DIVINE PROPORTIONS

Louise Morley Cochrane '40 turns out to have inherited a taste for numbers as well as words. The daughter of writer Christopher Morley and granddaughter of mathematician Frank Morley, she majored in political science at Bryn Mawr and says that only Professor K. Laurence Stapleton coaxed her out of her fear of putting pen to paper. Moving to Great Britain after her marriage, she was an interviewer and broadcaster for the B.B.C., a writer of children's books and career novels for teen girls (including Anne in Electronics, 1960), a Fulbright commissioner, and chicken raising enthusiast.

In her golden years, she became interested in the lay Benedictine, Adelard of Bath, who introduced Hindu/Arabic numerals and the concept of zero to the Western world. Her 1994 book, Adelard of Bath:The First English Scientist, was the first on the subject for the general reader in English. While traveling in the Middle East during the Crusades, Adelard obtained a manuscript of Euclid's Elements, which he translated from Arabic into Latin. Later, as a member of the court of Henry I, he collaborated on a translation of the Zij or Astronomical Tables of Al-Khwarismi, a 9th-century Arab astronomer and mathematician who gave his name to the words algorithm and algebra. Adelard's most important work, on the astrolabe, was written for the future Henry II whom he tutored as a boy of 12.

As part of the Class of 1940's millennium contribution to the College at last year's Reunion, Cochrane carried with her from Edinburgh a collection of brass and steel mathematical models used as teaching apparatus by her grandfather, first at Haverford College and then at The Johns Hopkins University from 1886 to 1937. In an essay, "Some Call It Serendipity," written at the request of her classmates, she tells the story of her grandfather, author of Morley's Theorem, one of the most sparkling gems in all of elementary geometry, and the return of his models to the bi-college community.

By Louise Morley Cochrane '40

The Millennium Reunion was an evocative experience for all who attended. We were very privileged and all are grateful. Memorable indeed for the Class of 1940 was that great gesture on the part of the Class of 1995, producing May baskets of flowers for us, still growing!

On Sunday morning, our class President Marian Kirk Appel was first to speak in the class gift rollcall. She remembered May 1940, sitting with friends, on a third storey window ledge in Rhoads, facing outwards, smoking feet in the gutter, and discussing the Nature of Reality. "Here we are now," she continued, " at our Sixtieth Reunion. We've been sitting in Rhoads, in armchairs, our feet firmly on the air conditioned floor, discussing the Nature of Reality, only," she paused, "It's virtual Reality." There was a roar of laughter and tumultuous applause. We all know what she was talking about. Nowadays Virtual Reality is the picture we see on a television screen, multi-dimensional on a slightly curved surface. Symbolically, however, there are many ramifications.

On the previous afternoon, I had presented to our Class a gift which would become part of our millennium contribution to the College. I had carried it with me from Edinburgh. It was a collection of brass and steel mathematical "models' which represented the teaching apparatus of my grandfather, Professor Frank Morley, first at Haverford and then at Johns Hopkins Unversity from 1886-1937. He had three sons, all Haverford alumni. Two Bryn Mawr granddaughters and one Haverford grandson established a continuing tradition. All the living grandchildren agreed that the models would make an admirable millennium gift, and as the member of a reunion class, I was authorized to arrange it with Bryn Mawr. As a result, the models are to be available in the Math Room of the Science Building, which our class had helped to endow in memory of Judy Martin Cheever.

As I went through a security check at Heathrow Airport, I groped desperately for an explanation of what all these queer metal objects were. One by one they had to be taken from their bubble wrapping. I had a flash of inspiration. 'They're what mathematicians used in the old days before Virtual Reality on television,' I explained. With a somewhat baffled look, the official waved me on. I was very relieved later to have Professor Victor Donnay present me with the Bryn Mawr math video on minimal surfaces as modern analogue' of my grandfather's work.

I had already been informed by Dr. Eric Pumroy, in charge of the Library Speical Collections, and Professor Donnay that the models would be a welcome gift. Included with them are the associated papers and offprints. These are to be held in the Morley Archive at Haverford. Thanks to the generosity of Steven Rothman, a Philadelphia book collector who is a member of both Library Boards, duplicate copies of Morley offprints can be kept with the models.

I had for many years wanted to mark in some way the early association of my grandparents, Frank and Lilian Morley, with both Bryn Mawr and Haverford. They began their married life in 1887 in Low Buildings, which was near the site of the present science complex. This was when my grandfather, an English Quaker, had been appointed to a professorship at Haverford and could afford to marry. My grandfather must have walked meditatively many times where the mathematical labyrinth is now planted. My grandmother was able to 'audit' lectures in Taylor Hall and left my father, her firstborn infant son, Christopher, in Merion with students while she did so. Merion, not surprisingly, was later my choice of Hall of Residence.

Most fortuitously this collection of models was discovered last summer when I began planning to attend Reunion. The shared interests of Professors Morley of Haverford and Harkness of Bryn Mawr, both former Cambridge 'wranglers," (indicating distinction in mathematics) produced the book entitled The Theory of Functions in 1893. The collaboration of the two mathematics departments has therefore had a long history of seminal scholarship.

There has recently been renewed interest in the Morley theorem on triangles, formally published in 1923 but first discovered at the turn of the century. My grandfather waited more than 20 years before publishing it, to be sure no one else had produced it. There have been a number of subsequent published proofs, one in 1953 by Roger Penrose, the distinguished Oxford mathematician now Emeritus professor Sir Roger Penrose, OM.

