THE FOLLOWING collection of simple projects illustrates the fact that many attractive and significant experiments and observations can be conducted with apparatus that calls for the expenditure of little time and money. The first project involves the construction of the handy instrument called the cross-staff. It is a primitive equivalent of the mariner's sextant.

The instrument measures the angle between distant objects. Its introduction into Europe during the 14th century by the Jewish philosopher Levi ben Gershon revolutionized navigation by enabling mariners to find their way at sea in rough weather. Unlike the astrolabe and the quadrant that had previously served navigators, the cross-staff does not require a plumb bob for finding the zenith. Its construction is described by I. L. Fischer, associate professor of physical science and mathematics at Bergen Community College (400 Paramus Road, Paramus, N.J. 07652).

"A year ago," Fischer writes, "I was attracted by the idea of having my astronomy students build their own instruments so that they could make observations at home instead of waiting for those rare occasions when the student was free, the weather was clear and the school's limited collection of instruments was available. Most students, including those without particular mechanical skills, managed to make and calibrate a cross-staff within two hours. Only a few spent money for materials.

"As its name suggests, the cross-staff consists of two elements: the staff, which resembles a yardstick, and the transversal, which is essentially a mask that makes a right angle with respect to the staff and slides along the staff [see illustration at right]. The staff is graduated in angular degrees.

"To make a measurement the user sights along the staff from one end while simultaneously adjusting the position of the transversal until the distant objects appear to line up with the ends of the transversal. The angle between the objects then corresponds to the graduation on the staff indicated by the position of the transversal. The students checked the accuracy of their instruments against a pair of lines that had been drawn on a blackboard with the aid of a sextant. Many instruments were accurate to within .05 degree, and virtually all came within .2 degree.

"When the cross-staff is employed in navigation, it is positioned so that the transversal is in the vertical plane. The cross-staff was the first instrument that involved the visible horizon in celestial observations. Latitude could be approximately determined by sighting simultaneously on the pole star and the horizon, thus eliminating the need to determine the zenith with a plumb bob.

"A serviceable cross-staff can be made with a sheet of stiff cardboard eight inches square, a smooth length of wood about the size of a yardstick, several rubber bands, scissors, a pencil and a ruler. The accuracy of the instrument can be increased by fitting the staff with transversals of two or more lengths, according to the size of the angles to be measured. The model I designed has transversals of two, four and eight inches.

"The transversals are formed from a single sheet of cardboard, which also is cut and folded to form a pair of braces that support the transversals at right angles with respect to the staff and constitute a cardboard sleeve on which the transversal unit slides on the staff. In the accompanying illustration [at left] the solid lines represent cuts and the dotted lines represent folds. The top inch of the cardboard serves as the four-inch transversal, the second inch is the eight-inch transversal and the bottom inch functions both as the brace to support the unit and as the two-inch transversal.

"One or two rubber bands hold the sleeve on the staff. The tension should be adjusted so that the transversal slides freely but remains in the position to which it is set. An additional rubber band holds the braces in position on the sleeve. Measure the length of the three transversals as accurately as possible.

"To calibrate the staff, mark one of its ends 'eye' on both sides. Measuring from the eye end, draw straight lines halfway across one side of the staff at distances listed in the accompanying table [] for an eight-inch mask and label each line with the number of degrees that correspond to that distance. Then turn the staff over and similarly label the other side according to the part of the table that applies to the four-inch mask.

"A third set of graduations can be added on each side with ink of contrasting color to measure angles with the two-inch portion of the transversal. I leave to the experimenter the fun of calculating this table of distances. The distances are calculated by dividing the width of the mask (in inches) by two times the trigonometric tangent of one-half of the desired angle. For example, with the two-inch transversal at what distance from the eye position should the graduation be placed to indicate an angle of five degrees? According to the formula, the distance (d) equals the width (W) divided by two times the tangent (tan) of half the angle : inches. The formula can of course serve for correcting the calibration scale if the transversals do not turn out to be exactly the specified four or eight inches and for deriving tables for transversals of any length

"The cross-staff is suitable for any application that would normally involve a sextant. Indeed, for some purposes I find it superior to a sextant. For example, during observations at night it is difficult to keep track of a particular star unless its relation to surrounding stars is constantly evident. The cross-staff facilitates measurements involving stars because there is always a large field of view except for the small portion obscured by the mask.

