Ma¯dhavan of Sangamagra¯mam (c. 1350–c.1425), Nı¯lakantha Soma¯yaji (1444–1544)
Astronomical research in India through the Middle Ages was initially based on the early findings and writings of Aryabhata and other mathematicians and astronomers; it was ultimately expanded by the creation of dedicated research and teaching groups like the Kerala school of astronomy and mathematics, founded in the fourteenth century by the mathematician Ma¯dhavan of Sangamagra¯mam.
Ma¯dhavan and subsequent Kerala mathematicians like Nı¯lakantha Soma¯yaji developed mathematical methods of estimating the motions of the planets based initially on geometry and trigonometry and later on newly developed techniques for modeling complex curves and mathematical shapes using combinations of functions. Among these shapes were parabolas, hyperbolas, and ellipses; their work on ellipses proved especially applicable to astronomy because they were able to show that Aryabhata’s earlier conjecture was correct: the paths of the planets could be described by elliptical orbits. The new mathematical methods developed at Kerala that focused on series of functions were early versions of calculus, predating the European development of calculus some 200 years later by scientists like Isaac Newton.
Nı¯lakantha’s work Aryabhatiyabhasya (a commentary on Aryabhata’s Aryabhatiya), published around 1500, further demonstrated that a rotating Earth and a partially heliocentric solar system provided a more accurate way of fitting the planetary orbits. In his model, Mercury, Venus, Mars, Jupiter, and Saturn all orbited the Sun, but the Sun orbited Earth. A similar model was adopted by the sixteenth-century Danish astronomer Tycho Brahe, and some aspects of Nı¯lakantha’s model are also consistent with the fully heliocentric cosmology proposed in 1543 by Polish astronomer Nicolaus Copernicus.
The contributions of the Kerala school, and perhaps of Indian mathematics and astronomy in general, may have previously been underappreciated in the West. It seems clear now that they should be counted among the “shoulders of giants” that supported the later discoveries of Copernicus, Newton, and others.
SEE ALSO Sun-Centered Universe (1534), Kepler’s Laws of Planetary Motion (1609), Development of Modern Calculus (1665).
Planetary orbital calculations by mathematicians from the Kerala school in southern India, active between the fourteenth and sixteenth centuries, fit a heliocentric model for the solar system. This figure shows some examples from modern Indian physicists reconstructing the geometry used by Kerala school astronomers.
1509
Golden Ratio • Clifford A. Pickover
Fra Luca Bartolomeo de Pacioli (1445–1517)
In 1509, Italian mathematician Luca Pacioli, a close friend of Leonardo da Vinci, published Divina Proportione, a treatise on a number that is now widely known as the “Golden Ratio.” This ratio, symbolized by ø, appears with amazing frequency in mathematics and nature. We can understand the proportion most easily by dividing a line into two segments so that the ratio of the whole segment to the longer part is the same as the ratio of the longer part to the shorter part, or (a + b)/b = b/a =1.61803. . . .
If the lengths of the sides of a rectangle are in the golden ratio, then the rectangle is a “golden rectangle.” It’s possible to divide a golden rectangle into a square and a golden rectangle. Next, we can cut the smaller golden rectangle into a smaller square and golden rectangle. We may continue this process indefinitely, producing smaller and smaller golden rectangles.
If we draw a diagonal from the top right of the original rectangle to the bottom left, and then from the bottom right of the baby (that is, the next smaller) golden rectangle to the top left, the intersection point shows the point to which all the baby golden rectangles converge. Moreover, the lengths of the diagonals are in golden ratio to each another. The point to which all the golden rectangles converge is sometimes called the “Eye of God.”
The golden rectangle is the only rectangle from which a square can be cut so that the remaining rectangle will always be similar to the original rectangle. If we connect the vertices in the diagram, we approximate a logarithmic spiral that “envelops” the Eye of God. Logarithmic spirals are everywhere—seashells, animal horns, the cochlea of the ear—anywhere nature needs to fill space economically and regularly. A spiral is strong and uses a minimum of materials. While expanding, it alters its size but never its shape.
SEE ALSO Projective Geometry (1639), Euler’s Number, e (1727), Transcendental Numbers (1844).
Artistic depiction of golden ratios. Note that the two diagonal lines intersect at a point to which all the baby golden rectangles will converge.
