From an illustration of a dial designed and built by Russell Porter and described in this department, Ronald F. Scott, Box 84, Johnson City, Tenn., made a similar dial with minor improvements. "A really good sundial had been the object of my search for years," he writes. "I tried several built on flat plates but they had many shortcomings. Porter's sundial, with its analemma curve and lens for focusing the sun's image, masters practically all the difficulties with remarkable simplicity. I have been gratified to find that this dial is accurate. The readings vary by no more than one minute from correct time, and often the two match."

Roger Hayward's drawing at the left was made from Scott's data and from familiarity with the prototype in Porter's garden. The tiny lens, only a quarter of an inch in diameter and equal in focal length to the radius of the analemma plate, is mounted in a 12-liter boiling flask obtained from the Corning Glass Works, Corning, N. Y. The curve of the "equation of time" or error of the sun (actually it is the earth that errs) is inscribed on a metal sheet inside the flask; it is the mirror image of the one that appears on a globe. To find the time, the flask is grasped by the neck and rotated on the stud in the lower right-hand part of the illustration and on the two supporting pads until the sun's image bisects the curve. This image then gives the date; the time is read on the scale surrounding the flask.

"The polar-axis stud is tapered," Scott writes, "to fit a tapered hole in the glass, and sits in an adjustable thrust-bearing. The two pads that constitute the north bearing are adjustable vertically, and final adjustment of the sundial is achieved with them. A second hole must be drilled in the flask for the post that holds the plate in position. I spent considerable time trying to figure out how to locate the plate correctly before I struck the happy thought of suspending a plumb bob through the stud hole so that the string represented the axis of rotation. Sighting through the center of the lens enabled me to align the longitudinal axis of the plate quickly, and the transverse axis was positioned by using a level line of sight through the lens. I marked a cross on the flask at the end of the axis opposite the stud hole to show the pointing of the bob in checking the inclination of the axis during adjustment. A lead weight was later mounted inside the flask behind the analemma plate to balance the weight of the neck, as the flask tended to creep in some positions."

Scott's improvement on the prototype is in waterproofing the dial by using copper instead of paper for the time band. Porter warned that extreme care must be taken to locate the stud hole accurately. The holes may be made with a hand drill, an inch of tubing and wet abrasive grains. Scott's tapered hole insures against looseness.

IN spite of a widespread impression, it is not easy to know the inventor and the date of invention of such things as the steamboat, the telegraph and the incandescent lamp. The same applies to the reflecting telescope. He who turns to history to find the solid ground of fact will soon find himself bogged in ooze. Was the inventor of the reflecting telescope the English astronomer John Hadley, who in 1723 made the first good modern reflector with a paraboloidal mirror? Or was it Sir Isaac Newton, who in 1668 built the first reflector with an eyepiece, though he knew no way to parabolize his spherical mirrors? Was it the Scottish mathematician James Gregory, who five years earlier proposed the type of reflector known today as the Gregorian, though for lack of manual dexterity he did not build one? Was it the French mathematician Marin Mersenne, often called Mersennus, who proposed the idea of a reflecting telescope to the French philosopher Rene Descartes in 1689, only to be told that it was fallacious? Was it the English mathematician Leonard Digges, who used concave mirrors before 1571, though probably without an eyepiece and as terrestrial telescopes? Was it the English philosopher Roger Bacon, who with Peter Peregrinus spent three years and the equivalent of $3,000 learning to make concave mirrors in approximately 1267? That Bacon even then understood spherical aberration is shown by his statement that the focal length is much less for rays from the outer zones of the mirror, and that it is half the radius of curvature. He left no record of the uses of the two mirrors that he made, but L. W. Taylor of Oberlin University tells us that a century later Peter of Trau recorded the tradition that with Bacon's mirrors "you could see what people were doing in any part of the world." He adds that the students at Oxford University spent so much time experimenting with these mirrors that the University authorities had them smashed. Perhaps their "experiments" were not limited to pure science.

