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Isaac Barrow

"Barrow completed a major phase of the Keplerian revolution in geometrical optics by creating a mathematical theory of optical imagery ... After more than a half a century, Barrow was the first to publish a complete solution to the problem bequeathed by Kepler, namely to find the focal and image points for all species of lens." Alan Shapiro

The first holder of the Lucasian Chair was Isaac Barrow (1630-1677), born in London. His life's goal was theology, but his path lay through science and mathematics. He was accomplished in the fields of mathematics, theology, physics and the classics.1 Barrow provided a very appropriate transition from the world of classical education that was Cambridge to the new world of mathematical understanding that was to come.

Isaac Barrow first came to Cambridge as a student attending Trinity College in 1646. Arriving in the midst of religious controversy, he began his studies just after the ejection of the majority of fellows and their replacement by loyalists.2 In 1649, Barrow was elected a fellow, then got into some mild difficulties over issues of loyalty, but was protected from expulsion by a master. He received his M.A. from Cambridge in 1652 and another from Oxford in 1653. In 1661 he obtained a second bachelor's degree in divinity. The degree of DD was mandated for him in 1670. He was appointed master of Trinity College in 1672, and was vice-chancellor of the university from 1675 to 1676.3

Later, he ran into more difficulty over loyalty and was encouraged by the same master to spend a few years, starting in 1655, on the Grand Tour of Europe. During his travels, he spent time in Paris, Italy, Turkey, Germany and Holland.4 Eventually, the fuss quieted down, so upon his return to Trinity, he was able to progress in rank. He also took holy orders at that time, only to renounce them shortly afterwards.

Barrow's original appointment was as Regius Professor of Greek. This was an appropriate appointment, considering that the classics were the source of mathematical knowledge at the time. He was able to master the ancient Greek texts of Euclid, from which he produced the authoritative texts for university study. In addition to Greek, he was also fluent in Italian, French and, of course, Latin.5

Barrow was also appointed as the Gresham Professor of Geometry in 1662, which carried little in the way of responsibilities, so it did not interfere with his other activities.6 The Royal Society elected him as a fellow in 1662, but he was rarely involved with its a activities.7

Barrow was not as interested in publishing as one might expect. His scientific publications were only compilations of his lectures, and were the result of efforts by others who strongly urged him to publish. His religious works were published posthumously in several volumes. Only two of his sermons had been published before his death.8 His science lectures as Lucasian Professor were mainly about optics and geometry. These lectures drew more students than his earlier lectures as the Regius Professor of Greek.

The general consensus at the time was that Barrow was an outstanding, well respected intellectual, most certainly an appropriate choice as first Lucasian Professor. However he did not produce the revolution of his successor, Isaac Newton. It appears that his real calling was to theology and not mathematics, in spite of his abilities. He was concerned with his students and teaching, often dedicating books to students, mentioning the importance of teaching in public lectures, and producing a textbook on Euclid that was the standard for many years. "Barrow's Euclid was used as standard textbook for at least a half century and its heavy use may account for the difficulty of establishing precisely the number of reprints."9 He helped to establish the tradition of excellence in mathematics at Cambridge. He was appointed college preacher in 1671.10

In 1673, after resigning the Lucasian Professorship, Barrow was appointed master of Trinity College, where he remained until his death in 1677, at the early age of forty-seven.

Although he was a very religious man, he received a special dispensation from the requirement that all faculty be ordained ministers11 , which started a tradition of the Lucasian chair holders being relieved from the normal requirements.

When Barrow became Lucasian professor, he insisted on the requirement that he and his successors were to leave the university ten written lectures per year.12 "... Barrow was required to lecture once a week during the term and to submit annually at least ten of these lectures to the vice-chancellor for deposit in the university library for public use."13

Barrow's main work was his Lectiones opticae et geometricae, published in 1669. The fundamental theorem of calculus is the recognition that differentiation and integration are the inverse of each other. This is a key realization in mathematics and it was provided by Barrow in his Lectiones.14 It is a critical building block in the development of calculus, which was later formalized by Newton.

