Sight any two bright stars. Adjust three micrometers on the Navigator-Sphere. Read your latitude and longitude directly, instantly. No plotting. No skillful graphics. No library of intricate tables. No mathematics. Such a feat is simple in sphero-metric navigation, using the precision instrument just completed at the laboratories of the University of Washington.
Today the tremendous range and speed of our modern aircraft have focused new attention on all methods of navigation. Even the special problems of celestial navigation have attracted widespread popular interest because of their importance in this post-war expansion of air service.
Over a dark countryside or high above the clouds, can the navigator tell his position from those myriads of remote glittering stars? Can he check his compass and set his course just from the heavens? Yes! But it takes training, instruments, experience, and precious time. In recent years, methods, training, and instruments have been steadily improved, greatly reducing the time and experience the navigator needs to do his indispensable job.
Today standards are higher. Our airplanes fly at more than 300 miles an hour. In two minutes, 10 miles have slipped away. Speed is in the air. The navigator in our big bombers has to work fast to keep track of his position. He must be accurate too. The size of the targets and airfields is not greatly increased. Determination of position within 2 miles is generally considered excellent for aerial celestial navigation. When action is rapid, mistakes creep in. Checks must be handy, quick, and simple. Many instruments and calculators, graphical and mechanical, have been devised to speed, simplify, and mechanize the work of the navigator.
In our Navy, since 1837, when Captain Thomas H. Sumner1 made his unique voyage on which he discovered the “Sumner line of position,” new and improved methods of celestial navigation based on this principle have been continually introduced. Because this fundamental principle is unchanged, the labor and ingenuity for each improvement was focused on devising short-cuts, on simplifying tabulations, and on pre-computing the elements of the astronomical triangle. Within the last twenty years, five successive methods of celestial navigation have been taught our officer candidates. Of these, Ageton’s method, and the new Altitude-Azimuth Tables2 seem outstanding today. These Hydrographic Office publications have reduced the intricacies of spherical trigonometry to simple interpolation in the tables.
Attacking the problem of celestial navigation from a different point of view, Captain P. V. H. Weems, U. S. Navy (Retired), prepared charts3 covering the earth showing lines of constant altitude for several selected stars in each region. Given the actual observed altitudes of these stars, the lines of position may be plotted directly by graphical interpolation on these charts. The newest tabular methods accomplish much this same end. The Astro-
Graph which has been used for navigating our Flying Fortresses projects these star altitude curves from photographic film directly onto the navigator’s table, to the scale of his chart. This device also provides mechanical adjustment between the star altitude curves and the longitude co-ordinates of the chart, corresponding to the sidereal motion of the stars and of their respective curves of equal altitude relative to the earth.
Recently, Drury A. McMillen, the Brazilian engineer, raised a minor tempest when he introduced his Sphero-graphical Navigation in sensational and provocative statements to the press.4 He returned to the fundamental sphere, and by the use of precise instruments was able to obtain results by simple and clear graphical construction approaching in accuracy the requirements of the navigator. The advantages in the pictorial representation of the relations in the problems of spherical trigonometry have long been recognized, both for the clear appreciation of the factors, and for teaching. However, precise graphics on a sphere require a skilled and steady hand, and ideal circumstances, conditions which do not always obtain in military aircraft. Difficulties in accuracy and manipulation prevent the enthusiastic adoption of this graphical method.
Mechanical instruments have been designed and some put on the market to simplify celestial navigation. Nathaniel Bow- ditch puts the situation nicely: “Such instruments are costly and subject to instrumental errors, but they are very useful for checking computations and may some day be devised in a form useful to the navigator.”5
The author has designed and built a simple, sturdy, and precise instrument which gives promise of fulfilling Bowditch’s predictions.
This Navigator-Sphere is a calculating machine for the problems of celestial navigation. It is designed specifically to give the latitude and longitude directly from simultaneous star altitude sights. Given two star sights, and Greenwich sidereal time, these three values are set on scales of the instrument, the selected stars plugged in, and the latitude and longitude read off their scales. That is all.