1543
De Humani Corporis Fabrica • Clifford A. Pickover
Jan Stephan van Calcar (1499–1546), Andreas Vesalius (1514–1564)
“The publication of De Humani Corporis Fabrica [On the Fabric of the Human Body] of Andreas Vesalius in 1543 marks the beginning of modern science,” write medical historians J. B. de C. M. Saunders and Charles O’Malley. “It is without doubt the greatest single contribution to medical sciences, but it is a great deal more, an exquisite piece of creative art with its perfect blend of format, typography, and illustration.”
Physician and anatomist Andreas Vesalius of Brussels performed dissections as a primary teaching tool and showed that many previous ideas about the human body, from such great thinkers as Galen and Aristotle, were demonstrably incorrect. For example, in contradiction to Galen, Vesalius showed that blood did not pass from one side of the heart to the other through invisible pores. He also showed that the liver had two main lobes. His challenges to Galen made him the enemy of many, and a detractor even claimed that the human body must have changed since Galen’s studies to explain Vesalius’s observations! In actuality, Galen had based nearly all of his observations on animal dissections, which led to significant errors about humans.
As a medical student, Vesalius braved feral dogs and horrible stenches in his feverish attempts to obtain rotting corpses from cemeteries or the remains of executed criminals hanging from beams until they disintegrated. He even kept specimens in his bedroom for weeks while dissecting them.
Fabrica, Vesalius’s groundbreaking anatomy book, was probably illustrated by Jan Stephan van Calcar or other pupils of the famous Italian painter Titian. The book revealed the inner structures of the brain as never before. Science journalist Robert Adler writes, “With the Fabrica, Vesalius effectively ended the slavish scholastic worship of the knowledge of the ancient world and demonstrated that a new generation of scientists could forge ahead and make discoveries the ancients never dreamed of. Along with a few other Renaissance giants such as Copernicus and Galileo, Vesalius created the progressive, science-driven world in which we live.”
SEE ALSO Paré’s “Rational Surgery” (1545), Circulatory System (1628), Morgagni’s “Cries of Suffering Organs” (1761).
Delineation of spinal nerves from Vesalius’s De Humani Corporis Fabrica.
1543
Sun-Centered Universe • Clifford A. Pickover
Nicolaus Copernicus (1473–1543)
“Of all discoveries and opinions,” wrote the German polymath Johann Wolfgang von Goethe in 1808, “none may have exerted a greater effect on the human spirit than the doctrine of Copernicus. The world had scarcely become known as round and complete in itself when it was asked to waive the tremendous privilege of being the center of the universe. Never, perhaps, was a greater demand made on mankind—for by this admission so many things vanished in mist and smoke! What became of our Eden, our world of innocence, piety and poetry; the testimony of the senses; the conviction of a poetic-religious faith?”
Nicolaus Copernicus was the first individual to present a comprehensive heliocentric theory that suggested the Earth was not the center of the universe. His book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres) was published in 1543, the year he died, and put forward the theory that the Earth revolved around the Sun. Copernicus was a Polish mathematician, physician, and classical
scholar—astronomy was something he studied in his spare time—but it was in the field of astronomy that he changed the world. His theory relied on a number of assumptions: that the Earth’s center is not the center of the universe, that the distance from the Earth to the Sun is miniscule when compared with the distance to the stars, that the rotation of the Earth accounts for the apparent daily rotation of the stars, and that the apparent retrograde motion of the planets (in which they appear to briefly stop and reverse directions at certain times when viewed from the Earth) is caused by the motion of the Earth. Although Copernicus’ proposed circular orbits and epicycles of planets were incorrect, his work motivated other astronomers, such as Johannes Kepler, to investigate planetary orbits and later discover their elliptical nature.
Interestingly, it was not until many years later, in 1616, that the Roman Catholic Church proclaimed that Copernicus’ heliocentric theory was false and “altogether opposed to Holy Scripture.”
SEE ALSO Egyptian Astronomy (c. 2500 BCE) Telescope (1608), Kepler’s Laws of Planetary Motion (1609), Measuring the Solar System (1672), Hubble Telescope (1990).
Orreries are mechanical devices that show positions and motions of the planets and moons in a heliocentric model of the solar system. Shown here is a device constructed in 1766 by instrument-maker Benjamin Martin (1704–1782) and used by astronomer John Winthrop (1714–1779) to teach astronomy at Harvard University. On display at the Putnam Gallery in the Harvard Science Center.
1545
Paré’s “Rational Surgery” • Clifford A. Pickover
Ambroise Paré (1510–1590)
The French surgeon Ambroise Paré is one of the most celebrated surgeons of the European Renaissance. Surgeon and biographer Geoffrey Keynes writes, “Ambroise Paré was, by virtue of his personality and his independent mind, the emancipator of surgery from the dead hand of dogma. There was no comparable practitioner, during his time, in any other country, and his influence was felt in every part of Europe. He left in his collected ‘Works’ a monument to his own skill and humanity which is unsurpassed in the history of surgery.” Paré’s humble credo of patient care was “I dressed him, God cured him.”
