We quote from a number of different sources about three different aspects of Project MATHEMATICS!.

These are:

1. Why Apostol undertook the project;

2. Why the name was chosen;

3. The videos.

Most of 1. and 2. are taken from: D J Albers and T Apostol, An Interview with Tom Apostol, *The College Mathematics Journal* **28** (4) (1997), 250-270. Extracts from reviews are referenced individually.

**1. Why Apostol undertook the Project MATHEMATICS!**

The mechanical universe is a project that encompasses 52 half-hour television programs, two textbooks in four volumes, teachers' manuals ..., videotapes ..., and much more.

The turning point for me was when I saw the first example of Jim Blinn's computer animation. It was clear that this was a new and powerful tool that could be exploited for mathematics education. It brought mathematics to life in a way that cannot be done in a textbook or at the chalkboard. I was hooked, and for the next five years I devoted nearly all my energies to this physics project, learning new skills such as scriptwriting, television production and editing, the use of original historical documents, and other techniques well known to the television industry but rarely encountered among research mathematicians.

**2. Why the name Project MATHEMATICS! was chosen.**

**3. The Project MATHEMATICS! videos.**

**3.1. Similarity.**

**3.1.1. Product description.**

Scaling multiplies lengths by the same factor and produces a similar figure. It preserves angles and ratios of lengths of corresponding line segments. Animation shows what happens to perimeters, areas, and volumes under scaling, with various applications from real life.

**3.1.2. Review by: Connie Buller.**

*The Mathematics Teache*r **85** (6) (1992), 496.

The Similarity videotape is outstanding. Although it would be excellent in a high school geometry class (the 25-minute length is just right for viewing plus discussion), I think portions of it might be helpful for reviewing the unit circle in calculus. The included booklet, with pictures of key screens, and the fact that the videotape is divided into numbered sections helped me to locate easily the part I wished to show. The two-minute section from 'Similarity' saved me explanation time, and the animations were better than static drawings I could have done myself. Although I was not used to considering videotapes as part of my repertoire of teaching aids, I plan to use portions of this one again in other courses.

**3.1.3. Review by: LAS.**

*Amer. Math. Monthly *98 (1991), 579.

One of a series of motivational and instructional videotapes intended primarily for students in grades 7-9. Production is consistent with normal TV fare: animation, stock footage, voice overlays, background music, etc. Focuses on scale models (moon lander) and scaling (maps) as an ubiquitous application of similarity - a "great triumph of Euclidean geometry." Uses computer graphics with gimmicks (ray guns for enlargement and reduction) to illustrate ratios, perimeters, areas, and volumes. Divided into "slates" coded with small on-screen numbers that are coordinated with the accompanying Workbook.

**3.2. The Theorem of Pythagoras.**

**3.2.1. Product description.**

Several engaging animated proofs of the Pythagorean theorem are presented, with applications to real-life problems and to Pythagorean triples. The theorem is extended to 3-space, but does not hold for spherical triangles.

**3.2.2. Review by: Linda Parrish.**

*The Mathematics Teache*r **83** (6) (1990), 488, 490.

The module contains some practical applications, several different proofs, and some history. The video tape itself is very well put together. The applications include women in non-traditional roles, as well as Crusaders trying to use ladders to scale a castle wall. Equations are cleverly balanced by sound and sight. Colour and sound are used effectively throughout. The entire videotape lasts about twenty minutes, making some of the segments quite brief. I believe that high school and community college students will find this activity both instructive and entertaining.

**3.2.3. Review by: LAS.**

*Amer. Math. Monthly *98 (5) (1991), 466-467.

A motivational and instructional video introduction to right triangles: opens with typical survey problems, covers standard algebraic formulation (including Pythagorean triples), and then gives three visual proofs (Chinese, Euclid, dissection). Concludes with extensions to other shapes and dimensions. Intended for middle school students. Accompanying Workbook contains discussion, extensions, and exercises.

**3.3. The Story of Pi.**

**3.3.1. Product description.**

Although pi is the ratio of circumference to diameter of a circle, it appears in many formulas that have nothing to do with circles. Animated sequences dissect a circular disk and transform it to a rectangle with the same area as the disk. Animation shows how Archimedes estimated pi using perimeters of approximating polygons.

**3.3.2. Review by: Elaine M Hale.**

*The Mathematics Teacher* **84** (4) (1991), 334.

