IN THE LAST stages of telescope mirror grinding the pencil mark test is sometimes resorted to in an endeavor to bring the mirror and its mating tool to full spherical contact, preparatory to polishing. This test is described in "Amateur Telescope Making," page 288, thus: "As finer sizes are used, make the simple pencil mark test for sphericity by carefully drying and cleaning tool and mirror, drawing pencil marks across each, working them dry - with a few very short strokes, and observing whether they are in contact by noting where the marks are rubbed thin or entirely off."

Your scribe used this test many times on telescope mirrors, had no troubles with it, and offered it in "A.T.M."- since when the mails have brought occasional wails about scratches and sticking mirrors. Ralph Dietz, of Pasadena, California, wrote that this test "is bad, a mistake, wrong, and you shouldn't do it." Before removing it from "A.TM." the opinions of others were solicited-still are. Dave Broadhead, Wellsville, N. Y., says: "My vote is to leave it in, as a help to beginners to understand their problem." Fred Ferson, Biloxi, Mississippi, votes: "I have used it often and have never traced a scratch to it." On the other hand, H. H. Selby, in "A.T.M.A.," 117, says: "The pencil test is not recommended, due to danger of scratch formation." Why this test should scratch for some, while for others it hasn't scratched yet, remains a puzzle. Some have testified that it pulled chunks right out of the glass. Can this be true?

To his comment, already quoted Dietz adds another which opens up a fresh aspect of the same subject. "Anyway," he writes, "two pieces of glass, such as a mirror and its tool, when ground together with abrasive between should not fit each other exactly. The bottom disk will have the radius of curvature of the top disk minus the thickness of the abrasive used. Therefore, if the pencil lines rub out, something is wrong, not right." This reasoning looks solid, but read on.

In Twyman, "Prism and Lens Making," 62, the same consideration is discussed. Two disks were ground with fine emery well rubbed down to 0.0002" thickness, then cleaned, dried, and put together. They were found to swing very noticeably around the middle. These disks had deep curves, f/3. Twyman points out that the same phenomenon would not be appreciable with shallower curves.

To demonstrate mathematically the truth of this last statement is now our mention. We choose a specific mirror, the common 6" f/8, its f.l. 48", its radius 96". How wide will the edge separation between this mirror and its tool theoretically be after they have been worked to a fit with No. 600 Carbo between them, washed up, dried, and put together? By the old standby, , this mirror's sagitta is 192/9, which figures out at 0.046875000". The average grain size of Carbo 600 being 0.00042" ("A.T.M.," 492), we subtract this amount from 96", leaving 95.99958 for the radius of the tool. Having a shorter radius, the tool will have the greater sagitta and so the cleaned up mirror will pivot at its center. (Theoretically, anyway.) We divide 9 again by twice the tool radius, or 191.99916" and get 0.046875205". Subtracting, we get the width of the gap between edges of mirror and tool and this is the amount, an insignificant one five-millionth of an inch, by which "something is wrong" if a mirror of common focal ratio and diameter fits the tool in the pencil test. Here it is:

In passing, note Ellison's cognate argument on page 369, "A.T.M.," where he refers to the loose fit between mirror and tool after a paper polisher 0.01" thick has been removed from the tool. Though the gap is here much larger than in the case worked out above, the misfit still comes out only 0.00001" for a 6" f/8 mirror, which is far from the "gross error" he terms it. It seems probable that Ellison made his statement only on a basis of intuition, as Dietz did, without calculating, and that is what your scribe did when inserting it. This, therefore, must come out of "A.T.M." Why has no one challenged it?

Incidentally, this department has heard extensively from a group of amateurs in Belgium who were making telescopes as a hobby right through the war and who have been using paper polishers, and they claim "at least as good" results with them as with pitch laps. Though most opticians look down the nose at the paper polisher, these Belgian amateurs' claim is so strong that a serious test is in order, not to prove them wrong or even to prove them right but to find the facts. Later we hope to find some workers to make these tests and to provide the correct polishing paper and technique. Will the reader volunteer as a guinea pig? Working instructions are available.

