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Alfonso's Rectifying the Curved

Ruth Glasner and Avinoam Baraness
Publication Date: 
Number of Pages: 
Sources and Studies in the History of Mathematics and Physical Sciences
[Reviewed by
Victor J. Katz
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It is well-known in the history of mathematics that similar ideas have appeared at different times and places, often separated by hundreds of years and thousands of miles. For example, we know that the Pythagorean Theorem appeared in Mesopotamian clay tablets well over a thousand years before the time of Pythagoras. The Euclidean algorithm for determining the greatest common divisor was used in both India and China for solving simultaneous congruences. The recreational problem of three men buying a horse appears abstractly in the work of Diophantus in the third century CE, but also in the work of Levi ben Gerson in fourteenth century France, while as a problem involving men and horses in such places as Leonardo of Pisa’s thirteenth century Liber Abbaci, around 1300 in an Arabic work in North Africa, and again in 1500 in a Hebrew book in Constantinople. And mathematicians in Kerala, in the south of India, devised power series for the sine, cosine, and other trigonometric functions hundreds of years before they appeared in early calculus works in Europe. The question a historian often poses when confronted with such a situation was whether there was any transmission across the centuries and the miles. In many such cases, it is virtually impossible to establish any transmission, partially because no documentation is available and also because writers in the past frequently failed to name their sources. Still, the similarities in the cases often cause one to wonder whether perhaps the ideas traveled orally — without there being any records.
Two new situations of similar ideas occurring appear in the book under review, a translation and commentary on a fourteenth-century Hebrew treatise by the Spanish mathematician and philosopher Alfonso of Valladolid (c. 1260–1347), known as Abner of Burgos before he converted to Christianity around 1320.  The treatise, The Rectifying of the Curved is known through only one copy, made in Mantua in the fifteenth century and owned originally by Mordechai Finzi, a Jewish scientist who was at the center of mathematical and astronomical activity in Mantua and had both Jewish and Christian scientific colleagues. The aim of Alfonso’s text was to“inquire whether the existence of a rectilinear area equal to a circular area can be truly established, ..., by way of true decisive demonstration, free of any approximation or sophistry.” Although Alfonso realized that one cannot establish equality between a rectilinear area and a circular area by Euclid’s method of superposing parts of one on parts of the other, he knew that there were other ways of establishing equality. For example, he noted that Hippocrates in his quadrature of lunes had stated that in a right triangle, the semicircle on the hypotenuse equals the sum of the semicircles on the sides. Also, Archimedes had proved that the surface of a sphere is equal to four times a great circle on the sphere. In neither case could one prove these results by superposition.
Unfortunately, we do not know how Alfonso intended to answer his “inquiry.” The manuscript was supposed to have five parts, where the fifth part was to answer the question. But only three parts and a few lines of the fourth are extant. Nevertheless, there is much of interest in the manuscript, particularly in part three, which is a collection of geometrical theorems, presumably all having some relevance to Alfonso’s quest. But the most fascinating of the results are proposition 33 and the set consisting of propositions 29-32.  Proposition 33 is not strictly geometrical, but deals with a question coming from astronomy as well as from some ideas of Aristotle. In that proposition, Alfonso describes how one can have a “continuous perpetual linear motion forth and back on a finite line without a state of rest between forth motion and back motion.”  Aristotle had claimed this was impossible, but Alfonso constructed this motion by the use of a mathematical device invented by Islamic astronomers at Marāgha in the thirteenth century under the leadership of Nasir al-Dīn al-T. ūsı̄. These astronomers felt that Ptolemy had violated the principle that heavenly motion must be composed of uniform motion of circles about their centers. Among their methods of “correcting” Ptolemy was the use of the Tūsı̄ couple, a construction in which the radius of the epicycle of a planet is half that of the deferent and its angular velocity is twice that of the deferent in the opposite direction. The resulting motion of the planet is then linear, back and forth along the diameter of the deferent, thus showing that a linear back and forth motion can be composed of circular motions without any “state of rest.” It turns out that Copernicus, too, also rejecting some of Ptolemy’s ideas, used the Tūsı̄ couple in his own De Revolutionibus to help return astronomy to its “proper roots.” For years now, historians have debated whether or how Copernicus learned of this mathematical device coming out of Persia. Although, we still cannot answer the question, now that we see the idea in a fifteenth-century manuscript available in Italy, it becomes more possible for us to believe that somehow Copernicus learned of this, probably orally, during his studies in Italy.
