Title:   Blade lapping: parabolizing at f1.
Abstract:   Fine grinding a parabola from a sphere, using a sheet metal blade 
cut to the profile of a diameter of the mirror.  Historical review of 
mechanical means for aspherizing surfaces.

Aspherizing a mirror is a complex process that has brought out the ingenuity in 
many a mirror maker, and there are a large variety of processes & machines that 
have been developed to meet this need.  This paper will review some of the 
techniques found in the references, but will begin with a procedure that I find 
particularly interesting.  Blade lapping is not a process that an amateur 
telescope maker is likely to want to use, so this is not a practical lecture, 
but is instead one that illuminates a technique that epitomizes the ingenuity 
and skill of the mirror maker.

Imagine slicing a basketball in half and cutting a cardboard template to fit 
inside, across a diameter, [viewed from the side & from above].  If you allow 
the template freedom of motion in the vertical direction, and try to slide the 
template across the open diameter of the ball, as the diameter becomes a chord, 
the template will ride up out of the sphere and contact the ball on the rim, at 
two points only.  The same geometry applies to a spherical mirror.  A chord 
across a spherical surface has a shorter radius of curvature than does the 
diameter of the sphere.  However, a parabola has a vertex with a radius that is 
shorter than the 'radius' of the wings of a parabola.  If you rotate that 
parabola to become a paraboloid, the same geometry applies.

To generalize: A three dimensional parabola is a surface of revolution (the 
surface defined when a parabola is rotated about its axis,), more correctly a 
paraboloid, and it has many interesting qualities.  Focusing parallel light to 
a point is its most remarkable quality.  Blade lapping reveals another very 
interesting quality of a paraboloid: chords will overlay the diameter.  [illus]  
A parabola is in fact the only surface of revolution that has this property.  A 
sphere has a diameter that is not similar in this way to its chord. [illus]
More abstractly, and expressed three different ways:  [demonstrate]
1. Given a paraboloid of revolution, if the surface is cut by planes parallel 
to the axis, the intersections of these planes with the paraboloid will be 
identical parabolas.
2. Any intersection of a parabolic mirror with a plane that is parallel to the 
mirror's axis is a parabola of the same curve.
3. When a straight blade is held parallel to the axis of a surface of 
revolution, is then displaced from the center while remaining parallel, and 
remains in contact with all parts of the curve in its traverse; the curve of 
the surface (and of the blade) can only be a parabola.
This means that the parabola can be fabricated by preferential wear, which will 
grind the blade and surface to matching parabolas, which are the only surfaces 
that will contact at all points under these conditions.  Parabolization 
proceeds by flattening the edge, not by the typical deepening of the center.

Blade lapping is a technique that has been used to aspherize very short focus 
reflectors, f0.4 in one example, and is appropriate when a 'medium precision' 
surface is needed.  Some uses would include illuminators and infra red optics.  
Concave or convex surfaces can be achieved.  The parabolizing process is 
described below, followed by notes on achieving other curves.
After rough grinding, one diameter of the spherical surface of a mirror is 
taken as a concave template, and one edge of a piece of sheet brass is cut to a 
convex profile to match it.  Thicker sheet reduces chatter, lasts longer, and 
speeds grinding; but reduces the contact area at the edge of the mirror, 
because the blade is contacting the surface at its corner only [illus].  5mm 
sheet, feathered to a knife edge, is used in one published account by John 
Larmer of Goddard Space Flight Center, Maryland, in the May 1966 Applied 
Optics.  Larmer fabricated a 30 cm diameter mirror of 23 cm focal length.
[illus 2] The mirror is first rough ground to a sphere, with a radius of 
curvature equal to the vertex radius of the parabola.  It is then placed on a 
horizontal grinding table, and above it is a jig to precisely & firmly hold the 
brass in a vertical position.  Larmer used a universal shaper machine..  The 
blade must be maintained parallel to the optical axis to very high tolerances.  
The jig is mounted so it can slide in one dimension on a rigid arm, parallel to 
the mirror disc.  As the mirror rotates, the blade is slid back and forth, and 
it rises at the end of each stroke and falls to mid stroke.  At the midpoint of 
each stroke, the brass contacts the sphere at all points along the blade.
As the blade slides off the center point, it becomes a chord instead of a 
diameter of the disc, and, as noted, the radius of the sphere across the chord 
is shorter than the radius across the diameter.  Therefore, the brass will 
contact the glass only at the edges of the glass.  Abrasive slurry is applied 
during the process.  The glass is abraded at the edge, and the brass is abraded 
across its length, as the two contact points travel along the edge.  The 
spherical glass surface is slowly parabolized by taking the edge down, 
increasing the focal length.  Lapping abrasive is used, and grit sizes would 
depend on how much glass needs to be removed.
After some period of lapping, the blade and glass grind each other to a 
matching parabola.
It is not necessary to take the blade all the way to the very edge of the 
sphere, and the effect of different stroke lengths is complex.  Larmer found 
that he got zones unless he made the stroke length a full radius of the mirror 
on one side and seven tenths radius on the other side of the stroke.
To illustrate the variables involved:  Larmer's mirror was a 30 cm parabola of 
23 cm focal length.  He rotated his mirror at 20 rpm and used 10 strokes per 
minute.  4.55 kg. was the measured weight on the blade.
To illustrate the tolerances involved: Larmer maintained centering of his 30 cm 
mirror while it was rotating & being lapped, to 2.5 microns, --centering to 
millionths of a meter.  Larmer's 30 cm, f 0.7 mirror had a disk of confusion of 
25 microns in diameter (this is the only published test result).

