time, when compared with the Saint-Hilaire method as commonly used, amounting to 31 minutes; it has also been officially recognized to the extent that the method is explained and the tables are reprinted in H. O. No. 200, "Altitude, Azimuth and Line of Position," published by the Hydrographic Office of the navy. In practicing this method I found in the interpolations a considerable delay which I believed could be largely eliminated by this new contrivance. The additional saving of time, it seemed, might commend the Aquino method as more than ever worth adopting, especially if the present activities of our ships are such that rapid determination of line of position would be of practical value.
On page 190 of H. O. No. 200, the "First Method" of using Aquino's tables is explained; this appears to be the easiest of the three methods to master. The procedure is simple and rapid up to the point at which the value of a to the nearest 30' is taken out. The next step is as follows: "Reentering the tables from above with this a and with the given d, by a simple interpolation the corresponding values of b and t are obtained.'' There are here two distinct interpolations, one for b and one for t, which if carried out completely involve eight separate arithmetical processes—three subtractions, two multiplications, two divisions and one addition—and the writing down of over 50 figures. The use of a slide rule will expedite this work considerably, but there still remain the addition and subtractions. The interpolations could be reduced to their simplest terms and performed mechanically by the use of a simple contrivance based on the principle of similar triangles. A rough sample instrument of brass and wood was therefore constructed in a machine shop with the aid of a milling machine, and the saving of time and mental labor resulting from its use seemed to warrant its further development.
The essential elements of the instrument are: (1) A rigid right-angle triangle; (2) a graduated sliding scale or ruler, free to move right, left, up or down, but always parallel to one side of the triangle, similarly graduated; and (3) a rotating arm made of transparent material pivoted at the point of intersection of the other two sides of the triangle and with a ruled line on its surface passing through the pivotal point. Fig. 1 illustrates the construction. The three sides of the triangle may be designated the base, the fixed scale and the hypotenuse. The triangle is mounted on a board, and raised above its surface high enough to allow space for the moving parts beneath it. Also fastened rigidly to the board is a fixed groove or track, parallel to the base. In this track slides a short bar on which is mounted at right angles another groove or track in which slides the sliding scale. Since the fixed scale is perpendicular to the base, the sliding scale must be parallel to the fixed scale. The sliding scale moves under the triangle, leaving just enough space between for the motion of the transparent rotating arm. The entire length of the fixed scale represents 1° divided into 60'. The sliding scale is twice as long, and bears on its right edge 2° divided into 120', each of the same length as 1' on the fixed scale. In this way it is adapted to the conditions obtaining in Aquino's tables. The graduation of the fixed scale is based on the fact that b is given in whole degrees. The graduation of the sliding scale is based on the fact that successive values of d never differ by more than 1°, and those of t only do so in a small portion of the tables, the treatment of which will be considered presently.
In the operation of the instrument the sliding scale is taken to represent values of d, while the fixed scale represents values of h; the sliding scale is then reset to represent t. The procedure is as follows: The values of d next above and below the given value are found in the tables. The sliding scale is moved up or down until the number of minutes in the value below the given value is at the intersection with the base; this number appears twice on the sliding scale, once in each of the two degrees; the lower should be selected. The sliding track (in which the sliding scale lies) is then moved to right or to left in the fixed track, without disturbing the up and down set of the sliding scale, till the number of minutes in the value of d above the given value is at the intersection with the hypotenuse. The rotating arm is then swung till the indicator line on its surface crosses the sliding scale at the given value of d. The indicator line will then cross the fixed scale at the mark which indicates the desired number of minutes in b, the number of degrees being given in the b column of the table. Then, without disturbing the set of the rotating arm, the sliding scale is reset according to the values of t in the table corresponding to the values of d below and above the given value, and the desired value of t is directly read from the point on the sliding scale at which it is intersected by the indicator line.
This procedure will perhaps be made clearer by an example. Suppose we have given a= 39° 0', d = 21° 11'; the problem is to find b and t. The part of the table in which these values are found is as follows:
b | d | t |
27° | 20° 40' | 42° 16' |
28° | 21° 24’ | 42° 32’ |
The sliding scale is set so that the base intersects its right edge at 40' in its lower half (the bottom being assumed as 20° 0', the middle point as 21° 0', the top as 22° 0'), it is then moved laterally till the hypotenuse intersects it at 24' in the upper half. The rotating arm is swung till the indicator line intersects the sliding scale at 11' in its upper half. Then it will be seen that the indicator line meets the fixed scale at 42.3', thus giving 27° 42.3' as the value of b. The sliding scale is then moved upwards till the base intersects it at 16', and to the left till the hypotenuse intersects it at 32' in the same half of the scale (i.e., the same degree). The indicator line now crosses the sliding scale at 27.3', showing the value of t to be 42° 27.3'. It will be seen from this that the entire operation of obtaining the values of h and t is performed with five simple motions, which with a little practice can be performed rapidly.
The accuracy of the instrument depends on the dimensions, the workmanship in construction, and the care with which it is manipulated. It is very easy to make an instrument of the dimensions of the sample, 7" x 5", or even smaller, accurate to less than 0.1'. With such an instrument readings can be made to this degree of accuracy if care is taken in setting the scale and the rotating arm. In practical navigation, however, where an error of 0.2' or even 0.5' is unimportant, time can be saved by rapid setting of the moving parts, which can be done very quickly to the nearest half minute, and results good enough for the purposes of the navigator obtained with great speed.