It is thus that the invention of the reflecting telescope is blurred. Like most inventions it was a gradual process. The English mathematician Robert Smith, in his Compleat System of Opticks, published in 1738, refers to Gregory as "the first inventor" of the reflecting telescope; but Sir John Pringle in his Discourse on the Invention and Improvement of the Reflecting Telescope, delivered in 1777 before the Royal Society of London, designated Newton as "the main and effectual inventor." Pringle looked down his nose at the telescope which the French sculptor Guillaume Cassegrain revealed in 1672, describing it as merely "a disguised Gregorian never put into execution by its author", but Louis Bell has pointed out in Popular Astronomy that "it is the irony of time that Cassegrain's form is the one that has survived, in the great telescopes."

It may be surprising that Mersenne, Digges and others did not build directly upon the advances of their predecessors. Because of the lack of facilities for disseminating such information, they no doubt remained unaware of those advances. Galileo, who lived from 1564 - 1642 and built the first astronomical telescope, did not know that Leonardo da Vinci, who lived in Italy a century earlier, had designed machines for grinding concave mirrors; none of Leonardo's scientific writings were published until 1880, and they have not all been published yet.

In 1885, two centuries after Cassegrain's invention, a modification of the optics of that telescope was proposed in English Mechanics by an unknown contributor with the initials A.S.L. He proposed to substitute a simple sphere for the hyperboloidal secondary and to shape the paraboloidal primary so as to balance the aberrations of the secondary. This would call for an ellipsoid, sometimes loosely called an "undercorrected paraboloid." So far as is known nothing tangible resulted from the proposal of A. S. L.

Since that time a modest number of Dall-Kirkhams have been built and have proved satisfactory. No claim was ever made that they are optically superior to the Cassegrainian. They are simply easier to make, and this is why the armed forces have recently had a number of them made at the Tinsley Laboratories and at the Frankford Arsenal. The older more difficult Cassegrainian paraboloid-hyperboloid combination still survives, partly from the momentum of tradition, and perhaps because it has been difficult to collect the fragmentary instructions for making the Dall-Kirkham, scattered as they are in several back numbers of SCIENTIFIC AMERICAN. Robert Turner Smith of 785 Cerrito St., Albany, Calif., an employee of the Tinsley Laboratories, has now prepared instructions that are complete in themselves. He writes:

THE compound telescope has an appeal which cannot be denied. The Cassegrainian in particular has advantages both in construction and in observation. Its long equivalent focal length is conveniently folded into a tube only about one fourth as long as that of the Newtonian, an arrangement which results in great stability, less vibration and little overhang of mass beyond the bearings of the mounting. The eyepiece is at the lower end of the tube, more easily accessible and with less sweep than the eyepiece of a Newtonian. In the common focal ratios the Cassegranian has one half the length but twice the power of the usual Newtonian. When high power is desired, a lower-powered eyepiece with greater eye relief can be used to obtain the same power where a short-focus eyepiece with uncomfortably close eye relief would be needed with a Newtonian. When good seeing prevails, the maximum useful power of 50 times the aperture in inches can be attained without resorting to an ultra-short-focus eyepiece.

Despite its advantages the true Cassegrainian presents many problems in the polishing and figuring of its mirrors. The primary, although usually an f/4 requiring a "strong" paraboloid, is not too much more difficult to figure than an f/8 mirror. On the other hand, the convex hyperboloidal secondary is difficult not only to figure but also to test. The high center and turned-up edge is the opposite of what usually "just happens," and the small linear diameter of the secondary for a moderate-sized primary merely adds to the figuring problem. The test of the secondary requires either a flat of the same diameter as the primary or a short-focus sphere of equal diameter for the Hindle test. All this adds up to a project that few amateurs are willing to embark upon. For those whose determination transcends the difficulties, disappointment usually follows when the perfection of figure of the primary is not equaled in the secondary, for a Cassegrainian is never any better than the figure of its secondary. As a result, the Cassegrainian is maligned and has become something to be avoided.