R. E. Lewis of Illinois Institute of Technology in Chicago published another 
variation on blade lapping in the Oct. 1952 Review of Scientific Instruments 
(vol. 23, #10).  This process was designed to allow for commercial production.  
[illus 3]  Lewis used two parallelograms in linkage, fixed at two leveling 
screws.  Problems included varying loads on the bearings, and irregularities 
when the stroke was reversed, and therefore two arms of the second 
parallelogram were re-made of flexible bar to provide smooth travel with no 
chatter.  The various operating speeds were found to be not critical.  The 
ratio of strokes to rotation was found to control focal length.  This mechanism 
was more lightly built than most, and torque from the rotating mirror was 
noticed in the action of the blade.  These problems could be observed in the 
action of the blade, since it acted as a wiper, cleaning the mirror surface, 
and the thickness of the slurry would vary with irregularities.  Lewis implies 
he was working on a 6 inch mirror of 3 inch focal length.  He used emery of 
mesh 2600 or 3200, and in 20 to 60 hours his machine parabolized the spherical 
blank, using one 3/32 inch thick blade, and no manual labor.  60 hours does not 
reflect a large amount of glass removed, but rather the small area of contact.  
Glycerin in water was used to slow evaporation.  Polishing after 3200 grit is 
minimal, and for polishing Lewis used blades made of felt impregnated with 
pitch, wrapped around a metal blade.  These felt wrappings would creep, and so 
felt pads that had been soaked in pitch were also used.  One unsolved problem 
concerned wear at the edges of the blade and the glass.  [illus 4]  Grinding 
wears the blade more towards the center than the edges, and the unground edge 
of the blade will skate on the rim of the mirror, causing misalignment and 
other problems.  This outer edge of the blade was manually profiled to remove 
the lip.  The rim of the blank requires attention as well, it must be centered 
and carefully ground to profile.  Spherical mirrors can be blocked by pouring 
pitch around them, but a paraboloid has an axis, not a simple center of 
curvature.  Lewis found that softening the pitch during operation was 
sufficient for self centering.  He also notes the significant influence of grit 
size.  The blade rides on a layer of the slurry, which in a parabola causes 
noticeable error that is reduced with finer grits.  In addition, grits that are 
scratch free in standard fabrication are not so in blade work.
Lewis notes that the slow procedure can be accelerated by using multiple 
blades, thereby permitting 'industrial production'.
Lewis polished with a blade of felt impregnated with pitch; also by pads of 
rubber and pitch.  The 6 inch mirror of 3 inches focal length, made on the 
machine without manual touch-up, was tested as follows: an image of a point 
source at 90 feet was imaged with 80 percent of the light within .004 inch, and 
all light within .0075 inch.

Polishing is performed by other opticians with rouge and a softer blade 
material (considered difficult), a flexible lap, or a membrane lap under 
hydrostatic pressure.  There are limits to the precision attained in this 
process.  The elasticity of the blade, movement between parts of the sliding 
arm, and wear to parts, mean that precision to fractions of a wavelength are 
not usually possible.  Zones are commonly introduced during the process.  
Therefore, much of the technique is directed towards smoothing irregularities 
and zones.

