### Wooden model produced by W. W. Ross to illustrate the Pythagorean Theorem

Chapter 4 of my book, Dissections: Plane & Fancy, starts with a description of several dissections of squares which illustrate the Pythagorean theorem. Recently, I was excited to learn about a wooden model in the collection of the National Museum of American History that illustrates two of these dissections. The model was produced around 1890 by W. W. Ross, a superintendant of schools in Fremont, Ohio. The photograph of it below appears in the on-line version of the exhibition Slates, Slide Rules, And Software: Teaching Math in America. (The exhibit is now closed.)

The model as shown in the photograph illustrates two dissections simultaneously. In the center is a right triangle whose sides are in the ratio of 3:4:5. On its upper right are three pieces that form a 5-square. The three pieces can be rearranged to form a figure congruent to a 4-square with a 3-square attached to it. Such a 3-piece dissection was suggested by a diagram in a book by Johann Sturm in 1700.
The second dissection is a 5-piece dissection by Henry Perigal (1873). In the photograph, we see the uncut 3-square to the right of the right triangle. Below the right triangle are the four identical pieces that form the 4-square.
A close examination of the photograph reveals that the pieces are held together by dowels that fit into holes on the sides of the pieces. Thus the 3-square has two holes on each side, into which the dowels from the four identical pieces fit, producing the 5-square in Perigal's dissection. The three pieces from Sturm's dissection are similarly held together when forming the attached 3- and 4-squares.
A nifty feature of both dissections is that the pieces can be moved from one figure to another without rotating them. Ross's model might seem to emphasize this feature by preserving the grid pattern while moving from one figure to the other. However, there are two identical pieces in Sturm's dissection and four identical pieces in Perigal's dissection. In both dissections, the grid pattern is the same for identical pieces. Thus pieces may be interchanged, effectively rotating them by multiples of 90 degrees, and still preserving the grid pattern.
The correctness of Ross's model does not depend on the fact that the sides of the squares are the very special Pythagorean triple (3,4,5). Actually, one could take advantage of that fact to produce a 4-piece model of a dissection of a 3-square and a 4-square to make a 5-square. In chapter 7, I identify two infinite families of Pythagorean triples for which 4-piece dissections are possible, and also give different dissection methods for them.

### References

• Peggy Kidwell, "American Mathematics Viewed Objectively: The Case of Geometric Models", in Vita Mathematics, ed. Ronald Calinger, The Mathematical Association of America, Washington, D. C., 1996.

• William Mueller, Mathematical Teaching Tools in the Department of Mathematics, University of Arizona, 1998.

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Last updated November 27, 2006.