1. Lens shapes and inconsistencies in the literature of Galileo' existing telescopes, in the drawings and actual data for the optical systems.
Most  optical diagrams for Galileo's telescopes, show  symmetric double convex  and double concave lenses.  We have  searched through our library and scanned samples of drawings of the optics of Galileo's telescope  in  text books on optics, texts on the history of telescopes, texts  in  general science  and especially on web pages put together for students meant as lectures by educators in science.  Our sources ranged over a time period of c 1800 to 2000,  and most of the figures illustrate the lenses as  symmetric bi- convex  and  biconcave lenses even though  none of the lenses, attributed to Galileo, are symmetrical . Very few even show plano curved lenses and  none show  asymmetric curved lenses.  See reference section 9. for figures from some of these sources.  The shape of both sides of a lens is important  to its performance and sheds light on  the inventor.

With great respect for the  very valuable and beautiful resource  "Catalogue of Early Telescopes" for the famous collection at IMSS,  we  find an interesting mixture of drawings on the optics.  On  page 30 the author, describing Galileo's  leather telescope in some detail,  shows  a symmetric biconvex / biconcave  diagram  for  both lenses although the objective lens is described as plano convex in the text.  Also the diagram of the objective lens in its lens housing is  portrayed as  biconvex.  In the appendix of the catalogue, pages 99 through 111 "The Optical Principals of Telescopes" written by another author are  a mix.   Fig 9 uses biconvex and biconcave  for the "Optical layout of the Galilean telescope".   By remarkable contrast many of the more sophisticated  optical designs further on in the catalog  use a  more accurate  mixture  of bi-lenses  and plano-lenses in the figures.

Last but not least,  in describing our very precise replication of these historic instruments on our  web site,  the authors, to save time, borrowed  and displayed a   drawing from an 1860 Natural Philosophy text which had the infamous symmetrically drawn  curved lenses.  Fortunately we were called on it by one of our readers and we changed it.

See our figure 1 above for the corrected drawing  based on the following data from the IMSS Catalogue  descriptions of the Galileo telescopes in their possession: Galileo's telescope, IMSS 2428,  consists of a "plano-convex" objective and a symmetric biconcave  eyepiece for the current lens  with  a suspicion on our part that the original was  a plano-concave eyepiece.  (Please see the earlier discussion on this subject which is linked from the home page.)  Galileo's IMSS 2427 has a non symmetric double convex objective and a plano concave eyepiece. Galileo's broken objective lens,  IMSS 2429, is  a non symmetric ,  nearly a  plano - convex lens.  See the figure 1 above for the drawing that, given our current knowledge, more correctly shows the actual lens shapes for all of the known Galileo lenses.

The bottom line:  None of  the five lenses  attributed to Galileo  at the IMSS are  actually shaped as symmetric double curved lenses (with the exception being the replacement eyepiece of #2428 - with the qualification given above) yet in contrast  most  of the drawings in the literature  consistently show them as symmetric double curved lenses.  Why are we compelled to do this when it is wrong yet easy to fix?
 

So!

It  could also make a difference in  the judgment by our peers and  our readers on our accuracy as science reporters.

It makes many of us wonder - can't we get the drawings of the two little pieces of glass, the heart of the instruments, closer to the actual  lens shapes for three of the worlds most famous telescopes?  It certainly made a difference for us in precisely replicating Galileo's telescopes and making choices for the lenses.

2. Differences in the measurements of other optical characteristics for lenses in Galileo' existing telescopes;
There is not only  confusion in the  popular literature but in the technical literature  by experts in the field of optics as well.   In the latter,  two scientists report  different  measurements for the same lenses.  The IMSS 2428 objective's focal lengths measurements  are given as 956 mm in one case and  980 mm in a later measurement. The IMSS 2427 rear radius of curvature measurements  differ  from -3465 in one case to          -2700 mm in the other.   The broken objective rear radius for IMSS 2429  is measured as -14363 compared with  -12000mm, and the refractive index measurements of  IMSS 2427 is given as 1.528, the other value as 1.58 , etc.  (Ref 1, 2, 2a).

  3.  Lens symmetry issues

Using data from Refs 1 & 2 we can take the ratio of the back to front surfaces and see how symmetrical the lenses are.

• IMSS 2427:  the ratio of the back radius of curvature divided by the front is 3.48/1.  The front radius is 995.5 mm.

• IMSS 2428:  the ratio of back to front radii is 93.5/ 1.  The front radius is 535 mm. The rear radius  is minus 50,000mm. in one paper and infinity in a second,
  
• IMSS 2429:  this is the broken objective that very likely was the objective Galileo used in his discovery of Jupiter's moons, etc. The back to front ratio of the radii is 15.26/1.  The front radius is 941.6 mm.