Convex parabolas can also be produced by blade lapping.  Conicoids such as 
hyperbolas and ellipses can be generated by a variation of the process.  [illus 
5]  Instead of sliding in a horizontal displacement, the blade is mounted on a 
fixed horizontal rod so that it can tip back and forth. This rod is placed at 
the focus of the mirror in Jeffree's machine.  The blade is oscillated in a 
small range of motion at about 1 cycle per second, and swung over the entire 
conicoid in a much slower oscillation.  The axis of rotation of the blade (the 
rod) can be placed at any distance from the mirror to vary eccentricity.  If 
the axis is closer to the mirror, more material is ground from the center of 
the surface.  A paraboloid can be generated in this manner if the rod is placed 
at infinity, or in other words mounted on a linkage that keeps the blade 
parallel with the optical axis.
================================
Some historical highlights of the quest for aspheres:
Johannes Kepler described spherical aberration in his 'Dioptrice' of 1611, and 
to correct it, he proposed a lens with a hyperbolic surface.

Descartes, circa 1638, [illus 6 > 9A, 7] designed a grinding wheel working on 
edge that is dressed by an oscillating point.

Christopher Wren is best known as an architect but also designed a very 
ingenious machine for grinding convex hyperbolic surfaces, publishing his work 
in 1669.  [illus 8] Two cylindroidal grinding tools are mounted at varying 
angles of incidence, and by mutual abrasion produce a hyperboloid that is 
ground into the surface mounted below.  Translated from the Latin, the machine 
worked by 'rotation and mutual attrition'.

In 1668, Francis Smethwick published his article 'On Grinding Optic and Burning 
Glasses of Non-Spherical Figures'.  Smethwick described a telescope he made, of 
about four foot focus with a singlet objective and three ocular lenses that 
were aspheric plano convex lenses.  The telescope was reported to be of 
superior quality.

[illus 8A] Isaac Newton was a lens grinder, and circa 1666, he sketched a 
machine for figuring hyperbolic lenses (Bedini notes? 'Opticks', 1718 edition, 
page 83.)  Newton writes, "Ye glass 'a' may be ground Hypebolical by ye line 
'cb', if it turns on ye Mandrill 'e' whilst 'cb' turns on ye axis 'rd' being 
inclined to it as was shown before." (Bedini -?)
Newton the mirror maker understood the need to parabolize the primary, but in 
his telescopes he used a spherical mirror because the fabrication & testing of 
a parabola was beyond the techniques of his time.  The surviving example of 
Newton's work, which is credited as the first reflecting telescope, has a two 
inch primary of 6.4 inches focal length, or f3.2.  Performance of a spherical 
f3.2 mirror would be poor, and the telescope was used with a field stop at the 
eyepiece.  Newton probably made three telescopes, and regarded them as 
prototypes for a larger instrument.  He contracted an optician to fabricate a 
glass mirror with spherical concentric front & rear surfaces, which was coated 
with mercury on the convex side, but the mirror was a failure due to 
inhomogenous glass.  Robert Hooke also experimented with glass mirrors, to no 
success -- the idea would have to wait 200 years.

The success of these early efforts at mirror making is an open question.  
Different sources credit the first successful parabola to John Hadley in 1721, 
James Bradley (Jan. 1721, Royal Society, 6 foot parabolic reflector), or James 
Short circa 1732.

