I have not had occasion to use this instrument in my own work, as a more simple, delicate, and efficient method was at my command, but for one measurement of convex surfaces I know of nothing that can take its place. I will briefly describe the method of using it.
The usual form of the instrument is shown in Fig. 4; a is a steel screw working in the nut of the stout tripod frame, b; c c c are three legs with carefully prepared points; d is a divided standard to read the whole number of revolutions of the screw, a, the edge of which also serves the purpose of a pointer to read off the division on the top of the milled head, e. Still further refinement may be had by placing a vernier here. To measure a plane or curved surface with this instrument, a perfect plane or perfect spherical surface of known radius must be used to determine the zero point of the division. Taking for granted that we have this standard plate, the spherometer is placed upon it, and the readings of the divided head and indicator, d, noted when the point of the screw, a, just touches the surface, f. Herein, however, lies the great difficulty in using this instrument, i.e., to know the exact instant of contact of the point of screw, a, on the surface, f. Many devices have been added to the spherometer to make it as sensitive as possible, such as the contact level, the electric contact, and the compound lever contact.
The latter is probably the best, and is made essentially as in Fig. 5.
I am indebted for this plan to Dr. Alfred Mayer. As in the previous figure, a is the screw; this screw is bored out, and a central steel pin turned to fit resting on a shoulder at c. The end of d projects below the screw, a, and the end, e, projects above the milled head, and the knife edge or pivot point rests against the lever, f, which in turn rests against the long lever, g, the point, h, of which moves along the division at j. It is evident that if the point of the pin just touches the plate, no movement of the index lever, g, will be seen; but if any pressure be applied, the lever will move through a multiplied arc, owing to the short fulcri of the two levers. Notwithstanding all these precautions, we must also take into account the flexure of the material, the elasticity of the points of contact, and other idiosyncrasies, and you can readily see that practice alone in an instrument so delicate will bring about the very best results. Dr. Alfred Mayer's method of getting over the great difficulty of knowing when all four points are in contact is quite simple. The standard plate is set on the box, g, Fig. 4, which acts as a resonater. The screw, a, is brought down until it touches the plate.
When the pressure of the screw is enough to lift off either or all of the legs, and the plate is gently tapped with the finger, a rattle is heard, which is the tell-tale of imperfect contact of all the points. The screw is now reversed gently and slowly until the moment the rattle ceases, and then the reading is taken. Here the sense of hearing is brought into play. This is also the case when the electric contact is used. This is so arranged that the instant of touching of the point of screw, a, completes the electric circuit, in which an electromagnet of short thick wire is placed. At the moment of contact, or perhaps a little before contact, the bell rings, and the turning of the screw must be instantly stopped. Here are several elements that must be remembered. First, it takes time to set the bell ringing, time for the sound to pass to the ear, time for the sensation to be carried to the brain, time for the brain to send word to the hand to cease turning the screw, and, if you please, it takes time for the hand to stop.
You may say, of what use are such refinements? I may reply, what use is there in trying to do anything the very best it can be done? If our investigation of nature's profound mysteries can be partially solved with good instrumental means, what is the result if we have better ones placed in our hands, and what, we ask, if the best are given to the physicist? We have only to compare the telescope of Galileo, the prism of Newton, the pile of Volta, and what was done with them, to the marvelous work of the telescope, spectroscope, and dynamo of to-day. But I must proceed. It will be recognized that in working with the spherometer, only the points in actual contact can be measured at one time, for you may see by Fig. 6 that the four points, a a a a, may all be normal to a true plane, and yet errors of depression, as at e, or elevation, as at b, exist between them, so that the instrument must be used over every available part of the surface if it is to be tested rigorously. As to how exact this method is I cannot say from actual experience, as in my work I have had recourse to other methods that I shall describe.
I have already quoted you the words of Prof. Harkness. Dr. Hastings, whose practical as well as theoretical knowledge is of the most critical character, tells me that he considers it quite easy to measure to 1/80000 of an inch with the ordinary form of instrument. Here is a very fine spherometer that Dr. Hastings works with from time to time, and which he calls his standard spherometer. It is delicately made, its screw being 50 to the inch, or more exactly 0.01998 inch, or within 2/100000 of being 1/50 of an inch pitch. The principal screw has a point which is itself an independent screw, that was put in to investigate the errors of the main screw, but it was found that the error of this screw was not as much as the 0.00001 of an inch. The head is divided into two hundred parts, and by estimation can be read to 1/100000 of an inch. Its constants are known, and it may be understood that it would not do to handle it very roughly. I could dwell here longer on this fascinating subject, but must haste.