  This shows that none of the these objective lenses are symmetrical

, therefore their symmetrical depiction in the drawings we have referenced are not representative of measured values.

 So our conclusion about drawings:

It is unreliable to assume that  drawings of  the lens shapes appearing in text books and published papers are correct and / or consistent with the actual shapes of the lenses used in the system.  It appears that generally for simple optical systems such as a telescope it is traditional for the artist to show all  the lenses as either  double concave or double convex. This  is certainly true  for most  illustrations of the earlier telescopes. The actual lenses more than likely  have one side of the lens  flatter than the other.   So again the reader should be on the alert that the diagrams outlining the optical layouts  are neither to scale nor are the lenses shaped as they are measured for the actual telescope.


 

4.  Calculation of the impact of a back surface radius of  the objective lens on the focal length of non symmetrical lenses.   Does the double radius rather than a plano - convex surfaces make a sufficient difference in the optical performance of the lens to make it an important issue in defining and illustrating  the optics? In general yes. The following figure shows a curve based on the calculation of how the focal length of a lens with a fixed front surface radius of curvature varies with increasing curvature of the rear surface. The objectives of IMSS 2427 and 2429 have about the same front radius, so they can easily be compared on the same graph.  So we chose a front surface radius of 954 mm,  a value approximating their radii,  and a refractive index of 1.54 for this calculation.  The graph is very compact but it collects most of the important data all in one place.  It does take a fair amount of time to digest it all.  One can use this graph to get  a feeling of how flat some reviewers feel is flat.   A one line summary for Galileo's lenses; Only  the rear surface of the objective lens of the IMSS 2427 makes a large difference in its  focal length as compared to a plano convex lens. A more detailed summary using the data in the graph is: For IMSS 2427 - even though the back curved surface is only about one third the radius of the front, it has a strong influence on the focal length of the lens.   IMSS 2428 is either a plano convex or very closes to a plano convex surface depending on which measurement in the literature one accepts. IMSS 2429  might be roughly referred to as a plano convex lens, because the  curve  alters the focal length by only 10% or less. An important observation from the curve in the figure is that when the radius of curvatures of the two surfaces are similar, then a slight variation in the back radius has a large impact on the focal length of the lens, and when the back radius is very large, a small error in the curvature will give a very small or negligible change in the focal length. The question one is left with: What was Galileo's intent? To curve or not  to curve  the second surface, that is the question!  The  question MIGHT BE  did Galileo understand the effect of curvature on spherical aberration? (see next section 5, below) 5.  The affect of lens shape has on Spherical Aberration The following figure demonstrates how radii of curvature on both sides of the lens impacts its spherical aberration. The figure below is from ref 3.   From this figure we can conclude that a slightly curved back surface, depending on the amount, can reduce the spherical aberration. It shows that a  plano-convex lens has less spherical aberration than a symmetrical bi convex lens (the kind most often drawn in the literature). Does this imply that Galileo knew about spherical aberration and made his lenses with the large radius of curvature on one side rather than making it planar in order to achieve a better image. We think not. But who can tell - perhaps his experimental skills were greater than we give him credit for!   Would  Galileo have been  able to tell the difference in the quality of the  image due to spherical aberration? Galileo improved the image quality by use of a small aperture, but achromatic and spherical aberration sources would be the  least detectable disturbance when one considers all the possible image disturbance sources including blurring from mechanical vibration in a 20 power telescope, turbulence of the atmosphere, and optical errors from the eyepiece which is not trivial to grind and polish.  So perhaps from a practical view point, Galileo probably shaped only one side of his lenses because this was easier and took less time.  If grinding and polishing one side gave pretty good visual results, why would he want to grind and polish the other? Only a touch up would suffice.   An interesting final note, Looking at the data in the  figure above shows that a lens of equal radii on both sides is not the best choice for reducing spherical aberration.  None of the three objective lenses attributed to Galileo had this shape!  So why would one want to make drawings of Galileo's lenses with symmetrical double convex surfaces  when his lenses had lower spherical aberration than these drawing imply.  Why isn't there more attention  paid to the eyepieces and their effect on the quality of image.  There is a  rumor  going around that the reason that Galileo did not use a positive eyepiece was that the glass transmission was so poor  and the thickest part of the lenses was in the center that no one would  have been able to see through it.  If that was true we would not have been able to see through the objective as well!  The eyepieces were smaller in diameter  than the objective so they would have been very close to the same thickness as the objectives which no one denies has reasonable transmission.