A very useful explanation of grinding machines from 1850 is found in Charles 
Holtzapffel, 'Turning and Mechanical Manipulation', volume 3, 'Abrasive and 
Miscellaneous Processes'.  Most of this 5 volume set is devoted to 
Holtzapffel's famous ornamental turning lathe, but workmanship of rotating 
objects of all sorts is covered.  I was intrigued to find on page 1278 a 
reference to an 'elliptical polisher'used for parabolizing by Reverend John 
Edwards, from his 1812 instruction book on mirror making.  However, this is 
simply a pitch tool with an elliptical outline [illus 8C], instead of a 
circular profile, which would reduce contact at the outer zones.  Holtzapffel 
goes on to describe the Earl of Rosse's machine, with more detail found in an 
anonymous booklet, 'The Monster Telescopes', with a frontispiece of 'The 
Monster'.  [illus 8B]  Rosse's three foot diameter speculum was fabricated as 
follows: [illus 8C]A steam engine powered this contraption through shaft A, 
driving two adjustable eccentrics B and G linked by a bar.  The speculum HI is 
immersed in water in a trough EF.  The 250 pound polishing tool KL is hooked to 
a counterpoise so that weight can be adjusted to about 10 pounds pressure.  The 
polisher has two adjustable motions, and for every revolution of the mirror, it 
makes 24 longitudinal strokes of up to 18 inches; and also 1 72/100 transverse 
strokes of about 10 inches.  Parabolization was effected by varying the stroke 
of the eccentric wheel G.
Rosse's large 6 foot speculum was fabricated on a similar machine, chain 
driven, substituting an elliptic wheel for the circular wheel G, changing some 
of the linkages from rigid to jointed rods, and of sturdier construction.
Rosse authored his own description of the process for the Philosophical 
Transactions [illus 8D], and if you look at the library stamp of this copy, you 
will see it belongs to the Steward Observatory of the University of Arizona, 
where to this day it is regularly consulted as a handbook for the figuring of 
8.4 meter mirrors.  After all, why reinvent the wheel? <  > In this article 
Rosse notes that Mr Whitworth, of some renown, and Mr. A. Ross, a legendary 
optician, believe that, since there is no way to ensure an even distribution of 
grinding slurry, the best precision of surface is attained by scraping rather 
than grinding, a process that is well known to any machinists in the audience.  
Of course, the scraping method of working an optical surface went out with 
speculum mirrors.

A visitor to Lord Rosse's workshop was William Lassell, a brewer and dedicated 
amateur astronomer who [illus 8E] discovered Neptune's satellite Triton, and 
Saturn's eighth moon Hyperion, using a 24 inch diameter Newtonian equatorial of 
his own design and construction.  He also built this 48 inch diameter Newtonian 
on a fork equatorial.  Lassell built a machine [illus 8F] similar to Rosse's 
but found it unsatisfactory.  To grind the 24 inch specula, a machine was 
designed by Lassell and built by James Nasmyth, who also produced these 
illustrations. [illus 8G]  It is a sign of their times that when this drawing 
was published by the Royal Astronomical Society, a note was added that the 
letters [illus 8H] identifying various parts were removed by the editor so as 
to not detract from the beauty of the drawing.
The power train drives both the mirror and the tool, via a gear train that 
produces both epicycloidal and hypocycloidal curves.  The machine parabolizes 
by flattening the edge, rather than by deepening the center.  Lassell was 
immensely pleased at eliminating manual labor by the use of steam, and wrote, 
"the curve can be changed almost at pleasure from the spherical side to the 
hyperbolic side of the parabola, and vice versa....commensurate with the 
adjustments of the cranks.  In fact, one of the most anxious and laborious 
operations is by this machine converted into an intensely interesting 
amusement." (p14, Memoirs RAS)  

The Reverend William Hodgson adapted Lord Rosse's design for the amateur of 
modest means, using a foot powered treadle lathe.  [illus 8I]  The specula is 
rotated at a slow speed, and a second power belt imparts an eccentric motion to 
the tool.  The eccentric can be shifted on its axis and relative speeds of the 
two tools adjusted by moving the power belts.  3.5 inch specula were produced 
on this modified lathe, described in Holtzapffel.

The 1800s were a century of mechanical ingenuity and experimentation.  Joseph 
Fraunhofer is credited (by Martin) with a suggestion for generation of 
aspheres.  A small lap is attached to a radial arm that turns on a symmetrical, 
aspheric pivot.  Other linkage mechanisms reflect the 19th century fascination 
with the mechanics & mathematics of linkages, the more simple examples being 
trammels or pantographs.  Sliding carriages and cams were used to fabricate 
'polar reciprocal' curves; by which is meant, a circular cam generates a conic 
'polar reciprocal'.  Also used were crankshafts, which generate linear motion 
at one end, circular motion at the other, and various points on the crank 
generate curves of the fourth degree that approximate ellipses.  Gullstrand 
published a study in 1919 in Swedish that analyzed convex aspheres in terms of 
the envelope of successive positions on a curve.  L.C. Martin in 'Technical 
Optics', notes that the design of such systems is very complex & has not been a 
productive area.
[illus 9]  In the mid 1900s, Grubb Parsons built this simple device to polish 
aspheres.