When the above calculations were shown to Dietz he said he was a little surprised at the smallness of the edge gap but still believed the pencil test bad. He also described an experiment which he and many others had often made. Grind two disks together, fine grind them, give them a quick shine polish and test them by interference. Both will be spherical, as shown also by the shine coming up evenly, but usually four or five fringes say 1/100,000"-from a fit. This is much more than theory calls for but that often happens in shop optics. The present discussion, until farther along, is quite frankly, unashamedly, unapologetically concerned with theory for the pure sake of theory, and this is where Dietz took off for an interesting flight. He pointed out that, strictly speaking, is the formula for the parabola not the sphere; though, as he added, it makes no practical difference in the present case of the pencil test. For the sphere we should, he added, use another formula which is

the appertaining figure being the one in "A.T.M.," 312. Perhaps, through using so often because its use is virtually always accurate enough to be practical, we sometimes forget this nice distinction.

Formulas for the sphere appear in numerous optical works and they appear to differ. Some examples:

which is more convenient to apply to a specific problem, even though it involves remembering how to do square root. Lazybones can, however, use tables or slipsticks, or resort to a calculating machine. This is what your scribe did since it had the advantage of giving correct answers.

It is satisfying to know that (parabola) and the above sphere formula are each complete, concealing no microscopic silent partners.

Where do such formulas as those in "A.T.M.," 257, and "A.T.M." 5, fit into this picture? They don't. These pertain to edge separations between parabolas and spheres that touch (osculate) as in "A.T.M.A.," 379, Figure 2. Here the formula is an expansion of the root in the sphere formula by Newton's binomial theorem, which gives Only the first two terms are significant in mirror making. This gives depth of glass removed in parabolizing by polishing the outer zones.

What also about things like, encountered in the literature of optics? These pertain to still another picture, that of longitudinal aberrations along the axis of mirrors, as in the Foucault test, and are a cat of another color; hence, pssst, cat, get out and don't confuse the present argument, which will stay on the paraboloid-sphere basis. (Anybody who wishes to stir up these particular cats may dig out Prof. Wadsworth's article in Popular Astronomy, Volume X (1902), 337-348, but, should then hunt up Prof. Hussey's sequel article on "The Longitudinal Aberration of a Parabolic Mirror," in the Publications of the Astronomical Society of the Pacific Volume XIV (1902), 179-188, because the latter pointed out basic errors in the former caused by misunderstandings concerning fixed and moving pinhole and knife-edge. Your scribe also has a file of TN exchanges dated 1933, on the longitudinal aberration cat which may be borrowed by any who suffer from the mathematical itch There also is an article by Charles G. Rupert, in Popular Astronomy, 1918 525-542, entitled "Mathematics of the Reflecting Telescope." These old articles are virtually unavailable to all except those having access to institutional library files and ought, perhaps, to be reprinted.)

Before going on, we have been told that it is bad mathematics to call a paraboloid a curve, it being a surface, but a parabola is a curve. Pedantry?

If we now apply the sphere formula the theoretically correct one for an unparabolized mirror, to our same 6" f/8 mirror we get-a lot of headaches from doing square root. Your scribe recalled the method but wasn't able to multiply and divide twice alike in trying to spin out the square root of 9207 (mirror sagitta by sphere formula) to a lot of decimal places and then, to get the tool's sagitta, square 95.99958, subtract 9, and take out the square root to an equal lot of places. So The Monroe Calculating Machine Company, Orange, N. J., was asked to do it on one of its machines. These robots don't sweat for two hours over such a problem but sass you right back with the correct answer, and this one came by mail strung out to 17 digits for good measure; these machines being so eager you can't stop them. Of the 17 enough are offered below to show the major deviation between results of the two formulas in the present instance. By the sphere formula

Yet let's not celebrate too hard, for that submicroscopic gap grows mightily with shortened focal ratio and soon enters the realm of the entirely practical. C. R. Hartshorn, Los Angeles, to whom the above data were shown, has explored some of the effects of varied focal ratio on the use of the parabola formula instead of the sphere formula and worked out (didn't lean on any lazy man's crutch) the following table for 6" mirrors of eight focal ratios:

Early in the above article your scribe said the pencil mark test hadn't scratched yet for him. The article was written, set aside for a long time and almost forgotten. A pair of 10" mirrors were made, and on one of them the pencil test pulled little hunks right out of the glass. Exuent test, with slow, sad music, Dietz and Selby grinning.

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