Eves argues that Barrow offered an early insight into calculus, the process of differentiation. He demonstrated how to find the tangent of a curve using the method of the differential triangle, although he used different terminology from the dx/dy that we use today and his methods were not as refined, but the fundamental methodology was clearly present. Struik points out that Barrow's methods are geometric, making it difficult to see the calculus in his work without some modernization.15 Child has also written convincingly that Barrow's writings contain a complete calculus if the insights are drawn out and the methods of geometry are modernized. He points out, along with others, that Barrow's self limitation to geometry prevented him from fully developing the calculus.16

It is a pity that Barrow did not make the leap from his geometrical techniques to the symbolic techniques of algebra and calculus. He gave credit to James Gregory for the fundamental theorem although each had independently discovered it. The geometric approach is different enough from the calculation methods that we can say that Barrow did not invent calculus, but he certainly discovered basic techniques of calculus.17

Use of the geometrical approach had strong proponents in Barrow's time. Because geometry represented real, physical things, the empiricists were satisfied in a philosophical sense. When it came to arithmetic, one could not point to numbers as real objects, so philosophically, arithmetic was thought to be on a lower level than geometry, and of considerably less value. This attitude affected the use of symbolic reasoning, slowing down progress in the development of algebra, because algebra involves the manipulation of symbols.18

Although Barrow was a good scholar, he was still a product of the late seventeenth century. Mathematics still had a long way to go toward developing the subdisciplines. Arithmetic and geometry were the most advanced of the subdisciplines, and even "... Barrow explicitly denied arithmetic an existence independent of geometry... and when discussing the new algebra, confused it with the methods of analysis and dismissed it as a part of logic."19 He would be amazed at the state of mathematics today, or even the path of development in the eighteenth century. He saw geometry as a pinnacle instead of one of many aspects of mathematics: "... for Barrow, geometry was the science of universal ideas which were derived by the mind from sensible objects and which ... were possibly little or no more than particular objects `understood universally.' "20

It was a difficult leap at the time to work with reasoning at very abstract levels, especially reasoning involving numbers. Barrow "... denied numbers any real independent existence and presented them as instead as mere `notes' or `signs' of magnitude."21 The ability to differentiate operations from the numbers, an important part of algebra, was still in the future. "If arithmetic emerged subservient to geometry in Barrow's scheme of mathematics, algebra fared worse. Barrow generally ignored algebra and in his few references to the subject, refused to recognize it as a science and presented it instead as an instrument of logic."22

Geometry was Barrow's mathematical speciality, which he applied successfully to optics. His general approach to mathematics was good, but his work showed the limitations of the times. Barrow, unlike most of those in the seventeenth century concerned himself with only the mathematics of optics. He ignored the real experience, applied optics, as a true mathematician would do. Nevertheless he achieved wonderful results. Barrow's achievement was the development of a method which showed where any image would be located on any surface, plane or curved, once it had been reflected or refracted. The modern theory of image formation in spherical mirrors is essentially Barrow's creation, although he never made such mirrors himself.23

Barrow believed in a mechanical theory of light, but he also realized that light had characteristics of both particles and waves. He believed that each idea had its problems, so that both were required to explain light's behavior. Barrow was the first to develop the concept of oblique pencils to describe the path of light. "According to Barrow an image of a point is located at the place from which a pencil of reflected or refracted rays entering an eye diverge." He used Kepler's approach, but he produced a mathematical theory to explain it. Barrow even wrote that "... he was the first one in print to enter the foundation of dioptrics on the sine law of refraction," which is a true statement.24 With this as a starting point Barrow "... created the mathematical foundation of general theory of optical imagery, and with his concept of oblique pencils of rays, began the exact study of astigmatism and caustics."25

In 1801 Thomas Young made the first contribution to the study of astigmatism, following Robert Smith's book Compleat System of Optiks in 1738. Young recognized that image points were in fact lines. Because he failed to cite Barrow, citing Newton's work, Lectiones XVIII instead, Barrow's contributions to the study of astigmatism remained largely unrecognized until P. Colman called attention to Barrow's work in Ueber den Astigmatismus (1904). (see Historische Notizen. B., 1904)

There has been some controversy over the influence of Barrow on Newton, but for the most part, authors agree that Barrow knew Newton as a student and certainly after that. Barrow's influence can be seen in Newton's Optical lectures , in which he fully adopts Barrow's approach and further develops his solutions. Newton even used the concept of the pencil, extending it to three dimensions. Because Newton took existing knowledge in fundamental areas to new heights, it is not surprising that he completed "... the erection of the mathematical foundation, which was to languish until the beginning of the nineteenth century."26 It is another, although unfortunate, truth that Barrow and Newton established a marvelous foundation that was not built upon for a long time by mathematicians at Cambridge or even in England. Mathematicians working on the Continent put their discoveries to far better use.

Barrow's main scientific contributions were in geometry and optics, but he had always wanted to be a theologian, a goal he achieved. He used mathematics and science as gateways to religion. He saw that in order to understand the Bible, he needed to understand time and astronomy. He also saw a need to understand nature and therefore he needed to learn chemistry. He traveled along a long path of human knowledge to reach divine knowledge. It was not uncommon during his time for men to use science to explain or support religious beliefs, he was just one of the best.