An intriguing mechanism is described by Heynacher as the evolute process. 
[illus 9A]  The grinding tool is carried on a steel band that wraps around a 
form that is the evolute of the generated surface.  Conic sections as well as 
other surfaces of revolution can be manufactured in this manner.  An evolute is 
formed by tracing the centers of curvature of sections of a plane curve, the 
normals (perpendiculars) to the curve are tangents to the evolute.  Here, you 
start with the evolute & place a grinding tool at the endpoint of the normal.  
Grinding is performed on the rotating surface of revolution.  Very fine grits 
can lead directly to polishing, and the technique can produce zone-free, high 
quality surfaces.

Hashimoto at Nikon Japan developed a cam guided machine using a grinding wheel 
that is continually dressed by a diamond form.)  Heel describes a Zeiss machine 
that uses a soft brass wheel, loose grit, that is continually dressed while 
working (?).  This is very similar to the Descartes machine.

[illus 9B]  In 1962, the English translation of K.G. Kumanin, 'Generation of 
Optical Surfaces' was published, with a wealth of information on fabricating 
aspheres.  These illustrations show some of the schematic drawings in that 
volume, and I stress 'schematic'.  These figures are mostly for toroidal 
surfaces.  Kumanin devotes a few pages to what he calls, 'method of linear 
contact', [illus 9C] including blade lapping and a method for generating 
hyperboloids whose utility is unclear to me......but is reminiscent of Isaac 
Newton's drawings.  

[illus 10]  Don Loomis of the University of Arizona's Optical Sciences Center 
developed a polishing machine in his home workshop, described in their undated 
newsletter, circa 1970.  This was to be used on a Schmidt camera for ultra 
violet light, which mandates another level of precision.  [illus 11]  The tool 
arm moves in two dimensions, both of which are on variable eccentric levers and 
have variable speed controls.  The tool can thus be moved in a variety of 
Lissajous patterns.  No follow up report was issued in the newsletter.

[illus 11A]  Also circa 1970 is this design by the ever-creative Lawrence 
Mertz.  This is the secondary of his proposed 15 meter telescope.

Takemaro Sakurai in Japan devised this linkage [illus 12], a form of 
pantograph, to generate paraboloids.  X is the mirror's optical axis, p is a 
parabola traced by one link with focus f which is a fixed point in the linkage.  
The directrix of the parabola is d, on which a sliding link D moves, 
maintaining the perpendicular link DP.  A grinding tool on the tip of PD cuts a 
paraboloid identical to the curve shown.  The focal length is changed by moving 
F along the optical axis.
Sakurai also fabricated a cam guided grinding tool, as illustrated. [illus 13]  
The grinding tool itself is an iron rod of 1 cm diameter, using emery abrasive.  
This machine was used to make a reflector for a 'sky projector' (ala Batman, 
projecting onto clouds, but for a candy company in Japan,) 75cm in diameter and 
36 cm in focal length.
[illus 14]  Not content to leave well enough alone, Sakurai improved the 
machine to fabricate an ellipsoidal mirror for an infrared spectrophotometer, 
which focused light from a slit onto a detector.  This grinding tool is a 
diamond point a few millimeters in diameter.  These cams are ground to 
exaggerated form and the displacement of the tool is reduced by a lever 
mechanism.
A final version of this mechanism [illus 15] uses a revolving diamond tool, and 
the cam is faced on the circumference with two rows of bolts that can be 
adjusted to trim the shape of the cam.  A series of levers reduces the 
displacement of the tool to 1/100 of the dimension of the eccentricity of the 
cam.  This machine was used to fabricate a paraboloid 30 cm in diameter and 28 
cm in focal length.  From spherical, only 20 minutes of grinding resulted in 
the needed paraboloid, accurate to +- 1 micron, and polished in only 40 
minutes.  The mirror was then cut into three segments, each of which had a 
circle of confusion under 0.02 mm in diameter.  These were used to collimate an 
infrared spectroscope.