How this module is used would depend on the mathematical back ground of the intended student audience. For students with a strong background, neither the advanced content nor the precise language would limit the value of the videotape to enhance partially developed concepts. For students with a limited mathematical background, this artfully produced videotape would also have potential benefits. Its perceptual approach in presenting the topic of pi with manipulatives tends to divert viewers from a rote mind set. The effective use of motion, colour, and sound captures the attention of any viewer! Connections of pi to real-world applications are also a special focus throughout the videotape. Further, the historical development of pi and various formulas involving pi are thoroughly explored and concretely demonstrated.

**3.3.3. Review by: LAS.**

*Amer. Math. Monthly *98 (5) (1991), 462.

A visual, historical introduction to π, produced in a thoroughly professional manner. Simple applications, Archimedes' discovery, computation, and extensions (lattices, random numbers, Buffon needle). Useful at many levels, from junior high to college classes in quantitative literacy. Workbook contains commentary and exercises.

**3.4. Sines and Cosines, Part I (Waves).**

**3.4.1. Product description.**

Sines and cosines occur as rectangular coordinates of a point moving on a unit circle, as graphs related to vibrating motion, and as ratios of sides of right triangles. They are related by reflection or translation of their graphs. Animation demonstrates the Gibbs phenomenon of Fourier series.

**3.4.2. Review by: Michael B Fiske.**

*The Mathematics Teacher* **86** (6) (1993), 506.

The videotape 'Sines and Cosines', Part 1 skilfully and artfully combines motion, colour, and sound to develop and illustrate properties of the sine and cosine functions. ... 'Sines and Cosines' is designed to augment teacher-led classroom instruction and is suitable for use in any classroom in which the sine and cosine are presented. The entire videotape can be shown as an introduction. Individual lessons can then be reviewed in the appropriate context. Materials include a workbook that both supplements and extends the videotape lessons. Each lesson contains exercises, suitable for individual or small group work and class discussion, that vary from the procedural to the conceptual The lessons are adaptable for use with graphing calculators. 'Sines and Cosines' should be in the library of every secondary school mathematics teacher. It successfully uses visual representations to illustrate fundamental mathematical concepts. The module makes mathematics accessible to all students.

**3.4.3. Review by: Jean Redfearn.**

*The Mathematics Teacher* **88** (1) (1995), 64.

The product is accurately described, and the content is mathematically sound and well sequenced. I found the material to be most helpful in my classroom after a concept had been introduced with manipulative activities, although some teachers may want to begin with the videotape segments or to review with a segment at the end of a unit. The most noteworthy feature of the videotapes is the clever integration of history, real-life examples, music, and animated graphs throughout each segment. For example, the description of a radial angle explains in an entertaining, unique way the origination of the name radian. Another example uses a visual approach; instruments in an orchestra help students understand a variety of sinusoidal waves. At times, the explanation with animation tends to be a bit confusing, but stopping the tape so that the increments are easier to under stand eliminates that confusion.

**3.5. Sines and Cosines, Part II (Trigonometry).**

**3.5.1. Product description.**

This video focuses on trigonometry, with special emphasis on the law of conies and the law of sines, together with applications to The Great Survey of India. The history of surveying instruments is outlined, from Hero's dioptra to modern orbiting satellites.

**3.5.2. Review by: Jean Redfearn.**

*The Mathematics Teacher* **88** (1) (1995), 64.

The videotape and workbook offer excellent introductions to, and reviews of, trigonometric concepts. The outlines and overviews guide teachers through the easily digested, bite-sized segments. The workbook exercises fit well with the various five minute program parts. ... I recommend these tapes to teachers of trigonometry at any level because the segments include easy as well as more difficult concepts and the program guide and workbook offers first rate instructions and student exercises that correlate well with any textbook.

**3.6. Sines and Cosines, Part III (Addition formulas).**

**3.6.1. Product description.**

Animation relates the sine and cosine of an angle with chord lengths of a circle, as explained in Ptolemy's Almagest. This leads to elegant derivations of addition formulas, with applications to simple harmonic motion.

**3.7. Polynomials.**

**3.7.1. Product description.**

Animation shows how the Cartesian equation changes if the graph of a polynomial is translated or subjected to a vertical change of scale. Zeros, local extrema, and points of inflection are discussed. Real-life examples include parabolic trajectories and the use of cubic splines in designing sailboats and computer-generated teapots.

**3.7.2. Review by: LAS.**

*Amer. Math. Monthly *98 (5) (1991), 461.

A half-hour videotape motivated by basketball shots that uses effective video graphics to illustrate linear, quadratic, cubic, and higher polynomials. An animated hand crank simultaneously varies parameters and shapes of graphs; on-screen algebraic choreography helps relate algebraic forms to visual representations, especially of factors and zeros. Accompanying Workbook provides reinforcement, extensions, and exercises.