"One of the interesting astronomical projects that can be carried out with a cross-staff is plotting the path of the moon or the planets against the background of the fixed stars. The position of the moon can be determined by measuring the angle between it and three stars. By repeating this determination over an extended interval the path of the moon can be plotted on a star map with impressive accuracy. The direct and retrograde motions of the planets can be recorded in a similar way.

"When the cross-staff is constructed carefully, it is capable of splendid performance. Accuracies of better than .1 degree can be attained easily for angles of up to about 20 degrees. Hence it is desirable to mark additional lines on the staff at .l-degree intervals to facilitate reading the scale. With the four-inch transversal, for example, the difference between an angle of 10 degrees and one of 10.1 degrees amounts to more than .2 inch on the staff. In making observations brace the eye end of the staff against the cheekbone to ensure that error will not be introduced by holding the instrument at an indefinite distance from the eye."

ROGER H. James, who is associated with Pratt & Whitney (400 Main Street, East Hartford, Conn. 06118), has developed a simple technique for making reasonably accurate paraboloids with a diameter of two feet or so. "As is well known," he writes, "the surface of any rotating fluid in a rotating container assumes the shape of a paraboloid of revolution. The height in inches (h) of a point r inches from the axis of revolution is approximately equal to the product of the square of the speed of the container in revolutions per minute (N) multiplied by the square of the radius in inches (r) divided by 70414: . The focal point of the paraboloid, which is the distance from the apex of the paraboloid at which impinging parallel rays are reflected to focus, is equal to 17604 divided by N.

"I made a paraboloid by pouring melted paraffin into a pan of very hot water. The pan was 18 inches in diameter and rotated at 45 revolutions per minute on the turntable of an old record player. Several hours later, when the rotating paraffin had solidified, I had a wax paraboloid, which served as a mold for making a permanent paraboloid of fiberglass. Fiberglass and an appropriate epoxy for embedding the fibers are available as a kit at most hardware stores. Both the concave and the convex surface of the paraffin mold have the same shape.

"The depth of my parabola is about 2.3 inches and the focus is 8.695 inches from the bottom. To find the slope of the paraboloid in angular degrees at any point from the apex multiply the focal length by 4 and divide 1 by the product. Multiply the resulting quotient by two times the radius in inches to the point of the desired slope. The product is the trigonometric tangent of the angle.

"For example, I have indicated that the focal length of my dish is 8.695 inches. The reciprocal of 4 x 8.695 is 1/34.78, or .02875. Hence the slope at the edge of my dish is arc tan 2 x .02875 x 9 = arc tan .517 = 27.36 degrees. The equations are valid for any liquid.

"For most practical purposes differences in surface tension can be neglected. Containers of any shape are satisfactory, and they need not be centered. The axis of the resulting paraboloidal surface will coincide with the axis around which the fluid rotates. With a cylindrical container that is centered the level of the fluid at the edge will rise by exactly the same amount that the level at the center drops. Containers of this shape can therefore be half-filled.

"The axis of rotation of the container must be exactly vertical, otherwise the fluid will not rotate uniformly. When the axis of rotation is truly vertical, a speck of dust resting on the bottom of the container near the axis does not move with respect to the bottom."

Incidentally, readers have called attention to three errors in the formulas previously presented in this department for constructing an approximate paraboloid with sectors of cardboard [see "The Amateur Scientist, SCIENTIFIC AMERICAN, December, 1973]. The rows of the table on page 127 should be numbered 0 through 10 instead of 1 through 10 The term (y-y) in the formula for should be squared. A closer approximation of a true paraboloid will result by changing the formula for the base of the sector to , in which N is the number of sectors.

MOST amateur astronomers have trouble focusing a telescope in cold weather. The rack-and-pinion adjustment is all but impossible to manipulate while the observer is wearing mittens. If the mittens are removed, fine adjustments are hard to make with stiff fingers. Charles W. Bowen of San Antonio, Tex., has solved the problem by providing the rack-and-pinion draw tube, which supports the eyepiece, with a motor drive.

