IN 1913 WILLIAM BROOKS CABOT AND Russell W. Porter together explored the St. Augustine River in Labrador, east of the route up which a railroad is now being pushed toward the great new iron-ore deposits. In later years they discussed the question whether an explorer could determine his latitude in the field without a precision instrument. Porter tried it and described the experiment in the journal Popular Astronomy.

Cabot, he wrote, "deplored even the addition of a pocket sextant and horizon to a camper's pack where dead weights must be shaved to a minimum, and argued that the tools irreducible to the explorer-a knife, hatchet and fish line- together with nature's available materials, would suffice to obtain latitude within a mile" by measuring the sun's altitude. Porter doubted this, but made the experiment at Springfield, Vt., and was greatly surprised to find that the specified precision could be leached if he used a steel tape instead of a stretching fish-line. By various refinements he found the latitude within a fifth of a mile. The method should interest explorers, Robinson Crusoes, escaped prisoners, amateur astronomers and others with intellectual curiosity.

Roger Hayward's drawings in Figure 1 describe the simple principle of the method. At C in the lower right-hand corner is a nail driven into a small log exactly opposite a chosen mark on the plumb line BC. The distance BC is known, and attached to the nail at C is a steel tape with a sliding sight. Always keeping BAC a right angle, the observer sights the sun at its culmination or highest angle (local apparent noon) and measures AC. In the right triangle ABC the distance BC divided by the distance AC will give a decimal fraction which is the sine of the angle ABC. The same decimal is then found in a table of sines, and on certain dates in March and September when the sun is over the Equator the angle shown opposite will be the observer's latitude.

On all other dates, corrections given in tables in the ephemeris of the sun must be added or subtracted. In either case a small correction, given by Porter as about one minute of arc in summer and two in winter, must be added for atmospheric refraction.

For convenience in calculation Porter used a tape with feet divided into tenths and hundredths, and interpolated to thousandths of a foot. A metric tape is just as convenient, and Hayward points out that an ordinary tape with 96-subdivisions per foot may not introduce significant error. An error of one thousandth of a foot affects the result about half a mile in latitude. For a sight Porter used a tiny hole in the sliding sight shown in the drawing, covered with colored glass to protect his eye. He found it possible to bisect the sun's disk with the nail at B reliably within less than one minute of arc. In a test of the method Porter made six sights between 11:41 a.m. and 12:08 p.m. on the same day, obtaining latitudes for Springfield varying from 48 degrees 17.2 minutes to 43 degrees 19.8 minutes. The mean of the six sights was 48 degrees 18.5 minutes. As a check he next determined the latitude with his theodolite and found it 43 degrees 13.3 minutes, Thus he had determined his latitude within .2 minute, or only 1,200 feet, without an instrument.

Theoretically the tape is unnecessary. Fish lines at AC and BC could be measured with any arbitrary, unknown unit of length, such as a stick, provided the same unit was used on AC and BC, but precision would be difficult.

John J. Ruiz of Dannemora, N. Y., recently tried Porter's experiment with more refined accessories and a modification of his own. Exactly halfway between B and C he pivoted a rod and hooked it to the sight, as shown at the left in the same drawing. "If you remember your Euclid," he writes, "you will note that the angle BAC is always 90 degrees." (An angle inscribed in a semicircle is a right angle.) The purpose of the rod was to exclude a possible source of error in Porter's method: "The observer's head," Porter wrote, "must be moved and the target shifted until the sun is bisected with the minimum length of tape."

Ruiz used a slotted pivot in place of the lower nail, and a sight with a vernier and 1/25-inch peephole (No. 60 drill) protected by two thicknesses of deep-colored cellophane. To average out the accidental errors he too made six observations and, since each took time, and since Joshua was not present to make the sun stand still, these sights were made at intervals before and after apparent noon and then reduced to the meridian. He came out neither worse nor better than Porter, that is, within about 1,200 feet of the true latitude.