[illus 16]  Alvin Sapp of NOAO in Boulder devised this unusual mechanism to 
generate aspheres.  If the bar was rigid and hinged, the surface generated 
would be a sphere.  A fixed, flexible bar generates an asphere.  A bar of 
rectangular profile causes most of the flexing to be done at the base, but if 
it is tapered, the flexing is distributed along the length, and the asphere 
will be generated with a deeper vertex.  A bar that was tapered in a 3 / 2 
ratio generated a curve that approximated a parabola.

[illus 16]  A final technique for aspherizing can generically be called 'bend 
and polish'.  Barr & colleagues at NOAO used this mechanism. [illus]  The 10 
meter Keck uses mirror segments fabricated in a similar manner.  Even lenses 
have been fabricated while deformed in a jig, then released to an aspheric 
configuration.
No doubt we can all look forward to the next 400 years of design & fabrication 
of aspheric optics.
================================
  Barr, L.D., J.H. Richardson, K-L Shu. "Bend-and-Polish" fabrication 
techniques. SPIE vol. 542, Optical Fabrication and Testing Workshop: Large 
Telescope Optics. Bellingham: SPIE, 1985.

  Bedini, Silvio. Lens Making for Scientific Instrumentation in the Seventeenth 
Century. Applied Optics vol. 5 no. 5 (1966) 687-694.

  Bender, John and Graham Flint. In-Process Measurement of Fast Aspherics. SPIE 
Vol. 171, Optical Components. Bellingham: SPIE, 1979.  (>wire test, 
sphereometer, interferometer)

  Cooke, Frank. Making a Concave f 0.4 Parabola. Applied Optics vol. 3 #10 
(1964), p1148-9. Reprinted in Optics Cooke Book; Optical Society of America, 
1991.

  Descartes, Rene. La Dioptrique, Paris, 1638.  Dioptrics, Discourse 10, On the 
Means of Grinding Lenses.  Translation: Olscamp, Paul. Discourse on Method, 
Optics, Geometry and Meteorology - Rene Descartes. Indianapolis: Bobbs-Merrill, 
1965.  Also, Descartes' correspondence with Constantyn Huygens, many 
references.

Deve, Charles. Rev. Opt. Theor. Instrum. 28 (1949) 212.

Deve, Charles. Le Travail des Verres d'Optique de Precision. (Ed. de la Revue 
d'Optique, Paris, 1949.) 3rd ed., p465. Similar to Jeffree.

Dourneau, F. Obtention de Surfaces Parabolique a Forte Courbure. La Theorie des 
Images Optiques (Ed. de la Revue d'Optique, Paris, 1949). Machine with single 
parallelogram for vertical motion & rollers on track for horizontal.

Dourneau, F. and J. Demarq. Revue d'Optique 28 (1949), 416. Two years operation 
of a blade type generator.

  Hashimoto, Hiroshi. Machine for Fabricating Axially Symmetric Concave 
Aspherics. Applied Optics 12:7 (1973) 1717-20. Nippon Kogaku, cam controlled 
grinding machine.

  Heel, A.C.S. Advanced Optical Technical Techniques. Amsterdam: North Holland, 
1967. p304-7, production of aspherical lenses.

  Heynacher, Erich. Aspheric Optics - The Reasons Why and How They are Made. 
Zeiss Information #88; vol. 24 (1979), p19-25. (Includes historic review.)

  Heynacher, Erich. The Production of Aspherical Surfaces by Mechanical Means. 
SPIE vol. 109, Advances in Optical Production Technology (London 1977), p71-5. 
(Zeiss machine).

  Holtzapffel, Charles.  Turning and Mechanical Manipulation, volume 3. 1850. 
Warwickshire, TEE Publishing, 1993 (reprint).

  Horne, Douglas. Optical Production Technology. N.Y.: Crane Russak, 1972. 
Chapter 6, Non-spherical Surfaces.

Jeffree, John H.  U.S. Patent 2,458,384. ca. 1950. 'Grinding Lenses or of Dies 
Therefor.'  Blade grinders.

  Kumanin, K.G. Generation of Optical Surfaces. N.Y.: Focal, 1962.

  Larmer, John and Emmanuel Goldstein. Some Comments upon Current Optical Shop 
Practices. Applied Optics, vol. 5, #5, May 1966, p677-685. Includes a 
description of a blade lapping machine built at Goddard Space Flight Center, 
Maryland.