The position is determined in the time it takes to tell it.
Using the development model, handmade by the author, results for any given data are repeated consistently with no difference greater than two minutes of arc. The position so determined is absolutely independent of any dead reckoning or plotting on the chart. Thus it gives the ideal check on the dead reckoning.
The instrument may also be used with similar speed and accuracy to determine the conventional lines of position, to calculate great circle sailing data, and to establish fix from long distance radio compass bearings.
The Navigator-Sphere, shown in Fig. 1, looks like an intricate maze of bright steel rings surrounding a black sphere. Essentially it is a scale model incorporating the relations and relative motions involved in the simplified astronomy of celestial navigation. The instrument provides the three customary systems of spherical co-ordinates—sidereal, geographic, and horizon—in the star sphere, the geographic scales, and the observer’s scales, respectively. It is mounted in a blued steel stand on a vertical axis with the zenith upright. The relationship of these parts is shown in Fig. 2.
Pivoted on the vertical axis are two star altitude scales. Perpendicular to this axis and mounted on the circle for the local meridian is the azimuth circle. Together the three provide two independent sets of co-ordinates for observations in the horizon system. All three circles arc graduated in degrees. The two star altitude circles read from zero at the horizon to 90 degrees at the zenith. Verniers and tangent screws are attached to the star bushings so that each may be set to the observed altitude to a minute of arc. The azimuth circle is also graduated in degrees from zero at the north limb of the meridian, east to south to west. A simple index is provided on one of the altitude circles to read the azimuth to its position.
In addition to the two star altitude scales and the azimuth circle are two other circles, one to measure latitude and one for longitude. In their basic construction, the latitude, longitude, and one of the star circles are made like ball bearings with an inner and outer member. The balls in the race between provide rotations between the two members in the plane of the ring.
The outer member of the latitude circle is pivoted on the vertical axis of the sphere. The inner member of the circle carries the pivot bearings for the polar axis. Thus the latitude circle, containing both the zenith and the polar axis, determines the plane of local meridian. It provides for the angular motion between the zenith and the pole which gives the latitude for the zenith. The inner circle is graduated from zero at the equator to 90 degrees at both poles, while the outer carries a vernier at the zenith with which the angle of the latitude there may be read to a minute of arc.
Fastened to the inner latitude circle perpendicular to the polar axis and circling the sphere near the equator is the circle which determines the longitude of the plane of the local meridian. The outer member of the circle is fixed perpendicular to the inner member of the latitude circle. It carries the vernier by which the local longitude, that of the zenith, is determined. The inner member of the longitude circle is graduated clear around in degrees from zero to 360 from east to west, and thus gives readings of west longitude from Greenwich.
Inside all these rings is the star sphere, made from a black, 5-inch bowling ball. It is freely pivoted at the poles and fits snugly within the longitude circle. Its position relative to the longitude is determined by the sidereal time for the observation. A brake and tangent screw between the sphere and the circle permit the accurate setting of the sidereal time vernier against the longitude scale, thus placing the Greenwich meridian in its proper position relative to the stars for the given instant. The sphere itself is a bakelite ball in which have been drilled small radial holes with the center of each at the exact right ascension and declination for the 61 principal navigation stars. A number stamped in the sphere surface near the hole, and a printed table, shown in Fig. 3, provide a positive index for unfamiliar stars. Two inlaid steel bushings provide a ball-bearing seat for the polar pivots.
Because, for purposes of navigation, the appreciable changes of stars’ positions in right ascension and declination are due to precession only, these changes are made on the star sphere by moving the polar axis the approximate 50 seconds of arc per year toward the vernal equinox, and by setting the sidereal time vernier ahead. On the development instrument this adjustment is accomplished by applying progressively more eccentric polar bushings as the years pass. Thus the pole is moved and the stars are fixed.
