Introduction

Soon after I began my current position in the Department of Mathematics, Computer Science, and Statistics at the State University of New York at Oneonta (SUNY Oneonta) in 2008, I began to think about how best to utilize my academic expertise in my classes. A Ph.D. graduate from Brown University's now defunct Department of History of Mathematics, my specialty is mathematical astronomy in ancient and medieval India. I felt strongly that incorporating the knowledge that I acquired in graduate school into my teaching would be valuable to SUNY Oneonta and its students, as well as to myself.

One persistent theme in my deliberations was the use of objects in the classroom. As a graduate student I was exposed to thorough analyses of a good number of astronomical instruments, and I spent time thinking through how they worked. I felt that similar exposure would be valuable to my students at SUNY Oneonta. I had already on some occasions brought objects to my Introduction to the History of Mathematics and Mathematics for the Liberal Arts classes at SUNY Oneonta, where each student would hold and briefly manipulate an abacus, a quadrant, or a replica of a Mesopotamian clay tablet containing a multiplication table. However, while I found that such brief handling of an object was definitely positive for the students' engagement and learning, it was not sufficient for them to get a deeper appreciation for the theory behind the object or for how it worked in practice. In other words, I was looking for a way to have my students engage with objects in a more substantial way, enabling them to understand both the theory behind an object and the construction and use of it.

During a conversation in April 2009, about a year and a half prior to the debut of the course that I was to design, Allen Anderson, SUNY Oneonta's Science Technician, helped me devise a plan for how to simultaneously achieve my goals of using my academic expertise in my teaching and greater student involvement with objects. I learned that SUNY Oneonta owns a 3D printer, as well as other machinery, by which parts for objects could be constructed using plastic, wood, metal, and other materials. This opened up an exciting pedagogical opportunity for me.  While students are not allowed to operate any of the machinery due to insurance issues, they could engage with the design of astronomical instruments in consultation with Anderson and myself, and the designs could subsequently be constructed by Anderson. More specifically, Anderson could construct the necessary parts for a given astronomical instrument, which the students could subsequently assemble.

Designing a Course in Ancient Mathematical Astronomy

During Fall 2009, I designed a new course at SUNY Oneonta with the title Ancient Mathematical Astronomy. The course was based on my academic expertise and the idea of incorporating design and construction of objects into my teaching. Ancient Mathematical Astronomy is a 300-level course that provides an introduction to and a survey of the history of the astral sciences from ancient Mesopotamia to Copernicus. Beginning with astronomical records kept by Mesopotamian priests, the course traces the history and development of astronomy with an emphasis on the role of both mathematics and observation in the formation of astronomical theories and models. As an important part of the course, the students, divided into groups, are asked to research and design an astronomical instrument of their choice. The parts for the instruments are constructed by Allen Anderson, after which the instruments are assembled by the students. The instruments should ideally be tested by simple observations, though the feasibility of this step depends on the instrument in question, time available, and the weather.

Course objectives

By incorporating the practical component of instrument construction into the course, I seek to root mathematical theory in practical application with the aim of providing my students with a solid understanding of how the models of mathematical astronomy work. The idea is that the course should provide an attractive elective for our Mathematics majors that draws on both the history of mathematics and applied mathematics. In fact, one of the justifications for adding Ancient Mathematical Astronomy as a permanent course was that it would provide an elective with a focus on applied mathematics, an underrepresented branch of mathematics in my department.

The main lesson that I want my students to take away from the course is knowledge of the history and development of the astral sciences and their role in world intellectual history. More specifically, I want the students to understand and appreciate the role played by mathematics in the formation of astronomical theories and models, and, more generally, in the formation of theories and models in the sciences. Another aspect that I seek to emphasize is awareness of natural phenomena that were hugely important to our ancestors in the past, but which now mostly pass unnoticed due to the use of modern technology.

Course prerequisites

Prerequisites are an important consideration for any new course, and I gave serious thought to the prerequisites for Ancient Mathematical Astronomy. On the one hand, I did not want the course to be too restricted; on the other hand, I wanted to have mature students with sufficient mathematical skills to handle the difficult material of the course. After some reflection on the matter, I decided to make Ancient Mathematical Astronomy a 300-level course and require that students enrolling in it must have passed either Calculus II or Discrete Mathematical Structures with a grade of C or better (the latter course introduces students to mathematical proof at SUNY Oneonta). I felt that completion of either of these courses would ensure that a student possessed sufficient mathematical maturity and capability. The reason for requiring only one of the courses and not both, which is generally the case for higher-level mathematics courses at SUNY Oneonta, is that I did not want to exclude Physics and Astronomy majors or minors, who are required to take Calculus II but only rarely take Discrete Mathematical Structures. (The Department of Physics and Astronomy at SUNY Oneonta offers a major in Physics and minors in Physics and Astronomy.) At the same time, I did not want Mathematics majors who had taken Discrete Mathematical Structures but not yet Calculus II to have to wait to take the course, which is offered infrequently.