"I installed a Model CA 1/2-RPM Hurst reversible motor on my eight-inch reflecting telescope," Bowen writes. "When I bought the motor, it cost $13.95. It is available from mail-order distributors.

"The motor can be switched on and off in either direction by a homemade control that has a push-button switch for each direction. The gearing allows the unit to develop a force of 150 inch-ounces at constant speed. I learned by experiment that a half-revolution per minute is the best rate at which to turn the focusing pinion. Focus is approached at a controllable speed with eyepieces in the focal-length range from 30 millimeters to four millimeters.

"The mechanical details involved in mounting the motor depend on the structure of the telescope. For my instrument, which is typical, I bent a bracket of sheet metal and drilled it to accept the mounting screws of the motor and to be attached by similar screws to the tube of the telescope [illustration above left]. One knurled knob was removed from the shaft of the pinion and: replaced by a short length of pipe that is the coupling to the motor. The ends of the pipe were drilled and threaded for setscrews that clamp the shafts.

"I cannot overemphasize the convenience of this focusing scheme. The observer can trim up the focus without touching the instrument and jiggling it. I made the installation in a few minutes less than two hours."

H. R. CRANE, professor of physics at the University of Michigan, suggests a way to view the sun safely. It is perhaps the simplest and safest method of all. It makes use of a pair of binoculars and should be kept in mind against the time when the observer becomes interested in examining a big sunspot or an eclipse.

"Textbook diagrams of binoculars," Crane writes, "always show the image to be virtual, at 25 centimeters or more in front of the eye. As a practical matter, however, the focusing range of binoculars is made so wide (in order to accommodate users having all kinds of visual abnormalities) that a real image can be cast at any distance behind the eyepiece-from infinity to as close as 10 or 20 centimeters. The size of the image of the sun cast by my binoculars (they are seven by 35 millimeters) is about six inches in diameter at eight feet. If the image falls on a white card that is shaded from direct sunlight, its brightness is just right for comfortable observation. In lining up the binoculars the experimenter must resist the temptation to look into the eyepiece. Concentrated sunlight could destroy the retina.

"I fastened my binoculars to a tripod. I cut a two-inch hole in an 18-inch square of cardboard and pushed it onto the front end of one lens barrel of the binoculars [right]. The cardboard served both to shade the image from direct sunlight and to mask one of the telescopes so that only a single image was cast on the screen.

"Much fun can be had with the setup even when there is no eclipse. I have viewed sunspots clearly with the apparatus and have also photographed the spotted solar image with an ordinary camera. This kind of photography requires care. The spots are small and the contrast is not great I did not succeed when I held the camera by hand. I made good pictures when I supported the camera with a second tripod and set the aperture small (f/22). The small aperture is necessary for increasing the depth of focus so that the camera can be positioned at an angle with respect to the plane of the screen.

"I found that it is important to maintain the optical axis of the binocular at a right angle in relation to the screen. Aberrations grow rapidly as the axis moves away from the normal. A way to increase the brightness of the image with respect to the background is to decrease its size by moving the screen closer to the binoculars, assuming that the camera can be focused to a short distance. Photographs of higher contrast can be made by enclosing the camera and screen in an improvised shadow box and by using high-contrast (copy) film."

MARTIN GARDNER'S recent illustration of hexagonal numbers as differences between consecutive cubes [see "Mathematical Games," SCIENTIFIC AMERICAN, July] brought to mind an identical cube that constitutes an optical illusion. The cube was devised a year or so ago by Fred Duncan (Box 264, Redway, Calif. 95560), who is a specialist in psychokinematic art. To make the cube cut from light cardboard a figure according to the accompanying pattern [left]. Slit the cardboard and fold it a specified.

Cement each lettered flap to the surface labeled with the matching circled letter. The result is a cube with one missing corner.