To correlate the observed altitudes with the definite position for the stars on the star sphere, cylindrical star pins are passed radially through bushings in the star altitude circles into the holes in the star sphere. For a particular altitude for a certain star, the setting is made on the star altitude circle and the star pin plugged into its proper hole. The arc of the altitude circle will then maintain the appropriate zenith distance between the star and the zenith. The several degrees of freedom for the star sphere are reduced to one, and the sphere may only move so the zenith traces a circle of position over the sphere. Co-ordinates for points on this circle may be read off as desired on the latitude and longitude scales, and the corresponding azimuth to the star on its scale.
With any other star, one can do the same thing with the second star-altitude circle. Setting both circles locks the sphere, so that the zenith is located correctly relative to the stars. The latitude of the observer is then already indicated on the latitude scale. To find the longitude, it is further necessary to set the inner member of the longitude circle at the correct sidereal lime. Then the longitude of the observer may be read at the local longitude index.
The local latitude and longitude provide a fix. This fix is at the intersection of the two circles of position described by the zenith for each of the two stars separately. On the Navigator-Sphere, this intersection is automatically established in the hinge at the zenith axis.
The Navigator-Sphere was designed primarily to solve the problem of simultaneous star altitude sights; it is also adapted to solution of other problems involving positions on a sphere, or measurements on a spherical surface. Those with which the author is acquainted are described herewith. With reflection, and some ingenuity, the reader may very well solve other such problems as he meets them.
A. Position from Simultaneous Star Altitude Sights
Given: Corrected altitude observed for two stars, and Greenwich sidereal time.
Find: Latitude and longitude of position.
Set Greenwich sidereal time in degrees and minutes of arc on the inner scale of the longitude circle, using the vernier on the sphere. Clamp and adjust precisely with the brake and tangent screw.
Set the given star altitudes on each circle with its vernier, using the clamp and tangent screw.
Insert star pins through bushings in the star altitude circles into the holes for the given stars.
The zenith axis is now mechanically locked in the required position at the co-altitude or zenith distance from the substellar point for both stars.
Latitude, and longitude west of Greenwich, for the fix are read directly from their scales.
This position corresponds to the point of intersection of the circles of position around the two stars. Theoretically there are two such points of intersection, but this second point offers no confusion in practice. Usually the spurious point is thousands of miles away from the known vicinity of the observer. However, if the approximate azimuth of each star is noted, say within 30 degrees, or the sphere oriented so that the stars are more or less in their obvious relation to the zenith, the correct fix is established without question.
The best intersection of the circles or lines of position is obtained when the azimuths of the two bodies differ by 90 degrees. For a difference of 0, or 180 degrees, the circles are tangent and thus do not intersect at a definite point. These relations are easily demonstrated on the instrument. They should be considered in choosing the stars for observation.
Checks of the fix are readily obtained with additional sights. Sights on three stars will give three independent fixes, four sights, six fixes, and so on. The instrument gives each fix for every pair of stars, by readings of latitude and longitude without the intermediate plotting of lines of position. However, these lines of position may be drawn, by connecting the several plotted fixes, or as in the following problem.
B. Line of Position Data for a Single Star
Given: Observed altitude of star, dead-reckoning latitude and longitude, and time.
Find: Line of position.
Set sidereal time, star pin, and star altitude. Turn sphere to nearest degree in latitude and read corresponding longitude (and azimuth). Plot this point on chart. Turn sphere to nearest degree of longitude and read corresponding latitude. Plot this point on chart. These two points determine the line of position. For check, this line should be perpendicular to azimuth noted above. Also, any number of other points on the line of position may be had by selecting appropriate values of latitude or longitude.
Alternate method:
Given: D. R. Latitude and longitude and time.
Find: Altitude and azimuth of any visible star.
Set sidereal time on inner scale of longitude circle with brake and vernier on sphere. Turn sphere to given longitude, tilt polar axis to latitude. Loosen star altitude circle, and set bushing over chosen star. Insert star pin through bushing into hole for star position, clamp star altitude circle and adjust star altitude tangent screw and azimuth to exact given latitude and longitude. Read altitude and azimuth for star, for this given position. Draw line of position in the usual way: Azimuth line through D. R. position, line of position perpendicular to azimuth line at a distance toward star equal to difference between observed and calculated altitude of star.