In propositions 29-32, Alfonso described and used the curve usually called the “conchoid of Nicomedes.”  Although we do not have any actual text of Nicomedes, who lived in the third century BCE, there are detailed reports of his use of the curve in both Pappus’s Collection and in Eutocius’s Commentary on Archimedes’ Sphere and Cylinder II. What is this curve? Given a straight line AB (the ruler), a point P outside it (the pole), and a distance d, the conchoid is the locus of all points lying at distance d from the ruler AB along the segment that connects them to the pole P . In modern symbolism, if P is the origin and AB is the line \( y = a \), then the curve is defined by the polar equation \( r = a \sec \theta + d \). Nicomedes used this curve in his solution of the two classical problems of trisecting an arbitrary angle and finding two mean proportionals between two given lines. Now as far as we know, neither Pappus’s nor Eutocius’s work was available to Alfonso, either in Greek or in Arabic. And until the discovery of Alfonso’s manuscript, it was believed that the conchoid was first mentioned in Europe after late antiquity only in seventeenth-century works of VièteDescartes, Newton, and others. But Alfonso himself used the conchoid to solve the same two problems.  And although his solution of the trisection problem was very similar to that of Nicomedes, his solution of the mean proportional problem was new. So the questions remain for the conchoid as well as for the Tūsı couple: how did Alfonso learn of these ideas and did his work somehow get transmitted to others in western Europe?
Of course, there is much more in Alfonso’s work than the two situations just described. Part one contains some philosophical ideas relating to determining the equality between rectilinear and circular areas. Part two contains Alfonso’s criticism of some earlier attempts to prove Euclid’s parallel postulate as well as his own attempt to do so. It also contains Alfonso’s discussion of the paradox of the rotating wheel: Suppose a wheel rolls along a track while a smaller wheel, attached to the first wheel at its center, rolls along another track above the first one. It would seem that both wheels would complete one revolution in the same time, hence the distance rolled along the two tracks, which should in each case be equal to the circumference of the wheel, would be equal, contradicting the fact that the smaller wheel has a smaller circumference than the larger wheel. Part three contains numerous theorems in plane geometry, mostly dealing with the relations between circles and inscribed or circumscribed polygons. There is also a detailed discussion of Euclid’s theory of proportions, including ideas on compound ratios. But in the end, we are left hanging, wishing we could know how Alfonso was going to finish his book and answer his original question.
Glasner and Baraness do an excellent job in not only translating Alfonso’s work but in explaining some of his technical terms, dealing with the history of some of his ideas, and, in general, putting Alfonso into the context of late medieval mathematics in Spain. As would be expected of a mathematician in Spain at that time, Alfonso knew the work of many Islamic philosophers and mathematicians, including al-Nayriziibn al-Haytham, ibn Sīnā, and ibn Rushd, as well as astronomers connected with the school of Marāgha.  On the other hand, he only mentions by name one Jewish author and one Christian author. Yet some of his ideas seem to have been influenced by work done by some Scholastic writers in western Europe during his time. Whether we will ever learn of any direct influence of Alfonso on later authors is, however, a question for the future to answer.
Victor J. Katz is Professor Emeritus of Mathematics at the University of the District of Columbia.  The third edition of his A History of Mathematics: An Introduction, appeared in 2008. He is the editor of The Mathematics of Egypt, Mesopotamia, China, India, and Islam:  A Sourcebook, Sourcebook in the Mathematics of Medieval Europe and North Africa, and three MAA books dealing with the use of the history of mathematics in teaching as well as two collections of historical articles taken from MAA journals. He directed two projects that helped college teachers and secondary teachers learn the history of mathematics and its use in teaching. Professor Katz was also the founding editor of Convergence, the MAA’s online history of mathematics magazine.