Copyright © 2007, Institute of Navigation, www.ion.org

Vol. 4, No. 6 Jacobs: Navigational Aspects of the let Stream 249

ice, APO 953, San Francisco, California, Pan-American World Airways, San Francisco Mnnunl 55-1, Tokyo-Honolulu C-97 Non- International Airport, November 1953. Stop Procedures, October 1954. lOStaff Members, Department of Meteorology, ‘Ruskin, R. E., et al, Development of the NRL University of Chicago, “On the General Cir- Ax&Flow Vortex Thevnzomete,; Report culation of the Atmosphere in Middle Lati- 4008, Naval Research Laboratory, Washing- tudes,” Bulletin of the American Meteoro- ton, D. C., 1952. logical Society, Vol. XXVIII, June 1947. “Serebreny, S. M., and Wiegman, E. J., The llBellamy, John C., “Four Dimensional Flight Characteristic Properties of the /et Stream Planning,” Navigation, Vol. 4, No. 3, Sep- OzJer the Pacific: Case History No. 1, Part 1, tember 1954.

ON THE GEOMETRICAL SOLUTION OF THE NAVIGATIONAL TRIANGLE JOHN A. RUSSELL University of Southern C&f ornia, Los Angeles, California

The Ukted States Naval Institute Proceed- line of position problems).” The triangular ings for March, 1952, contain a paper by pyramid that Mr. Breed discusses, although Joseph B. Breed III entitled “The Navigational sufficient for the determination of computed Triangle: How to Solve It by Drawing It.” In altitude or great circle distance, will not yield this paper Mr. Breed capably expresses the con- the azimuth or the great circle course. The fol- viction that although the supplanting of the lowing construction extends Mr. Breed’s solu- trigonometric solution of the navigational tri- tion to include the azimuth, without which, the angle by tabulated solutions represents a great solution of a line of position problem is not forward step in celestial navigation, instruction complete. in the newer methods often leaves the student In Figure 1, E is the center of the earth; 2 in ignorance of the basic relationships in the is the navigator’s position; P is nearer geo- navigational triangle. Mr. Breed does not pro- graphic pole; S is the geographic position of pose that spherical trigonometry be reinstated in the celestial object observed. ZP’ and ZS’ are the navigational curriculum. Instead, he sug- tangent to the earth at 2 and intersect the ex- gests, as an instructional tool, a geometrical tended radii EP and ES respectively. The de- solution of the navigational triangle that re- sired azimuth angle is the angle at 2 in the quires only a protractor, a straight edge, and a plane triangle P’ZS’. This triangle is the base pair of compasses. The paper describes with of a triangular pyramid whose vertex is at E. admirable clarity how to obtain the computed The three angles between the three radii to P, altitude of a celestial object by constructing the 2, and S are the complements of the triangular pyramid whose vertex is the center of 2, of the altitude of the celestial object of the earth; whose three sides are the extended viewed from 2, and of the declination of the radii of the earth through the nearer geo- celestial object. graphic pole, the geographical position of the Figure 2 shows how a flattened version of celestial object, and the navigator’s position; the pyramid of Figure 1 can be constructed and and whose base is the intersection of the pyra- the azimuth angle obtained. First, draw EZ to mid with a plane tangent to the earth at the any convenient scale. Next, draw EP, so as to nearer pole. make an angle of 9o”-L, with EZ, and ES SO Mr. Breed suggests that his readers “Work as to make an angle of 90°-H, with EZ. (H, out several navigation problems with this will already have been obtained from Mr. graphic method (azimuth, great circle course Breed’s construction.) Extend EP, and ES until and distance, and identification as well as they meet the perpendicular to EZ drawn Copyright © 2007, Institute of Navigation, www.ion.org

250 Navigation June 1955

Figure 1 Figure 2

through 2 at P: and S’. Next draw EP:, equal of spherical trigonometry, but they have not in length to EP:, so as to make an angle 90°- been as widely exploited in the manner sug- Dec. with ES’. Then strike an arc from 2 with gested by Mr. Breed as their instructiveness radius ZP{ and an arc from S’ with radius S’P:. merits. The author has found Mr. Breed’s ap- Angle PGZS’ is the desired azimuth angle. proach, as extended in this paper, a very useful Geometrical solids of the sort used in this manner of presenting the derivation of the tab- and Mr. Breed’s constructions are often em- ulated data in HO 214 and HO 249 without ployed in deriving the fundamental formulae introducing the concepts of trigonometry.

PUTTING THE MOON IN THE POCKET ALMANAC ROBERT W. BYERLY

The omission of all moon data from the gator his true position is a moon sight taken Pocket Almanac was recognized by the author at the time of a sight. In the night, the of the Almanac as a ground for criticism. TWO moon may provide a line of position which can reasons were given for omitting moon data.l be obtained in no other way. When the altitude One of the reasons was that “the moon is of the moon is not too high, the moon some- little used.” It is no doubt true that the moon times lights the horizon below it sufficiently is not much used in routine navigation, but in to permit the taking of a sight. It thus provides some situations the moon is of critical impor- a means, and the only celestial means, by which tance to the navigator. In the daytime, the moon a marine navigator may be able to obtain accu- enables the navigator to obtain a point fix. rate data as to his position during the some- When dead reckoning is uncertain because of times long interval between evening twilight an unknown current or wind, two or even more and morning twilight. Moon data thus has a position lines obtained from observations of very practical value for the navigator. the sun taken several hours apart may have Mr. Wylie’s other reason for omitting moon apparent crossing points several miles from the data was: “To maintain our standards of accu- true positi0n.l In this situation, the only racy, inclusion of the moon would involve, on astronomical means which can give the navi- account of that body’s rapid movement among