C. Running Fix
The running fix is available by the usual methods of plotting on the chart, by advancing a previous line of position for run. Generally the run between sights is not sufficient to introduce serious error from the curvature of the surface.
D. Observations on the Sun
Though the instrument was designed primarily for fixes from simultaneous star altitudes, it may also be used to reduce observations on the sun and planets and for problems of great-circle sailing. Auxiliary fittings are provided for the solution of these problems.
Line of position from the sun may be determined just as for any star except that the position of the sun, in right ascension and declination, must be established on the star sphere, for the instant of observation. A movable locating pad is provided which is set in position under the zenith axis, measuring declination and right ascension on the latitude and longitude circles, respectively. In this development model, the position of the longitude circle may require the use of a bridge attachment for the 8-degree band near the equator.
Line of position from sun sight may be found more directly using solar time. The position pad for the sun is set at the declination for the instant, along the index meridian on the sphere. (This corresponds to the equinoctial colure for stars and sidereal time.) Greenwich Apparent Time in degrees and minutes of arc, instead of Greenwich Sidereal Time, for the instant of observation is set on the inner longitude scale with the vernier on the sphere. This places the sun at the correct hour angle for every longitude. The line of position is then determined just as described above for the stars. Considering the importance of sun sights in actual navigation, an additional scale, inlaid in the sphere on the index meridian, may well be provided on which to set off the variable declination of the sun.
E. Observations on Planets
Line of position may be found from sights on planets or moon using adaptions of the methods described for the sun.
If position by simultaneous altitudes is desired, the first method employing sidereal time and position would be chosen. After locating the movable pad for the planet, it serves just as a star.
F. Great Circle Sailing
Given: Origin and destination
Find: Course and run for each heading, plot of route.
Clamp longitude circle relative to sphere in any arbitrary position. Set pads at origin and destination, using zenith axis to locate geographic latitude and longitude co-ordinates. Insert star pins, and set each star altitude circle at 0 degrees. The zenith axis is now at the pole for the desired great circle. Set another pad to mark this spot. Shift one of the star pins to the pole for the great circle, keeping its star altitude circle locked at 0 degrees. The other star altitude circle now lies in the plane of the desired great circle. By releasing the clamp, the zenith may be set at any desired point on the great-circle route, reading the distance from the fixed point, origin or destination, from the graduations on the great-circle scale, reading the bearing of the course from the azimuth circle, and the latitude and longitude from their usual scales. Any desired number of points may be taken, and a flight table prepared. From these data, using the Navigator- Sphere, the relative motion plot for the flight may be traced through the heavens. Also, stars near the horizon may be chosen to steer by.
In the construction and the development of the first model of the Navigator-Sphere, several peculiar problems were encountered. Because of the war, no skilled machinists could be found in Seattle to help with this project. However, the author was allowed to use the equipment and facilities of the College of Engineering of the University of Washington.
Materials were also hard to get with the priority red tape and depended in part upon the size of the projected instrument. Considering the precision desired—to one minute of arc, one mile—and the probable skill and accuracy of the author and the available machines (a 13-inch lathe and a new mill), a 5-inch bowling ball was selected for the star sphere.
These balls had not been manufactured for months and were completely out of stock. One of the local bowling alley proprietors had several cached. Caught off guard in a sober moment, he parted with one, “for the advancement of science,” and at twice the list price. Scraps of mild steel plate were begged from a sheet metal fabricator. It remained only to machine the steel into a precise instrument.
The ball-bearing arrangement of the rings was selected for accuracy and ease of manufacture. This design turned out well. Two rings acting together greatly improved their strength and stiffness. Some difficulty was encountered in attaching fittings to these rings without warping them, but these difficulties were mostly due to the restricted materials and restricted facilities at hand. Dovetailed rings were used for one of the star altitude circles but the ball-bearing device proved much the better.