The religious beliefs of Barrow were Anglican, Trinitarian, anti-Calvinist, and with a Royalist's coloring. When he left the Lucasian Professorship, it was with the idea of devoting himself to his religious studies. He wrote extensively about his religious beliefs, which helps us to understand what he believed, but at the same time, one can find support in his writings for many ideas that were accepted by groups that he did not follow. He was very much a man of his times when it came to religion. Faith was grounded in reason for Barrow. He relied on reason to prove the existence of God, but then looked to faith for elements in it like the trinity, Christ's resurrection, and ascension. "... [H]is theological position is that asserted... by the Anglican Church from Elizabethan days to his own, and his style, when minutely analysed, conforms readily to the reforming ideals of his time."27 Barrow was a staunch defender of the established Church and argued against Rome in the Treatise of Pope's Supremacy. Even King Charles II chose Barrow as one of his chaplains. Barrow supported an episcopal system of government in the Church and believed in obedience to our spiritual guides and governors, and supported tithing.

Known as one of England's great pulpit orators, most of Barrow's religious works were sermons. "Before Burke, he is one of the great orators produced by England."28 His sermons were considered long even by standards of his own time, on occasion lasting for many hours. Barrow exemplified the new trend in pulpit oratory that prevailed among Anglican divines after the Restoration. Most of his sermons were delivered in the chapel of Trinity College, but some were given at Westminster Abbey.

Barrow's beliefs were simplistic. He believed that the plain meaning of the scripture is the true meaning, believing the literal scripture when it stated God created the universe out of nothing by his mere will and command. Barrow showed his Anglican side, by believing in the visible church where all who believe faithfully in the promises of God and live accordingly may be saved. He rejected the Puritan teaching of assurance of election. He rejected Calvin's teachings, and did not seek any accommodation with Dissenters, thus showing his Royalist leanings.

Barrow thought that Lutherans were incorrect in their belief in consubstantiation and that Catholics were incorrect in the belief in transubstantiation. For him Christ was not corporeally present on earth, though everywhere present in his divinity. Furthermore he believed that Mary should not be worshipped, just respected. These views are consistent with the views of ministers in England at the time. He was a man of peace in an age that was given to animosity and censuring.

Footnotes

  1. Howard Eves, An Introduction to the History of Mathematics (New York: Holt, Rinehart and Winston, 1969), 329.
  2. Mordechai Feingold, "Barrow: Divine, Scholar, and Mathematician," in Before Newton: The Life and Times of Isaac Barrow, ed. Mordechai Feingold (Cambridge: Cambridge University Press, 1990), 5.
  3. John Ward, The Lives of the Professors of Gresham College (London: John Moore, 1740), 162.
  4. Dictionary of National Biography (London: Oxford University Press, 1882).
  5. Feingold, "Isaac Barrow," 17.
  6. Feingold, "Isaac Barrow," 61.
  7. Feingold, "Isaac Barrow," 62.
  8. Feingold, "Isaac Barrow," 90.
  9. Diana Simpkins, "Early Editions of Euclid in England," Annals of Science 22 (December 1966): 237-243.
  10. Irene Simon,"The Preacher," in Before Newton: The Life and Times of Isaac Barrow, ed. Mordechai Feingold (Cambridge: Cambridge University Press, 1990), 305.
  11. Feingold 1990.
  12. Ward 1740 161.
  13. Alan Shapiro, "The optical lectures and foundations of the theory of optical imagery." in Before Newton: The Life and Times of Isaac Barrow. ed. Mordechai Feingold. (Cambridge: Cambridge University Press, 1990) 109.
  14. Eves, History of Mathematics, 330.
  15. D. J. Struik, A Source Book in Mathematics, 1200-1800 (Princeton: Princeton University, 1986), 253.
  16. James Child, The Geometrical Lectures of Isaac Barrow (Chicago: Open Court Press, 1916), 252.
  17. Struik, 262.
  18. Helena Pycior, "Mathematics and Philosophy: Wallis, Hobbes, Barrow and Berkeley," Journal of the History of Ideas 48 (April/June 1987): 268.
  19. Pycior, 275.
  20. Pycior, 275.
  21. Pycior, 275.
  22. Pycior, 276.
  23. Alan Shapiro, "The optical lectures and foundations of the theory of optical imagery," in Before Newton: The Life and Times of Isaac Barrow, ed. Mordechai Feingold (Cambridge: Cambridge University Press, 1990), 138.
  24. Shapiro, "The Optical Lectures," 130
  25. Shapiro, "The Optical Lectures," 105.
  26. Shapiro, "The Optical Lectures," 136.
  27. W. F. Mitchell, English Pulpit Oratory from Andrewes to Tillotson (London: Society for Promoting Christian Knowledge, 1932), 401.
  28. Mitchell, 400.