I may add that if this spherometer is placed on a plate of glass and exact contact obtained, and then removed, and the hand held over the plate without touching it, the difference in the temperature of the glass and that of the hand would be sufficient to distort the surface enough to be readily recognized by the spherometer when replaced. Any one desiring to investigate this subject further will find it fully discussed in that splendid series of papers by Dr. Alfred Mayer on the minute measurements of modern science published in SCIENTIFIC AMERICAN SUPPLEMENTS, to which I was indebted years ago for most valuable information, as well as to most encouraging words from Prof. Thurston, whom you all so well and favorably know. I now invite your attention to the method for testing the flat surfaces on which Prof. Rowland rules the beautiful diffraction gratings now so well known over the scientific world, as also other plane surfaces for heliostats, etc., etc. I am now approaching the border land of what may be called the abstruse in science, in which I humbly acknowledge it would take a vast volume to contain all I don't know; yet I hope to make plain to you this most beautiful and accurate method, and for fear I may forget to give due credit, I will say that I am indebted to Dr. Hastings for it, with whom it was an original discovery, though he told me he afterward found it had been in use by Steinheil, the celebrated optician of Munich. The principle was discovered by the immortal Newton, and it shows how much can be made of the ordinary phenomena seen in our every-day life when placed in the hands of the investigator.
We have all seen the beautiful play of colors on the soap bubble, or when the drop of oil spreads over the surface of the water. Place a lens of long curvature on a piece of plane polished glass, and, looking at it obliquely, a black central spot is seen with rings of various width and color surrounding it. If the lens is a true curve, and the glass beneath it a true plane, these rings of color will be perfectly concentric and arranged in regular decreasing intervals. This apparatus is known as Newton's color glass, because he not only measured the phenomena, but established the laws of the appearances presented. I will now endeavor to explain the general principle by which this phenomenon is utilized in the testing of plane surfaces. Suppose that we place on the lower plate, lenses of constantly increasing curvature until that curvature becomes nil, or in other words a true plane. The rings of color will constantly increase in width as the curvature of the lens increases, until at last one color alone is seen over the whole surface, provided, however, the same angle of observation be maintained, and provided further that the film of air between the glasses is of absolutely the same relative thickness throughout.
I say the film of air, for I presume that it would be utterly impossible to exclude particles of dust so that absolute contact could take place. Early physicists maintained that absolute molecular contact was impossible, and that the central separation of the glasses in Newton's experiment was 1/250,000 of an inch, but Sir Wm. Thomson has shown that the separation is caused by shreds or particles of dust. However, if this separation is equal throughout, we have the phenomena as described; but if the dust particles are thicker under one side than the other, our phenomena will change to broad parallel bands as in Fig. 8, the broader the bands the nearer the absolute parallelism of the plates. In Fig. 7 let a and b represent the two plates we are testing. Rays of white light, c, falling upon the upper surface of plate a, are partially reflected off in the direction of rays d, but as these rays do not concern us now, I have not sketched them. Part of the light passes on through the upper plate, where it is bent out of its course somewhat, and, falling upon the lower surface of the upper plate, some of this light is again reflected toward the eye at d.
As some of the light passes through the upper plate, and, passing through the film of air between the plates, falling on the upper surface of the lower one, this in turn is reflected; but as the light that falls on this surface has had to traverse the film of air twice, it is retarded by a certain number of half or whole wave-lengths, and the beautiful phenomena of interference take place, some of the colors of white light being obliterated, while others come to the eye. When the position of the eye changes, the color is seen to change. I have not time to dwell further on this part of my subject, which is discussed in most advanced works on physics, and especially well described in Dr. Eugene Lommel's work on "The Nature of Light." I remarked that if the two surfaces were perfectly plane, there would be one color seen, or else colors of the first or second order would arrange themselves in broad parallel bands, but this would also take place in plates of slight curvature, for the requirement is, as I said, a film of air of equal thickness throughout. You can see at once that this condition could be obtained in a perfect convex surface fitting a perfect concave of the same radius. Fortunately we have a check to guard against this error.
To produce a perfect plane, three surfaces must be worked together, unless we have a true plane to commence with; but to make this true plane by this method we must work three together, and if each one comes up to the demands of this most rigorous test, we may rest assured that we have attained a degree of accuracy almost beyond human conception. Let me illustrate. Suppose we have plates 1, 2, and 3, Fig. 11. Suppose 1 and 2 to be accurately convex and 3 accurately concave, of the same radius. Now it is evident that 3 will exactly fit 1 and 2, and that 1 and 2 will separately fit No. 3, but when 1 and 2 are placed together, they will only touch in the center, and there is no possible way to make three plates coincide when they are alternately tested upon one another than to make perfect planes out of them. As it is difficult to see the colors well on metal surfaces, a one-colored light is used, such as the sodium flame, which gives to the eye in our test, dark and bright bands instead of colored ones. When these plates are worked and tested upon one another until they all present the same appearance, one may be reserved for a test plate for future use.