  Lassell, William. Description of a Machine for Polishing Specula. Memoirs of 
the Royal Astronomical Society 18 (1850) 1-20.

  Lewis, R.E et al. The Generation of High Aperture Parabolic Surfaces of 
Revolution. Review Scientific Instruments 23:10 (1952),555-8.  Describes blade 
lapping machine & operation, at Ill. Inst. Tech., Chicago.

  (Loomis, Don). Loomis Designs New X-Y Motion Polishing Machine. Optical 
Sciences Center News vol. 1, #1, p17-19. (Univ. Arizona, Tucson. For an f 1.5 
UV camera, polishing head traces a variety of Lissajous-type patterns.)

  Martin, L.C. Technical Optics. N.Y.: Pitman, 1950. vol. II, p321-327. 
(Illustrates a blade lapping mechanism where the blade pivots instead of 
traversing, this generates conicoids.  Also discusses evolute cams, linkages, 
the Zeiss machine, and more.)

(Memoirs of the Academy of Berlin, Tom, III, 1710), describes machine

  Mertz, Lawrence. Design for a Giant Telescope. (Dickson, J. Home, ed. Optical 
Instruments and Techniques 1969, Proceedings of Conference at U. of Reading 
7/14-19/1969. Newcastle: Oriel, 1970.)

  Mills, A.A. & P.J. Turvey. Newton's Telescope: An Examination of the 
Reflection Telescope Attributed to Sir Isaac Newton in the Possession of the 
Royal Society. Notes & Records of the Royal Society, 33 (1978) 133-155.

  Perry, W.H. Simple Aspherical Surface Generator. Applied Optics, vol. 5, #5, 
May 1966, p741-5.  Rod extending through hole in middle of mirror raises lap 
off center of mirror to edge.

  Rosse, The Earl of. On the Construction of Specula of Six Feet Aperture. 
London: Taylor and Francis, 1862. (Philosophical Transactions 1861.)

  (Rosse) The Monster Telescopes Erected by the Earl of Rosse. Sheilds and Son, 
Parsonstown, 1844.

  Sakurai, Takemaro and Koro Shishido. Study on the Fabrication of Aspherical 
Surfaces. Applied Optics vol. 2, #11, Nov. 1963, p1181-90

  Sapp, Alvin. simple Method for Machining Aspherical Surfaces. Applied Optics 
12:1 (1973) 168.

  Scott, R.M. (   ) Applied Optics and Optical Engineering (ed. R. Kingslake.) 
N.Y.: Academic Pr., 1965.  Vol. 3, p87. (not useful)

  Shea, William. The Magic of Numbers and Motion: The Scientific Career of Rene 
Descartes. Canton: Science History Pub., 1991. (p194-201: Grinding lenses. 
Describes & illustrates an aspheric grinder that shapes a grinding stone to use 
on the lens. Excerpts from letters to artisan Jean Ferrier. Descartes 
instructed Ferrier, who then made good hyperbolic lens, unsuccessful at concave 
lens.)

  Shishido, Koro and Masao Sugiura. A grinding apparatus for making a middle 
size parabolic mirror using the link method. SPIE vol. 817, Optomechanical 
Systems Engineering (1987), p171-9.

  Smethwick, Francis. An Account of the Invention of Grinding Optick and 
Burning Glasses of a Figure not Sphericall. Philosophical Transactions, vol. 3 
(1668) p631 (p226 facsimilie).

  Stahl, H. Philip. Aspheric Surface Testing Technique. SPIE Vol. 1332, Optical 
Testing and Metrology III. Bellingham: SPIE, 1990. (profilometer, null, 
autocollimation, Hindle, Sawtooth, interferometer)

  Twyman, F. Prism and Lens Making. London: Hilger & Watts, 1952 (2nd ed.)  
p355-63. (Zeiss machine, Burch machine)

  Wren, Christopher. Corporis Cylindroidis Hyperbolici. Philosophical 
Transactions 4 (1669), p961.

  Wren, Christopher. Of Dr. Christopher Wren's Engin, designed for grinding 
Hyperbolical Glasses.  Philosophical Transactions 4 (1669), p1059.  (See 
Heynacher ZI.)

  Zschommler, W. Precision Optical Glassworking. London: Macmillan, 1984. 
(transl. Feinoptik-Glasbearbeitung. Muenchen: Carl Hanser Verlag, 1963.