Course modules

SUNY Oneonta operates with a semester system, each of the two semesters per year being 15 weeks long, including a week for finals. Regular classes meet three times per week for 50 minutes per meeting.

The theoretical parts of the course are divided into six modules:

1. Introduction to ancient astronomy (approx. 3 weeks), which makes students familiar with ancient astronomy and cosmological models, and introduces them to various astronomical instruments.
2. The birth of mathematical astronomy (approx. 2 weeks), the focus of which is the formation of ancient Mesopotamian astronomy, including a discussion of how belief in astral omens led to careful observational records.
3. Ancient Greek and Hellenistic astronomy (approx. 3 weeks), which explores the formation of Greek geometrical models of the heavens, starting with early Greek astronomy and ending with Ptolemy and the Almagest.
4. Indian astronomy (approx. 2 weeks), which explores classical Indian astronomy, though pre-classical Indian astronomy is also discussed.
5. Islamic astronomy (approx. 2 weeks), which explores the history and development of astronomy in the Islamic world.
6. Astronomy in Medieval and Renaissance Europe (approx. 2 weeks), which examines the reception of astronomy from the east in Europe and the development it took there.

The mathematical content of the course includes numerical models of planetary motion from ancient Mesopotamia, geometric models of planetary motion from ancient Greece and other cultures, spherical geometry, and spherical trigonometry. These topics are by no means simple, and it requires maturity and dedication on the part of the students to master them.

Course textbook

For the course textbook I decided on James Evans' excellent The History and Practice of Ancient Astronomy (Oxford University Press, 1998). It is a book that I have benefited from in my own studies and research, and one that I felt would be sufficiently accessible to the students in the course. In fact, the students who have taken the course have expressed to me that they enjoyed the textbook, which they found informative and helpful.

Teaching a Course in Ancient Mathematical Astronomy

The course was offered for the first time in Fall 2010. At that time, the course ran as a special-topics course due to a delay in final approval over uncertainty over how to handle a potential student fee for the materials used to construct the instruments. Soon after the special-topics course started, the course received full approval as a permanent course at SUNY Oneonta. Four years later, in Fall 2014, the course was offered again. As such, the course has been offered twice at SUNY Oneonta. The present article details my experiences from teaching Ancient Mathematical Astronomy these two times.

Both times it was offered, the course had an excellent enrollment for an elective in the Department of Mathematics, Computer Science, and Statistics at SUNY Oneonta. The following table shows the enrollment in Fall 2010 and Fall 2014, including the majors of the students enrolled. The data for Fall 2010 includes an auditor, an Adolescence Education: Mathematics major, who, while not required to do any of the coursework and unable to receive a final grade, participated fully in the course.

In Fall 2010, the course attracted a more diverse range of majors than expected, whereas in Fall 2014, the range of majors was limited to precisely those majors that I originally had in mind for the course.

Required course work

Ancient Mathematical Astronomy has no sit-in tests. Instead it is required for the students to hand in five assignments, an instrument paper, and a final research paper. In addition, homework problems and readings are assigned on a regular basis, and students are required to make presentations before the class.

The idea of the instrument paper is to familiarize the students with the instrument that they are undertaking to design. In addition to the history of the instrument, the instrument paper must include a theoretical section on how the instrument works, as well as notes on the design of the instrument. Ideally, the students should also include sketches of their design. More often than not, however, the papers did not include useful sketches. One reason for this is that it is hard to produce the sketches before understanding the theory behind the instrument, an understanding that the instrument paper is meant to bring about.

For the part of the course dealing with design of instruments (including working on the instrument paper), the students are divided into groups of two or three, who in consultation with me agree on an instrument to work on. In Fall 2010, each student wrote an individual instrument paper, whereas in Fall 2014 each group handed in one paper for all the members. I felt that requiring individual papers was unnecessary given that the students work so closely together on researching and designing the instrument. After receiving feedback from me on the instrument papers, the groups began actively designing the instrument. During this phase, the students worked closely with Anderson, who advised on what is feasible and offered feedback on the ideas of the students. Anderson would later construct the parts required by the students' designs.

