Saturday, March 7, 2009

Gerald Rottman: The Geometry of Light: Galileo’s Telescope, Kepler’s Optics

Guest Review by Kit Bradley

The United Nations and the International Astronomical Union have declared 2009 to be the International Year of Astronomy, commemorating the 400th anniversary of Galileo’s use of a telescope to study the skies, and Kepler’s publication of Astronomia Nova. Through my membership in the Eugene Astronomical Society, I learned of a book self-published by Gerald Rottman, The Geometry of Light: Galileo’s Telescope, Kepler’s Optics, and I ordered a copy. This is a good year to learn about optics as Galileo and Kepler understood it in 1609.

Kepler published his work on optics in a short book titled Dioptrice. Rottman’s book presents the ideas in Dioptrice in a form accessable to us today. He explains refraction, convex lenses, concave lenses, what happens as we look through a lens, and how to put it together to make a telescope that magnifies objects.

Early on, Rottman recommends we read the Appendices first if we’re rusty on high school geometry. I am, and I did. I appreciate his including the math refresher.

It is interesting tracing rays of light through a lens and understanding the refraction that occurs. This all made sense to me until I got to the section that explains what happens when your eye is in the picture. Now you have to deal with the aperture of the eye (pupil), an internal lens, and focusing on the retina. I read one paragraph three or four times, gave up, and read the rest of the chapter. At the end of the chapter Rottman tells us to pick up any lens we have lying around (huh? oh yes, a magnifying glass), and he has us prove to ourselves that things work as he described. I went back to that paragraph, and this time it made sense!

We’re now ready to learn how to put two lenses together to make a telescope. It’s straightforward to understand how to magnify objects, but keeping the objects in focus requires more thought (at least for me). Galileo used a convex lens for the objective (the star end of the telescope) and a concave lens for the ocular (the eye end of the telescope). This works pretty well, but it has a very narrow field of view, that is, you see only a tiny section of the sky through the telescope. Kepler came up with a better design that uses convex lenses at both ends and has a much wider field of view. But it turns the images upside down (which is not much of a problem when looking at things up in space).

I went to a local star party last night, right after I finished reading this book, and coincidentally someone brought his homemade Galilean telescope. It was not much more than a cardboard tube and two lenses he had lying around. And indeed, the field of view was tiny – we could see only about half of the moon at a time through the telescope. And the edges were sometimes colorful (more on that in a moment).

Today I came across an International Year of Astronomy project that is producing an educational and very low cost “Galileoscope,” which can be configured for either the Galileo design or the Kepler design. Only $15 plus shipping! I ordered one (

I had to read The Geometry of Light very carefully to learn what was there for me to learn, and I’m glad I did. Rottman achieved his goal of explaining Kepler’s understanding of optics. Perhaps beyond the scope of this goal, however, there are three additional areas I would have liked to see discussed.

First, I wondered the whole time I was reading how things have changed since Kepler’s time. Is the information in the book still relevant today? Rottman partially answers this in the last section of the last chapter, where he shows the difference between Kepler’s approximation and the math used today to calculate the angle of refraction. They’re close.

Second, I wondered if Kepler had any understanding of chromatic aberration, in which a simple lens acts like a prism and refracts the different colors contained in white light by different amounts, which results in extraneous colors around the edges of objects. This wasn’t discussed in the book.

Third, I wondered if Galileo and Kepler understood the importance of the size of the objective lens for gathering light—the bigger the lens, the more light gathered, and the better the image. This wasn’t discussed in the book.

Given the limited mathematics of the day and the limited technologies for forming clear glass and grinding it smoothly, these three topics might have been beyond Kepler’s capabilities to address.

I had fun mastering this little book, and it will lead me into more explorations. It’s available from Gerald Rottman at his web site,

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