**3.8. The Tunnel of Samos.**

**3.8.1. Product description.**

This video describes a remarkable engineering work of ancient times: excavating a one-kilometer tunnel straight through the heart of a mountain, using separate crews that dug from the two ends and met in the middle. How did they determine the direction for excavation? The program gives Hero's explanation (ca. 60 A.D.), using similar triangles, as well as alternative methods proposed in modern times.

**3.8.2. Review by: Frank J Swetz.**

*The Mathematics Teacher* **95** (2) (2002), 158-159.

In the sixth century B.C., a Greek engineer, Eupalinos of Megara, designed and built an aqueduct to supply water to the city of Samos on the island of Samos, the birthplace of Pythagoras. The remarkable feature of this aqueduct was that it contained a one-kilometre tunnel carved straight through a mountain. This tunnel marked the work of Eupalinos as an engineering marvel of the ancient world. This feat was accomplished by using the mathematics of similar triangles. 'The Tunnel of Samos', a teaching module consisting of a thirty-minute-long videotape and a sup porting Program Guide and Workbook, takes the viewers into a fascinating adventure that involves applied mathematics, engineering, and history. This set is one of the best that I have seen that uses history to teach mathematics. Innovative computer animation combines with good mathematics, on-site action filming, and excellent historical storytelling to produce a wonderful teaching resource. A problem is proposed: measurements must be taken through an obstacle. A solution is obtained: the properties of similar triangles are employed. Possible solution techniques are examined. Intrigue and drama are injected into the story when alternative theories are pro posed, for example, Did Eupalinos physically measure around the mountain or over the mountain? Site exploration, simple mathematics, and common sense sup ply the answer. I highly recommend this well-crafted, thought-provoking mathematics resource for classroom use and for teacher training.

**3.9. Early History of Mathematics.**

**3.9.1. Product description.**

This video traces some of the landmark developments in the early history of mathematics, from Babylonian calendars on clay tablets produced 5000 years ago, to the introduction of calculus in the seventeenth century.

**3.9.2. Review by: Frank J Swetz.**

*The Mathematics Teacher* **95** (2) (2002), 158-159.

The videotape is divided into ten sections, or slates, each devoted to specific aspects of the history of mathematics. After an introduction and a brief survey of mathematical events up to the seventeenth century, the units describe topics in or about numeration systems, number theory, the Pythagorean theorem, irrational numbers, pi, the evolution of trigonometry from astronomy, simple analytic geometry, and some fundamental calculus. Although the initial units do survey the history of mathematics, the remaining units focus on mathematical ideas and supply some historical context. ... This extremely rich teaching resource includes computer animations, action frames, and visual images of historical documents and artefacts, all of which make for a lively and engaging presentation. It is certainly well done. ... 'Early History of Mathematics' is an innovative use of technology and an excellent teaching resource. I recommend it highly for classroom use and for teacher training.

**3.9.3. Review by: Karen Michalowicz.**

*Convergence* (September 2008).

http://www.maa.org/press/periodicals/convergence/project-mathematics

I used the videotape with my students after reading a complimentary review (see review above). I had high expectations. However, I did not share the other reviewer's enthusiasm for the videotape. I first previewed the entire program. Then with the careful planning as recommended in the review, I used the videotape with my high school geometry classes and in a graduate mathematics methods class. ... My students found the portion of what I showed them on the Early History of Mathematics to be interesting. In particular, the computer enhancements used to demonstrate the proof of the Pythagorean Theorem were worth while and highly recommended. However, all my student viewers found the background music-track to be distracting. My adolescent honors geometry students began to play "Guess the Tune" during the playing. As a teacher, I believe there is a need for good history of mathematics videotapes. ... The Early History of Mathematics videotape ... covers a broad period of mathematics history too superficially. Thus, important events are not even mentioned. For example, when the videotape briefly addressed algebra, it did not refer to Diophantus (although he is mentioned in the Program Guide), nor Hypatia, or even Al-Khwarizmi. Rather, Omar Khayyam was the only mathematician mentioned. ... there were a few historical inaccuracies, for example, when our number system was described as "originating from the Arabs;" however, later in the tape it was referred to as the "Hindu-Arabic Numbers." In my opinion, the strength of the videotape was not in the history it presents, but in its computer animations. It can be a useful tool to the teacher if the teacher is knowledgeable about the mathematics history, and does very careful planning. I leave it to the viewer to decide how worthwhile it is in the classroom.