Using the crude instrument, I have made comparative tests to determine the approximate saving of time. To this end a number of values of a and d were written down at random. The time was taken from the moment these values were looked at and the search in the tables begun to the moment the values of b and t to the nearest minute were written down in columns prepared for them. One series of 10 was done with the interpolation after very little practice, and another series of 10 using arithmetic; a third series of four was later carried out with a 10-inch slide rule. The results were as follows: Time with interpolator varied from 120 to 85 seconds, average 101 seconds; time with slide rule 2 minutes 55 seconds to 2 minutes 14 seconds, average 2 minutes 35 seconds; time with arithmetic, 6 minutes 27 seconds to 1 minute 50 seconds, average 4 minutes. This shows a saving of time over the slide rule amounting to approximately 1 minute, and over arithmetic amounting to 2 minutes 20 seconds. The test, moreover, does not show the full possibilities of time-saving, since with practice one could greatly reduce the time required to operate the instrument; in my own case far more than would be possible with either of the other methods. With practice it should be possible to perform with the interpolator the entire operation (i.e., beginning with the search in the table and ending with writing down the desired values) in a minute. I doubt if many could perform the same operation, using a slide rule, in less than 2 minutes, except in those cases in which the figures happen to be such that it can be done readily in the head. In short, it is possible with an instrument not much bigger or more expensive than a slide rule, to increase the saving of time by Aquino's method from 3 ½ to 4 ½ minutes.
It was noted above that in a small portion of the tables successive values of t differ by more than 1°, a fact which would make it impossible to use the scale as described in dealing with values in this part of the table. This difficulty is of minor importance as it never occurs when d is less than 45° and only in a minority of combinations when d is more than 45°. The only heavenly bodies used by navigators in the northern hemisphere whose declination exceeds 45° are Capella and four second magnitude stars including Polaris. Thus in the use of the sun, moon and planets this difficulty could not arise, and even in the use of these few stars it could arise only in a minority of observations. But most of this minority can be simply dealt with by engraving on the left-hand side of the sliding rule a different scale with smaller degrees and minutes, so that the space available may comprise more than 1°. For instance, this extra scale could be made to include 6° in its entire length, thus making a span of almost 3° available between the hypotenuse and base with the scale at the extreme right. This would adapt the instrument to declinations up to 71°, which would suffice for all the brighter stars except Polaris, and Polaris is usually treated with a different formula. It would be unwise to make the scale smaller for the sake of the insignificant number of instances in which the difference between successive values of t exceeds 3°, both because the minutes would be too small to read accurately, and because in this part of the table t is no longer a linear function of h.
It was mentioned in the beginning that this instrument could be readily adapted to interpolation with other tables than Aquino's, e. g., logarithms and all the trigonometric functions. To this end there could be marked on the outer edge of that side of the triangle designated the "fixed scale" another scale in which the whole distance was divided into 100 parts instead of 60. To go with this there could be supplied as many differently graduated sliding scales, to replace that described, as might be needed.
SUMMARY
An instrument is described, based on the principle of similar triangles, with which interpolations from mathematical tables can be performed mechanically with a minimum of labor. A sample has been made, designed especially for the interpolations required in Aquino's method of obtaining line of position. This method has been reported as saving 3 ½ minutes when compared with the usual method of applying the Saint-Hilaire procedure. It is estimated that with the aid of this instrument at least a minute more could be saved.
DISCUSSION
Commander E. B. Fenner, U. S. Navy.—As an instrument for general use in the interpolation of linear functions Lieutenant Forbes' machine has the serious fault of requiring a number of different scales for use with different tables and the shifting of scales necessary would seem to deprive it of its one claim to value, increased speed.
For use with the Aquino tables this difficulty is practically non-existent and "for those who like that sort of thing, it is doubtless just the sort of thing they would like." Personally, I do not like the Aquino tables and have never been able to find the gain of 3 ½ minutes in time claimed for them in a single sight, while in working out and plotting a cross of three star sights I have found that the Aquino requirement of plotting a different assumed point for each sight wipes out most of the gained time.
The Aquino method of computation is admittedly somewhat less accurate than the standard Marcq Saint-Hilaire formulse, the plotting of long altitude differences necessitated by this method introduces further chance of error, and the use of the interpolating instrument seems to result in still another rather small inaccuracy, but the results are claimed to be "good enough."
In the past 10 years a multiplicity of new methods and new gadgets for use in navigation have appeared, practically all of which have involved a possible gain in speed and reduction of labor at the expense of a certain loss in accuracy, and it seems to me that this tendency should be checked. With the present size and value of our fleet units and the unavoidable difficulties of navigation under war conditions no amount of labor is too great if it will, even in slight degree, reduce the danger of catastrophe. If the multitude of odd jobs that have been piled on the shoulders of the navigator make it impossible for him to devote sufficient time to his primary duty, then some one else should take some of the odd jobs. One battleship is too large a factor in our battle strength to make "good enough" a proper motto for our navigators so long as "better" is practicable.