Since it is impossible to evaluate all the contributing factors in the two observers' experiments, with the hidden chance errors, it cannot be known without more meticulous repetition by a third person whether or not Porter had already exhausted the precision inherent in the method. If he had, added accuracy in technique might bring only fictitious improvement.

The ephemeris of the sun is in the Observers Handbook of the Royal Astronomical Society of Canada, 3 Willcocks St., Toronto, Ont., Canada (price 40 cents). Ruiz provides refraction corrections which he says "are good enough for you and me." For 10-degree altitude of sun, add 5 minutes angle; for 20 degrees, 2.4 minutes; for 30 add 1.5; for 40 add 1; for 50 add .7, for 60 add .5; for 70 add .3; for 80 add .2, and for 90 add nothing.

ALAN R. KIRKHAM points out two omissions from the round-up on the modified or spherical-secondary Cassegrainian telescope (Dall-Kirkham) in the September, 1951, issue. He writes:

"The great difficulty in testing the hyperboloidal secondary of the straight Cassegrainian is almost eliminated by changing to the modified form with its spherical secondary, and using the King test or others. But the primary remains a bugaboo, with its zonal testing. No one seems aware that this, too, can be avoided. Since the primary is very nearly if not exactly an ellipsoid, it may be tested by the direct focal test (Amateur Telescope Making, page 271), which 'feels' just like figuring a spherical mirror by testing at the center of curvature. The problem is to find where to put the knife-edge and pinhole to make the desired ellipsoid look flat.

"The formula for finding their distances from the mirror is

in which R is the radius of curvature of the mirror and e its undercorrection found either from my original formula (SCIENTIFIC AMERICAN, June, 1938) or from the formula in the September, 1951, article in which N is the per cent correction. e = 1 - N; for example, a 70 per cent corrected paraboloid is .30 undercorrected, and it is the .30 that is used for e. By taking the +/- sign first as plus, then as minus, and solving for each, both distances are obtained. It is theoretically immaterial which distance is used for the pinhole and which for the knife-edge, but better results are obtained with the pinhole distance the greater. A diagonal or prism is necessary for viewing the mirror.

"The second omission is the fact that the straight Cassegrainian is afflicted with coma, and the modified Cassegrainian is somewhat more so. If the telescope is to be used photographically, with the wide field of the plate, this will be damaging, but when we come to the visual I instrument-and the majority of amateurs' telescopes are built for 'just lookin''-the case is very different. The field of stars intercepted by, say, a one-inch eyepiece is less than 3/4-inch in diameter, and over that small area the coma will be undetectable in either telescope, probably covered up many times over by the faults of the eyepiece. The modified Cassegrainian may even outperform the other, since it will probably be a little better figured.

"Modified Cassegrainians rigorously investigated by ray tracing have been found to be within or very near the boundary of the coma tolerance for high-quality visual instruments such as microscopes and binoculars."

IS the focal ratio of a mirror a determining factor in the exposure required for a focogram? No, says Lieut. Col. A. E. Gee of the Frankford Arsenal, who was asked to answer the question as a referee.

He demonstrates this as follows (see drawings): "Obviously, the same exposure that would give a focogram for an f/4 mirror would give an equally good focogram of the central half of the mirror if it were stopped down to f/8. The mirror is the object being photographed, not the lens taking the photograph.

The lens of the focogram camera is the image of the pinhole produced by the mirror. The light source is the pinhole itself. Ignoring for the moment the reflectivity and figure of the mirror and the presence of the knife-edge, the necessary exposure is solely dependent upon the intensity of the light source, size of the pinhole, sensitivity of the film and distance from the pinhole image to the film. If these factors are constant, all mirrors, whether large or small, long focal length or short, will require the same exposure. Increasing the focal ratio will simply mean a larger picture. Increasing the mirror size at the same focal ratio would make no change whatsoever.

"All other factors being the same, the diameter of the focogram will be directly proportional to the distance from the pinhole to the film. Exposures will vary as the square of this distance. If the pinhole image to film distance is left constant, the diameter of the image on the plate will be inversely proportional to the focal ratio of the mirror. That is, that of an f/4 would be twice as big as an f/8.

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