Several years ago Kirkham and Dall investigated the possibility of leaving the secondary spherical and adjusting the correction of the primary to compensate. Since leaving the secondary spherical amounted to overcorrection, it followed that the primary would have to be undercorrected, which is an easier job than full correction. Parabolizing corrects longitudinal spherical aberration. If the secondary is left spherical it introduces a calculable amount of longitudinal spherical aberration into the system. This spherical aberration is negative, and the secondary would be said to be spherically overcorrected. The primary must therefore contain positive spherical aberration in an equal amount and be spherically undercorrected. All that remains is to determine the exact amount of undercorrection necessary.

The formulas for determining the longitudinal spherical aberration and the percentage of undercorrection are not complex, but they do contain sign conventions which must be strictly adhered to, and are therefore subject to error. They can be simplified into the single formula shown at right center in the drawing [above]. In this all quantities are considered to be positive and no sign convention errors are likely. N is the fraction of r²/R for any zone, R is the radius of curvature of the primary, R' is the radius of curvature of the secondary, and p and p' are the conjugate focal distances of the secondary, as shown in the upper part of the drawing.

To clarify the percentage calculation let us take as an example a 6-inch spherical-secondary Cassegrainian in which the primary has a focus of 24 inches, an amplifying ratio of 4, with the focus falling 10 inches behind the surface of the primary. The radius of curvature of the primary is 48 inches and, from the compound telescope formulas on page 216 of Amateur Telescope Making, the radius of curvature of the secondary is 18.13 inches, p is 6.8 inches and p' is 27.2 inches. Substituting these figures in the formulas we have the example worked out there.

Now that we are armed with the information on the percentage of correction necessary in the primary, the question arises: How do we produce a good convex sphere? The problem is relatively simple compared with the hyperbolic secondary of the conventional Cassegrainian, both in figuring and testing. The best means of testing a convex sphere is with a concave spherical master, or test plate. When only one convex surface is to be produced the test plate should be one that is quick and easy to make. The glass grinder on which the secondary mirror has been ground need be given only a quick shine, and a suitable test is at hand. Being concave, it can be tested directly by knife-edge and checked for radius with a steel tape. It need only be polished front and back sufficiently to be seen through for the interference tests. In fact, the shorter the polishing period on the concave sphere, the more likely that the curve will remain truly spherical, provided it received a good grind.

Since the grinder has been used for the test plate it will not be available for making the polisher. The best substitute is a concave metal tool, which can be turned to shape in the lathe [Amateur Telescope Making-Advanced, page 55]. A truly spherical and smooth surface is not necessary, since the surface will be covered with pitch in making the polisher. As an alternative a polisher back can be made by greasing the face of the fine-ground secondary mirror, circling it with gummed paper tape and filling this with plaster or dental stone. After the polisher is made and the pitch formed with the mirror, a few minutes of polishing will produce a shine sufficient to obtain the first interference test. This test may show several fringes of difference between the mirror and the test plate. If the fringes are circular and uniformly spaced, then the surfaces are spherical and differ only in radius o f curvature, and polishing can be continued. If more than six to ten fringes are apparent, fine grinding was not controlled closely enough to bring the two surfaces coincident and it had best be redone; otherwise prolonged polishing will be necessary to correct the difference.

The final figure of the secondary should be within one-eighth wavelength of truly spherical [Amateur Telescope Making, page 261], but this does not require the same appearance under test as two flats that differ by one-eighth wave length. In making a flat accurate to one-eighth wavelength there is one and.] only one surface which is flat, or plano, and until this particular surface is arrived at the flat is not accurate to the tolerance specified. In producing curved surfaces accurate to the same standard:

there is much greater leeway, since the radius of curvature is not critical and may vary by many wavelengths so long as the spherical surface of the final radius does not deviate from truly spherical by more than one-eighth wavelength. This allows a multiple choice of radii, whereas the flat allows a single choice. Therefore, the spherical surface under test may show as many as four or five fringes and still be classed as accurate to one-eighth wavelength, if no fringe is distorted from symmetrical form more than one fourth of a fringe. (One fringe equals one-half wavelength, one half fringe equals one-fourth wavelength, one fourth fringe equals one-eighth wavelength.)