Hold the cube in one hand and with one eye look diagonally into the cube at the missing corner. Keep the other eye closed. After a second or two the missing corner will seem to be replaced by a smaller cube projected from the corner of the larger one. Keep staring without changing your viewing angle. After perhaps a minute or two the larger cube will vanish, to be replaced by three mutually perpendicular planes, like two walls and the floor of a room. In the far corner of the "room" you will see the smaller cube that formerly occupied the near corner. The structure therefore comprises three illusions in one.

CORDLESS appliances ranging from electric hand drills and grass trimmers to shavers and hand-held calculators abound these days. Most of them run on nickel-cadmium storage batteries, which offer a number of opportunities for amateurs. Nickel-cadmium cells come in all the standard sizes of flashlight cells. They cost only from six to eight times more than expendable cells and can deliver far more current. They are available from the mail-order distributors of electronic supplies.

Notwithstanding these advantages, the batteries have not become popular with amateur experimenters. One explanation may be the difficulty of getting information about the characteristics of the cells, particularly their charging characteristics. The following data were accumulated by persistent interviews with specialists and by extended correspondence with manufacturers over the past six months.

First, there is no simple test by which you can determine the state of the charge in a nickel-cadmium cell. The rate at which a cell is charged depends on a number of factors and is usually specified by a data sheet supplied by the manufacturer. The sheet also states the capacity of the cell in terms of ampere-hours. The charging rate in amperes is usually specified as the quotient of C/T, in which C is the capacity of the cell in ampere-hours and T is the charging time in hours.

Most cordless appliances that include a battery charger have the charging rate fixed between C/15 and C/20. Makers of some appliances specify the number of hours the cells should be charged, but most appliances come without any advice on the subject. Many experimenters have assumed that cells for which no charging time is specified can be charged indefinitely without damage. Supposedly the charger includes a regulator that reduces the rate automatically when the cells reach full charge.

All efforts to design a regulated charger have failed. In the absence of other instructions charge all nickel-cadmium cells at the C/20 rate. For example, a battery of one-ampere-hour capacity would be charged at the rate of 1/20, or 50 milliamperes. Assume a charging efficiency of roughly 50 percent. The fully discharged battery would recharge (at the rate of 50 milliamperes) in about 40 hours. It could be kept on a continuous charge without significant damage for 60 hours. One manufacturer states that its cells can be kept on continuous charge indefinitely at the C/15 rate, but the company does not mention what the cost is in terms of the life of the battery.

When a nickel-cadmium cell reaches full charge, the energy put in thereafter is expended partly to liberate oxygen, which oxidizes the cadmium electrode, and partly as heat. If the charging rate is more than about C/15, gas may evolve more rapidly than it can combine chemically with the metal. Pressure then increases explosively. Depending on the structure of the cell and the rate of charge, the pressure can build up to 125 atmospheres within minutes.

Most nickel-cadmium cells have pressure safety valves. When the valves open, the cells may lose their electrolyte (potassium hydroxide). This caustic substance can burn tissue and destroy apparatus. The lye can be neutralized with a 3 percent solution of boric acid. Usually excessive pressure ruins the cell.

Nickel-cadmium batteries can be discharged at much higher rates than conventional dry batteries but should never be fully discharged. The reason is that full discharge would risk applying current of reversed polarity to the weakest cell of the battery. Reverse charge permanently damages the cell.

On the other hand, every cell must be fully discharged at least once a year and fully recharged, otherwise the cells deteriorate chemically, with the result that ampere-hour capacity is permanently reduced. To discharge a battery safely, disconnect the cells from one another and connect each cell to a separate resistor load until it is fully discharged. Cells can be stored indefinitely without damage in the fully discharged state. A fully charged but idle nickel-cadmium battery will lose approximately 20 percent of its charge within two weeks at room temperature.

In this department's description in September of a quartz-crystal oscillator for a pendulum clock the two integrated COS/MOS circuits labeled CD 4020 AE in the upper right corner of the illustration on page 195 should have been labeled CD 4024 AE.

Bibliography

AMERICAN PRACTICAL NAVIGATOR. Nathaniel Bowditch. U.S. Navy Hydrographic Office, U.S. Government Printing Office, 1966.

FLUID MECHANICS. Victor L. Streeter. McGraw-Hill Book Co., 1971.

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