The final research paper is an open and flexible paper meant to allow each student to dive into a topic that they find especially interesting. The final research paper can be focused on the history of astronomy, formulae and derivations from mathematical astronomy, or philosophical questions relevant to the development of astronomy. Among the topics chosen by students are the computation of square roots in the ancient world, archeoastronomy, Ptolemy's Almagest, and the use of sine tables in the ancient world.

Planetarium and Observatory visits

SUNY Oneonta has a planetarium with a digital projector system that can provide views of the stars, planets, luminaries, and constellations, as well as 3D movies. The system is based on Evans & Sutherland Digistar 4 software. The Planetarium is available for class visits by appointment.

The college also has an observatory. The observatory has a large number of telescopes, both mounted and mobile. In particular, the SUNY Oneonta Observatory owns a 1-meter JMI NTT-40 mobile telescope, the largest optical telescope in the state of New York. The Observatory hosts open nights, and is also available for class visits by appointment.

I naturally sought to make use of these two excellent resources in Ancient Mathematical Astronomy. In Fall 2010, the class visited both the Planetarium and the Observatory. At the Planetarium, the class saw a presentation by a SUNY Oneonta student on astronomical coordinate systems. The visit to the Observatory, guided by former Planetarium and Observatory Director and astronomy professor Dr. Quinn Minor, focused on observing the moon by telescope.

In Fall 2014, the class made two visits to the Planetarium. A planned visit to the Observatory had to be canceled due to bad weather, and it was not possible to find another date that worked for all the students. The first visit to the Planetarium featured a presentation on astronomical coordinate systems. At the second visit, the class watched the 3D documentary Stars of the Pharaohs from 2004, produced by Evans & Sutherland. Both presentations at the Planetarium were given by Planetarium and Observatory Director and astronomy professor Dr. Joshua Nollenberg, who was also scheduled to guide the Observatory visit that was canceled.

Course fee

It was originally unclear how expensive the construction of the instruments would be, and I expected that there would be a small course fee per student, perhaps $20 or so. However, it turned out that the cost of constructing the instruments was low enough that the Science Technician's budget covered most of it. Students paid for minor things here and there (mirrors for the sextants, for example), but the costs of most of the materials were covered by the Science Technician.

Design and Construction of Astronomical Instruments

I will not go into the the technical details of the astronomical instruments that were designed and constructed the two times that Ancient Mathematical Astronomy ran at SUNY Oneonta. For each instrument described below, there are one or more links to Internet pages with more information, including images of historical specimens. There are furthermore plenty of good sources on the topic of astronomical instruments, including James Evans' book listed above, which has an extensive bibliography. Thus, the following is a general presentation of the instruments that were designed and constructed in Fall 2010 and Fall 2014.

Instrument design phase

For the design phase of the course, the students in the class were divided into several groups. In Fall 2010, there were six groups with two students each and one group with three students (the third being the auditor), and in Fall 2014 there were three groups with three students each and one group with two students. Each group decided on an instrument to design. The choice of instrument was made in consultation with me and with feedback from Anderson, who provided information on whether the group's first ideas for designing the instrument were feasible with the equipment available.

The following table shows the choices of instruments and how many groups undertook to design each of them in Fall 2010 and Fall 2014.

After the formation of the groups and final decisions on the instruments, each group began to research their instrument. This research culminated in an instrument paper, as described above, after which the group began the actual design process.

Eight class periods of the course were used exclusively for the students to work on their designs. Anderson and I were both available at those times (or an appointment could be made to meet with one or both of us at another time). When meeting with Anderson or me, the students would explain what parts would be involved, provide sketches and dimensions, and justify that the parts would fit together so as to make the instrument functional. Some of the instruments required only a few parts, others more. Anderson and I would discuss the students' ideas and designs, address mistakes, and make suggestions. Students were expected to follow a timeframe that allowed for Anderson to construct a specimen for each group by the last week of the semester for presentation to the other students in the course.

Machines and materials used

As has already been mentioned, SUNY Oneonta owns a 3D printer, which serves the School of Natural and Mathematical Sciences (in Fall 2010, before the restructuring of SUNY Oneonta from two large academic divisions to five schools, it served the Division of Science and Social Science) and is operated by the college's Science Technician, Allen Anderson. This was the machine used for most of the parts of the astronomical instruments designed by the students in Fall 2010. However, in contrast to Fall 2010, the 3D printer was not used at all to produce parts for the instruments designed in Fall 2014. This was partly due to the 3D printer not being operational at the beginning of the semester, prompting the students and Anderson to consider alternatives.