Testing should be done only after both pieces are thoroughly clean and dusted free of particles which could scratch the surface or hold the pieces apart to cause extra fringes. In the final interference test the test plate should be gently rocked on the mirror so that the center of the fringe pattern moves to all zones of the mirror. In all positions of the circular fringe pattern the fringes should show the same circular form and spacing. Any deviation of one half a fringe is readily apparent if there are fewer than six circular fringes in the diameter of the mirror, but detecting a quarter-wave difference becomes difficult and impossible as the number of fringes increases. The fewer the fringes the more accurately the deviations can be estimated.

In order to observe the fringe pattern easily a moderately monochromatic light source is needed. Fringes appear quite clearly under a fluorescent light, and t hey stand out even more sharply under a sodium or mercury-vapor light. The light should be diffused, and the lamp is best fitted under the top of a black box with the front open so that the angle of the eye is kept small.

All that remains is the figuring of the convex secondary to a spherical surface. The same techniques are used for correcting convex surfaces as for concaves. Low centers or long over-all radius (concave to the test plate) calls for short strokes or inverted rose laps. High centers or short over-all radius (convex to the test plate) calls for long strokes or straight rose laps. During the polishing-out period appropriate steps can be taken to keep the number of fringes small, and it is quite possible to have a good spherical surface at the same time it is polished out. The secondary need never be tested in conjunction with the primary with which It is going to be used, and large flats or spheres are unnecessary, The primary should be figured and tested to the same degree of accuracy as if it were fully corrected. The allowable error in the accuracy of parabolizing should be adhered to, so that the primary will also meet the one-eighth wavelength criterion.

In calculating the zonal readings for the undercorrected primary it is best to solve r²/R for the various zones to be tested, subtract to obtain the difference between zones, and then apply the percentage figure, obtained from the formula, to each zonal difference amount. For example, let us say that we will test three zones on the six-inch primary for which we determined the percentage correction to be 66.8 per cent, rounded off to 67 per cent. (Three zones will suffice for our example, but more might be desirable for better control of the figure.) We will take zones .75-inch r, 1.75-inch r and 2.75-inch r. For these zones r²/R will be .012-inch, .064-inch and .158-inch. Subtracting, we get.O52-inch for the difference between zones 1 and 2, and .094-inch for the difference between zones 2 and 3. Taking 67 per cent of these figures, we get .035-inch and .063-inch, and these new differences are those to which the mirror is figured. When the zonal test of the mirror shows that it agrees within the greatest allowable deviation from a perfect figure, in this case 5.5 per cent [see "Accuracy in Parabolizing," Table I, Amateur Telescope Making, page 257], the mirror will be undercorrected by the proper amount, and it will perform as well with its secondary as a fully corrected primary will with a hyperboloidal secondary.

A complicated ray trace of the spherical secondary would undoubtedly show some higher-order differences of correction when compared with the hyperboloidal secondary system. Coma is probably increased slightly, but for all practical purposes it can be considered negligible. Moreover, most of us are not concerned with higher-order coma or the like. We want a convenient, high-powered telescope that doesn't take 10 years of experience to build. We want to split that double star the book says this diameter should split, we want to find the Great Wall on the Moon, Syrtis Major on Mars, Cassini's division in Saturn's ring, see a transit of a satellite of Jupiter, locate Mercury in the twilight, pick out the Ring- Nebula by knowing where to point in Lyra, and accomplish it all with a pair of mirrors that were fun to make in the first place. If, after our apprenticeship on the standard beginner's six-inch Newtonian, we are going to build only one telescope and then see the sights, we might as well make the one that has the highest power and is easiest to use. Even if we just like to make telescopes, the spherical-secondary Cassegrainian is an intriguing one to add to the list. It works fine. I know because I've got one.

Suppliers and Organizations

The Society for Amateur Scientists
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Phone: 1-401-823-7800

Internet: http://www.sas.org/

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