The theorem as stated is quite straightforward. If you trisect the three angles of any triangle, the outer trisectors intersect to make the vertices of an equilateral triangle. The theorem provides an extremely useful demonstration of the pattern formed by the 3, 4, 5 triangle and the 'divine proportion.' It therefore has implications not only in mathematics and science but also in art, architecture and music. Those who are fortunate enough to have mathematical genes seem to be aware of this pattern intuitively. The Morley theorem provides a good rule of thumb to help those who have not. (For a geometric virtual device that demonstrates the theorem and further discussion, see http://members.aol.com/Windmill96/morley/morley.html and http://www.geocities.com/poetsoutback/morley.html)

It has only been in later life that I really made the effort to understand how it worked, and the significance of its relationship in the development of non-Euclidean geometry. It was extremely useful for my research on Adelard of Bath, a 12th-century mathematician. It helped me to appreciate how the planispheric astrolabe could be used as both calendar and clock as well as a primitive computer. It has also made me extremely aware of the mathematical imagery in my father's writing, imagery of which I had been previously unaware.

In the early twenties my father wrote two best-selling novels which explored the time dimension. For Where the Blue Begins, he deliberately made a special study of spherical trigonometry. In Thunder on the Left, the hero projects himself into his own adult world. The manscript/typescript from which Thunder on the Left was published is in the Canaday Library, the gift of an alumna from Hawaii.

Just before my trip to Bryn Mawr I was idly checking through the bibliography of my father's writings with a view to illustrating this imagery. I came across a reference to a short book I'd never heard of called When We Speak of a Tenth, privately published by the Hamilton Watch Company in Lancaster, Pennsylvania in 1931, a limited edition. I hope to procure a copy for the Morley alcove in the Haverford Library, which is a memorial to my father, endowed by the family of Page and Mary Allinson and members of his class of 1910. The title of this book is a quote from the engineer who showed my father round the watch factory. While he was studying the design of a watch magnified 500 times, he was told that the tenth of which the engineer spoke was a tenth of a thousandth. That was the tolerance to which they were working. My father recalls:

"I once found a fleet of baby starfish, each the size of a pinhead, in a clump of seaweed. The tiny cog-wheels of a watch remind me of them, but more perfect, each little tooth true to a tenth."

The image of tiny five-pointed starfish was important to my father. He had used it previously in his semi-autobiography, John Mistletoe.

"There is some notable virtue in owning a small frontage on actual tide-water, because to feel some proprietary right in the perpetual movement of tides is to put one in relation with huge things. The whole turn and tension of the cosmos is there on your own shore; and that of itself is enough to keep you aware of the enormity. On this drowsy afternoon, white the family sprawled on the sand or capered among boulders, he was thigh-deep in warm golden water, pulling up masses of seaweed…. Those masses of seaweed were full of innumerable small five-pointed stares. September is kindergarten time among the echinoderms, and every tress of seaweed carries in it dozens of baby starfish, perhaps a quarter of an inch across. He could not help believing that there was considerable meaning in this - they were like tiny pentameter epigrams." Passages like this reflect my father's awareness of the mathematical world his father inhabited but adapted to his own purposes. He sees a starfish and its shape take poetic form.

One of my father's greatest friends was Buckminster Fuller, best known for the design of the geodesic dome. Having just completed a sonnet, my father once remarked that a sonnet was to him what a logarithmic spiral was to Bucky. Towards the end of his life, Bucky drew diagrams for me to show the relationship between his geodesic design and my grandfather's theorem. It is not generally realized that Bucky Fuller first demonstrated his geodesic map projection at Haverford in October 1942, when my uncle Felix Morley was the President there. Bucky had been an admirer of my father's writing for some years before they met in 1933. He told me he had found in my father's writing of confirmation of his own mathematical ideas.

At or meeting, my classmates were enthusiastic and supportive about the models, delighted to hear about current website pages which explain the Morley theorem and to know they would prove useful to new generation of Bryn Mawr mathematicians. There are new applications which my grandfather could never have dreamt of, for example, the design of geometric folding for delicate materials which can enfold satellites in Space. Of particular interest to my classmates was the relationship to perspective that the models reveal and the images that are conjured by simple geometric relationship on a plane surface. My father's use of mathematical imagery conveyed this for those who found the mathematical formulae difficult to understand.

One of the great joys of Reunion is taking up conversations with friends where we left off five or even 10 years ago. One of the exciting aspects is to discover the new interests which we pursue in what used to be called 'declining years.' Mine has obviously been mathematics. Isota Epes Tucker and I had a spiralling conversation about her latter day career as an artist, stimulated by Sheila Isham's brilliant exhibitions of painting in the Library. Both Isota and Sheila had been influenced by Anna Ahkmatova's poetry. Subsequently, when Isota looked at my grandfather's models, she instantly thought of perspectives in modern art. When Victor Donnay looked at them, he reflected on his mother's early appreciation of Max Escher.

These models had been carefully preserved by the youngest of the three Morley brothers, F.V. Morley who worked with his father on mathematics as well as being a director of Faber and Faber and an associate of T.S. Eliot. He did so with the thought that at some time in the future, a new generation might find them suggestive. There could be no place where this was more likely to happen than at either or both of the two institutions which have had such a formative role in the lives of the family members.

There is always serendipity at Bryn Mawr and Haverford reunions. Read more about Reunions 2000 and 2001.

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