The SUNY Oneonta 3D printer operated by Anderson is a Dimension BST-768. The machine prints in a very strong ABS plastic and has an \(8''\times 8''\times 12''\) build volume. The models created by the machine are functional and even machinable. The machine requires an STL file to generate a model. At SUNY Oneonta, Anderson uses SolidWorks to generate the required STL file, though there are also scanning devices that can be used to generate the file directly. At this point in time, the Dimension BST-768 is an old machine in terms of material and technology. It was one of the first commercially available 3D printers, and there have been many improvements made since then. The biggest issue with SUNY Oneonta's 3D printer is the resolution of detail, which is limited by the size of the material “stream” that is used to create whatever model is being printed. The resolution of the Dimension BST-768 is on the order of \(0.003''.\)

In addition to the 3D printer operated by Anderson, the Department of Art at SUNY Oneonta owns a 3D printer as well. The Department of Art's 3D printer is also a Dimension and similar to the one operated by Anderson, though it is a newer model. Recently, in Fall 2014, the Department of Mathematics, Computer Science, and Statistics also acquired a 3D printer. This 3D printer, a MakerGear M2, is a low-end desktop 3D printer. However, the 3D printers of the Department of Art and the Department of Mathematics, Computer Science, and Statistics have not been used for the construction of instruments in Ancient Mathematical Astronomy.

Besides the 3D printer, other tools were used as well to construct parts out of metal, wood, and plastic. For example, the construction of the wooden quadrant in Fall 2014 was done with a handheld electric saw.

Notes on the design and construction phases

Overall, no insurmountable obstacles were encountered while designing and constructing instruments. Certain things took longer than anticipated and others were more difficult for the students than expected. In particular, given difficulties and time constraints, we had to change plans for a few instruments, namely, the astrolabe and the telescope. Still, overall we were on track with the timeframe that I had in mind, and in both Fall 2010 and Fall 2014 the students were able to present their instruments to the class at the end of the semester.

Due to a combination of time constraints and cloudy weather at the end of the course in both Fall 2010 and Fall 2014, we unfortunately did not end up doing much in terms of using the instruments practically. Had the time been available, I would have liked to, for example, mark the shadow of the sundials at regular time intervals during the day and to measure the angular distance between two heavenly bodies with the Jacob's staff and the sextant at night. However, it turned out to be too difficult to organize it during the end-of-semester rush.

Library Exhibitions

The course and the construction of the instruments in Fall 2010 attracted some attention on campus, and some of the instruments (the three sundials and the successful armillary sphere) were put on display in an exhibition at SUNY Oneonta's Milne Library together with books on astronomy. The exhibition was on display from February 3, 2011 to March 27, 2011.

Similarly, an exhibition of two instruments (the quadrant and the sextant) was put up at Milne Library following the Fall 2014 course. The exhibition was on display from April 3, 2015 to May 1, 2015.

In 2011, three of the students from the class wrote the text accompanying the featured astronomical instruments, while I wrote the general description of the course and its instrument design phase, which was printed on an exhibition poster. In 2015, two students from the class wrote the descriptions of the two instruments on display, while I again wrote a general description.

Photos of the 2011 and 2015 exhibitions can be seen in Figures 15 and 16.

Figure 15. Exhibition of astronomical instruments built by Fall 2010 Ancient Mathematical Astronomy class (photograph by Heather Beach, SUNY Oneonta Library)

Figure 16. Exhibition of astronomical instruments built by Fall 2014 Ancient Mathematical Astronomy class (photograph by Toke Knudsen)

Conclusion

Overall, the course was a success. Some students put in more work than others, as is always the case, but all of them expressed to me that the design process had been a valuable experience and that it had increased their appreciation of the course. The hands-on approach energized many of the students and they enjoyed the learning process, whereas similar enthusiasm rarely comes from purely lecture-driven courses. I personally experienced that the students' theoretical understanding of the course was enhanced due to the design process and their many conversations and discussions with Anderson and me. This was helped along by the fact that the pre-Copernican, geocentric model that we worked with in most of the classes permeated the theories of nearly all of the instruments considered in the course. It was an enjoyable and very educational process for all, and several students were excited about taking a specimen home after the course.

Incorporating a design-and-construction component is certainly an approach that I would like to repeat, in future iterations of Ancient Mathematical Astronomy, of course, but also in other courses, such as Introduction to the History of Mathematics, which I teach regularly at SUNY Oneonta. It is furthermore something that I would highly recommend to anyone with an interest in history or in a hands-on approach in the classroom.

Acknowledgments

I am thankful to Amy Shell-Gellasch, James Evans, and Janet Beery for their help and encouragement in bringing out this article.