Carl B. Boyer

The Foremost Textbook of Modern Times

(1951). The Foremost Textbook of Modern Times. The American Mathematical Monthly: Vol. 58, No. 4, pp. 223-226.

The History of the Calculus.

All too often, the history of a field is little known to its present practitioners. Therefore, we hope this section will provide information that is both useful and entertaining to those concerned with the two-year college.We feel especially fortunate to be able to reprint in its entirety Chapter 7 of the NCTM Yearbook, by Carl B. Boyer, since it so nicely exemplifies these criteria. In the future we hope to attract other notable contributions and establish this section as “must” reading for our subscribers.

The Tertiary Rainbow: An Historical Account

IT is customary to speak of the rainbow, even though no two observers see precisely the same bow, and in spite of the fact that a single observer may simultaneously behold more than one arc. That the so-called primary rainbow often is accompanied by an exterior concentric bow must have been known to very early man, and Aristotle referred to the secondary rainbow as though its existence were common knowledge. What could be more natural, then, than to inquire whether a third bow is possible? Aristotle, posing this problem in the Meteorologica, answered in the negative:

A History of Mathematics.

Origins. Egypt. Mesopotamia. Ionia and the Pythagoreans. The Heroic Age. The Age of Plato and Aristotle. Euclid of Alexandria. Archimedes of Syracuse. Apollonius of Perga. Greek Trigonometry and Mensuration. Revival and Decline of Greek Mathematics. China and India. The Arabic Hegemony. Europe in the Middle Ages. The Renaissance. Prelude to Modern Mathematics. The Time of Fermat and Descartes. A Transitional Period. Newton and Leibniz. The Bernoulli Era. The Age of Euler. Mathematicians of the French Revolution. The Time of Gauss and Cauchy. Geometry. Analysis. Algebra. Poincar? and Hilbert. Aspects of the Twentieth Century. References. General Bibliography. Appendix. Index.

Refraction and the Rainbow in Antiquity

T HE earliest theories of the rainbow and the halo ascribed these phenomena to the reflection of light (or of visual rays) by a vaporous medium. The definitive form of such views was given in Aristotles Meteorologica, a work which virtually dominated thought on the subject for about two millennia. In the thirteenth century, nevertheless, rival theories of the rainbow attributed it in part to refraction. The introduction of refraction into the explanation of the rainbow generally has been ascribed to Witelo; but Robert Grosseteste a generation before had adopted a refraction theory in his De iride.1 It is the purpose of this note to point out that Grosseteste himself seems to have been anticipated in this connection, for it would appear that the ascription of the rainbow to refraction is of ancient, rather than medieval, origin. Dozens of medieval Latin paraphrases of the Meteorologica are extant, but from the earlier Greek period there are only two of significance. The first of these two surviving Greek commentaries was by Alexander of Aphrodisias,2 the Athenian philosopher, who became the head of the Lyceum between 198 and 21 IA.D. Perhaps the greatest of all Aristotelian commentators, Alexander wished to purge the school of rival influences. In this connection he sought to free the orthodox Peripatetic theories from errors which had arisen. Among the heterodox theories which Alexander noted was one concerning the halo (or corona). After explaining the position of Aristotle and Posidonius that the halo is caused by reflection in a medium in which the minuteness of the reflecting surfaces results in the production of color rather than form, Alexander reported that most others ascribed it instead to refraction.

William Gilbert on the Rainbow

A qualitatively satisfactory explanation of the rainbow was given shortly before 1311 by Theodoric of Freiberg and Qutb al-dīn al-Shīrāzī, but this work had disappeared shortly before Gilbert was born. The quasi-Aristotelian theory which Gilbert espoused was akin to early medieval ideas according to which rays from the sun are tinged as they traverse a thin dewy vapor and then are reflected to the eye of an observer by a dark cloud or dense object. Gilbert had the happy thought that a spherical magnet might serve as a miniature earth, but he missed entirely the notion of a globe of water as a magnified raindrop. Unaware of the role of the spherical drops in the refraction of solar rays, the fuzzy speculations on the rainbow which Gilbert gave in De Mundo stand in marked contrast to the experimental philosophy which he advocated in De Magnete.

Essay Review: The Making of a Mathematician: The Mathematical Papers of Isaac Newton

Newtonian studies do not constitute a new field, and it probably is safe to say that more has been written about Newton and his work than about the life and work of any other scientist. If, however, one were to divide historians of science into two procrustean categories-(r) the frontiersmen who are directly involved in the study and publication of the primary sources, and (2) the practitioners who comment upon and interpret the findings of the frontiersmen-one would find over the past two and a half centuries a heavy preponderance, among Newtonian authors, of those in the second category. This is not unnatural in the world of scholarship, inasmuch as primary materials are accessible to the small number of research workers for whom time and space are available, and who are equipped by linguistic, paleographic, scientific, and historical training to profit most from analysis of the extant documents. The miracles of photoduplication today have vastly expanded the availability of precious primary materials; but persistence and stamina, as well as training and extended periods of time, nevertheless remain essential for effective frontiersmanship. In the history of medieval science this was made abundantly clear during our century by such scholars as Pierre Duhem, Lynn Thorndike, Anneliese Maier, and Marshall Clagett; and for pre-Hellenic science a similar conclusion is inferred concerning the researches of Francois Thureau-Dangin and Otto Neugebauer, among others. Through the efforts of such frontiersmen the accounts of preHellenic and medieval science accepted today differ toto caelo from pictures which had been presented less than a century ago. Cuneiform tablets and medieval manuscripts continue to provide frontiers for diligent scholars, but they no longer are the uncharted challenges that they once were. It may appear inappropriate to compare the frontier in Newtonian research with those in ancient and medieval science, for in this case the body of primary materials has been nearer at hand for two and a half centuries. True, it has not always been readily accessible, for the bulk of Newtons papers remained within the Conduitt family for well over a century; but as far back as r872 the Earl of Portsmouth had entrusted the collection to the University of Cambridge where a substantial portion of the papers have since been available for study. The accessibility of these primary sources notwithstanding, most papers and books concerning

Eighty-First Critical Bibliography of The History of Science and Its Cultural Influences (To 1 January 1956)

The divisions and subdivisions of each of these main sections are listed below in the Table of