Here is a small test plate made by the celebrated Steinheil, and here two made by myself, and I may be pardoned in saying that I was much gratified to find the coincidence so nearly perfect that the limiting error is much less than 0.00001 of an inch. My assistant, with but a few months' experience, has made quite as accurate plates. It is necessary of course to have a glass plate to test the metal plates, as the upper plate must be transparent. So far we have been dealing with perfect surfaces. Let us now see what shall occur in surfaces that are not plane. Suppose we now have our perfect test plate, and it is laid on a plate that has a compound error, say depressed at center and edge and high between these points. If this error is regular, the central bands arrange themselves as in Fig. 9. You may now ask, how are we to know what sort of surface we have? A ready solution is at hand. The bands always travel in the direction of the thickest film of air, hence on lowering the eye, if the convex edge of the bands travel in the direction of the arrow, we are absolutely certain that that part of the surface being tested is convex, while if, as in the central part of the bands, the concave edges advance, we know that part is hollow or too low.
Furthermore, any small error will be rigorously detected, with astonishing clearness, and one of the grandest qualities of this test is the absence of "personal equation;" for, given a perfect test plate, it won't lie, neither will it exaggerate. I say, won't lie, but I must guard this by saying that the plates must coincide absolutely in temperature, and the touch of the finger, the heat of the hand, or any disturbance whatever will vitiate the results of this lovely process; but more of that at a future time. If our surface is plane to within a short distance of the edge, and is there overcorrected, or convex, the test shows it, as in Fig. 10. If the whole surface is regularly convex, then concentric rings of a breadth determined by the approach to a perfect plane are seen. If concave, a similar phenomenon is exhibited, except in the case of the convex, the broader rings are near the center, while in the concave they are nearer the edge. In lowering the eye while observing the plates, the rings of the convex plate will advance outward, those of the concave inward. It may be asked by the mechanician, Can this method be used for testing our surface plates? I answer that I have found the scraped surface of iron bright enough to test by sodium light.
My assistant in the machine work scraped three 8 inch plates that were tested by this method and found to be very excellent, though it must be evident that a single cut of the scraper would change the spot over which it passed so much as to entirely change the appearance there, but I found I could use the test to get the general outline of the surface under process of correction. These iron plates, I would say, are simply used for preliminary formation of polishers. I may have something to say on the question of surface plates in the future, as I have made some interesting studies on the subject. I must now bring this paper to a close, although I had intended including some interesting studies of curved surfaces. There is, however, matter enough in that subject of itself, especially when we connect it with the idiosyncrasies of the material we have to deal with, a vital part of the subject that I have not touched upon in the present paper. You may now inquire, How critical is this "color test"? To answer this I fear I shall trench upon forbidden grounds, but I call to my help the words of one of our best American physicists, and I quote from a letter in which he says by combined calculation and experiment I have found the limiting error for white light to be 1/50000000 of an inch, and for Na or sodium light about fifty times greater, or less than 1/800000 of an inch.
Dr. Alfred Mayer estimated and demonstrated by actual experiment that the smallest black spot on a white ground visible to the naked eye is about 1/800 of an inch at the distance of normal vision, namely, 10 inches, and that a line, which of course has the element of extension, 1/5000 of an inch in thickness could be seen. In our delicate "color test" we may decrease the diameter of our black spot a thousand times and still its perception is possible by the aid of our monochromatic light, and we may diminish our line ten thousand times, yet find it just perceivable on the border land of our test by white light. Do not presume I am so foolish as to even think that the human hand, directed by the human brain, can ever work the material at his command to such a high standard of exactness. No; from the very nature of the material we have to work with, we are forbidden even to hope for such an achievement; and could it be possible that, through some stroke of good fortune, we could attain this high ideal, it would be but for a moment, as from the very nature of our environment it would be but an ignis fatuus.
There is, however, to the earnest mind a delight in having a high model of excellence, for as our model is so will our work approximate; and although we may go on approximating our ideal forever, we can never hope to reach that which has been set for us by the great Master Workman.A paper read before the Engineers' Society of Western Pennsylvania, Dec. 10, 1884.