One member of each pair of rings was graduated all around the circle in full degrees. The other member carried the corresponding vernier, reading to two minutes of arc. Because the rings were offset from the axis of their pivots, they were right- or left- handed, depending on which semicircle was used. Thus, though only 180 degrees is needed to cover the whole range of latitude and only 90 degrees for altitude, the graduations are carried on around so the rings may be used on either hand if any ring interferes with another.
All the scales were graduated in the lathe, scribing the lines with the horizontal motion of the lathe tool. The 360-degree circles were marked off using a 360-tooth gear attached to the outboard end of the lathe spindle, by inserting a dog into the successive teeth.
The trick used for graduating the verniers with this same equipment may have practical value to the reader in the construction of some similar device. Thirty equal divisions were required in the space for 29 degrees of arc. To accomplish this end, the ring for the vernier scale was set eccentric in the lathe so that these 30 divisions could be marked with the 30 teeth on the gear in the space for 29 degrees on the ring. This geometric relation is shown exaggerated in Fig. 4. Thus by simple trigonometry the exact amount of the eccentricity may be calculated and the work set accordingly off-center in the lathe. On the six verniers graduated thus by the author, no error could be detected when the 30 spaces on the vernier were matched with the 29 on the corresponding graduated circle. Strictly speaking, the 30 divisions made thus are not exactly equal. However, a rough calculation showed the maximum cumulative error to be about one-hundredth of a minute of arc, hardly significant in this instrument.
The Navigator-Sphere incorporates in one simple instrument adaptions of the valuable features of previous celestial navigation computers and has several new features of its own which add to its convenience and usefulness.
First, a two-star fix is produced mechanically, with all the essential elements in their true relations.
Second, the position of the principal stars is plotted on the sphere. Thus their position is always available without reference to the almanac or to any other tables. Also, this arrangement provides an ideal star finder, since there will be no projection distortion and the stars, in their true relation, stand out before the observer. Literally, one holds the heavens in his hand.
Third, mechanical adjustment is provided for the processional movement of the poles. With this single adjustment all the stars are continuously preserved in their correct relation to the axis of rotation. This feature also eliminates the use of tables of star positions from year to year.
Fourth, the convenience with which the Greenwich meridian may be used as the origin or reference eliminates entirely the use or computation of the hour angle for the heavenly bodies.
Fifth, the mechanical arrangements of the scales and parts combine to provide a sturdy, simple instrument, strong and accurate, and easily made. For navigational purposes where an error of one minute of arc might be tolerated, this instrument can be produced by ordinary mass production standards to give imperceptible error. Thus it could be made cheaply and in quantity so that every navigator in our air forces, Navy, and Merchant Marine could have his own Navigator-Sphere.
Finally, celestial navigation by the use of this instrument is so simple and obvious that it can be taught to almost anyone. Just like the riding of a bicycle, once the basic physical relations of time and space are impressed on the manipulative and visual senses, they are simply not forgotten. Also, as in the case of the bicycle, it is difficult to learn from the printed pages without the machine, but understanding comes quickly from experience with the device in hand.
One of our very green freshman engineering students exclaimed, when he saw the instrument, “Why, even the bombardier can be a navigator.”
1. Bowditch, Nathaniel, American Practical Navigator, U. S. Hydrographic Office #9, Revised edition, 1938, p. 159.
2. U. S. Hydrographic Office #211, Dead Reckoning Altitude and Azimuth Table, A. A. Ageton, 1938. #214, Tables of Computed Attitude and Azimuth, 1942. #218, Altitude and Azimuth for Selected Stars, 1943.
3. Weems, P. V. H., Star-Altitude Curves, Weems System of Navigation, 1928.
4. “Air Navigation for Global War”; D. A. McMillen’s spherographical system, Fortune Magazine, 27: 74-7, January, 1943.
5. Bowditch, N., op. cit., p. 118.