 

6.   Differences in and problems related to the measurements of the overall lengths of Galileo's existing telescopes. 

Both IMSS 2427 and 2428 are reported to have the same length as  the focal lengths of their  objectives. This is an optical impossibility if the telescopes are to focus at infinity. The Galilean design, with the negative eye piece lens, requires  the overall length of the telescope to be  shorter  than the focal  length of the objective  i.e. If the lengths of the telescopes are correct as reported then the objectives would have to have been of longer  focal lengths, suggesting that the present lenses are not the originals!

Optical telescope length = focal length objective  minus  the focal length of the (negative) eyepiece. 

•  So if the telescope IMSS 2428  is 980 mm. long, the focal length of  the objective would have to be 1027.5 mm.  The longest focal length measurement we found for this lens  was 980 mm.
   
• So either the objective is not the original or the telescope  needs to be  shorter than 980 mm !
   
• The other answer could be that the original eye piece would have to have been  a convex lens. This configuration would  give an inverted image and a much better field of view!!!

The same issues hold for telescope number 2427.

We will leave it to the readers to explain why scientific data should be described so unscientifically.

Near the end of this page are examples. Click on the thumbnails below to see larger pictures.

7. The bottom line: Looking through the telescope. The real field of view and how it changes. 

The field of view is not always 15' as  advertised but it's a dynamic field of view depending on the individual observer (see the explanation below).

Additionally what  one  sees  looking, up into the sky, through Galileo's telescope is dramatically different from what one expected to see after hearing so much about Galileo's telescope. To most  the view is down right disappointing, a tiny spot of light way at the end of a dark tunnel. In this spot of light the image races around and in and out of view with every tiny vibration of the instrument.  Then slowly with concentration  the spot seems to grow  into a larger  image with more detail becoming  more recognizable with time.

9, The field of view issue,  Comparing the field of view with a singlet positive lens versus  a negative lens for the eyepiece.

Remember that each of these photographs is extracted out of the center of a much larger picture frame with the rest having a  black background.

Above the field of view using a negative eye piece -47mm focal length and  a 980mm focal length objective. The image is right side up and moves same way telescope moves. Even though the real field of view is  smaller the observer can  obtain a much larger virtual field of view by scanning their eye back and forth and up and down over the eye piece without moving the telescope. the observer automatically does this with time to  get a larger field of view .

Above the field of view using a single lens positive eyepiece +47mm focal  length and a 980mm focal length objective. . The image is upside down and moves opposite the direction of the movement of the telescope. At the higher powers requires a trained observer to manage the telescope.

Historians tell us that one of Galileo's activities was making engineering instrument for sale so it is reasonable to assume that Galileo designed his telescope for terrestrial use by untrained observers,  not astronomical use.  At the time there was no reason to design a telescope for looking up at the heavens because people had been looking up for thousand of years and nothing much was new. Which gives those with  product development experience  a fundamental insight into Galileo's development of his telescope.

During Galileo's development period of the telescope his most immediate source for lens was the local optician who made eyeglasses and magnifiers for the visually challenged. The focal length of these lenses would be quite short and most certainly would not have included many if any weak lenses in neighbor of  a half of a diopter of 1000mm f.l. or greater because they would have been of little use to the buyer. The other issue that many may not have considered is that eye glass and magnifiers would have been available for hundreds of years before Galileo and it would be inconceivable that at least one optician would not  have tested from his box of lenses  various combinations of them stumbling over the telescope and microscope time and time again.

The product trade off's 

1, The magnification is the same for both eye pieces.  The field of view is larger for the positive eye piece but  the image is upside down and moves in the opposite direction of  the movement of the telescope, a decided disadvantage at high power. Galileo as a manufacturer and seller of scientific / engineering instruments recognize that   these would be considered to be  flaws to his  untrained costumers.

2, On  the other hand one can see about the same area of the image with a negative lens by scanning one's eye over the eyepiece, plus one has  an upright image . Combine this with the fact that  the telescope with the negative eye piece is considerably shorter than the one with the positive eyepiece  make the  telescope easier to carry around.  Last but least his goal was to make a  marketable product  to make his money in a hurry before  the competition from eye glass producers got there first. He had to be  aware of the possible commercial value of selling and licensing the production of  high power telescopes. Other eyepiece problems?
 

The scientific trade off's

Out of natural  curiosity Galileo  did  look up into the sky with one of his telescopes and was struck by the opportunity of new discoveries. At the same time,  we're are told by historians that within weeks, other technologist were looking up.  The race was on.  Galileo as a scientist was sharp enough to recognized that there was  no time to be wasted designing better telescopes. There appeared to be enough discoveries to be covered by his  existing instruments to make a sizeable name for himself.

The bottom bottom line.

The authors  from their background believe That Galileo was a renascence man of science and engineering,  a technologist that can only be best deciphered using an understanding of product development, marketing, experimental physics, and mathematics especially analytical geometry.