Paré lived during a time when physicians generally considered surgery beneath their dignity, and cutting of the body was left to the less prestigious “barber-surgeons.” However, Paré elevated the status of surgeons and spread his surgical knowledge by writing in French rather than the traditional Latin.
Paré made his first significant medical discovery while treating gunshot wounds, which were considered to be poisonous and were usually dealt with by pouring boiling oil into the wound to burn it closed. One day, Paré ran out of oil and was forced to improvise with an ointment that contained turpentine. The next day, he discovered that the soldiers treated with boiling oil were in agony, with swollen wounds. However, the patients who had been treated with the more soothing ointment rested relatively comfortably with little signs of infection. From that day on, Paré vowed never again to use the cruel hot oil to treat wounds.
In 1545, Paré popularized his wound treatments in his Method of Treating Wounds, thus leading the development of the humane “rational” practice of surgery. Another important contribution to medicine was his promotion of the ligature of blood vessels (e.g., tying off with twine) to prevent hemorrhage during amputations, instead of the traditional method of burning the stump with a hot iron. Paré also facilitated progress in obstetrics, using practices that ensured safer delivery of infants.
SEE ALSO Sutures (c. 3000 BCE) Antiseptics (1865), Heart Transplant (1967).
Artificial hand, from Ambroise Paré’s Instrumenta chyrurgiae et icones anathomicae (Surgical Instruments and Anatomical Illustrations), 1564, Paris.
1572
Imaginary Numbers • Clifford A. Pickover
Rafael Bombelli (1526–1572)
An imaginary number is one whose square has a negative value. The great mathematician Gottfried Leibniz called imaginary numbers “a wonderful flight of God’s Spirit; they are almost an amphibian between being and not being.” Because the square of any real number is positive, for centuries many mathematicians declared it impossible for a negative number to have a square root. Although various mathematicians had inklings of imaginary numbers, the history of imaginary numbers started to blossom in sixteenth-century Europe. The Italian engineer Rafael Bombelli, well known during his time for draining swamps, is today famous for his Algebra, published in 1572, that introduced a notation for , which would be a valid solution to the equation x2 + 1 = 0. He wrote, “It was a wild thought in the judgment of many.” Numerous mathematicians were hesitant to “believe” in imaginary numbers, including Descartes, who actually introduced the term imaginary as a kind of insult.
Leonhard Euler in the eighteenth century introduced the symbol i for —for the first letter of the Latin word imaginarius—and we still use Euler’s symbol today. Key advances in modern physics would not have been possible without the use of imaginary numbers, which have aided physicists in a vast range of computations, including efficient calculations involving alternating currents, relativity theory, signal processing, fluid dynamics, and quantum mechanics. Imaginary numbers even play a role in the production of gorgeous fractal artworks that show a wealth of detail with increasing magnifications.
From string theory to quantum theory, the deeper one studies physics, the closer one moves to pure mathematics. Some might even say that mathematics “runs” reality in the same way that Microsoft’s operating system runs a computer. Schrödinger’s wave equation—which describes basic reality and events in terms of wave functions and probabilities—may be thought of as the evanescent substrate on which we all exist, and it relies on imaginary numbers.
SEE ALSO Euler’s Number, e (1727), Riemann Hypothesis (1859), Fractals (1975).
Imaginary numbers play a role in the production of Jos Leys’s gorgeous fractal artworks that show a wealth of detail with increasing magnifications. Early mathematicians were so suspicious of the usefulness of imaginary numbers that they insulted those who suggested their existence.
1608
Telescope • Clifford A. Pickover
Hans Lippershey (1570–1619), Galileo Galilei (1564–1642)
Physicist Brian Greene writes, “The invention of the telescope and its subsequent refinement and use by Galileo marked the birth of the modern scientific method and set the stage for a dramatic reassessment of our place in the cosmos. A technological device revealed conclusively that there is so much more to the universe than is available to our unaided senses.” Computer scientist Chris Langton agrees, noting, “Nothing rivals the telescope. No other device has initiated such a thoroughgoing reconstruction of our world view. It has forced us to accept the earth (and ourselves) as merely a part of the larger cosmos.”
In 1608, the German-Dutch lensmaker Hans Lippershey may have been the first to invent the telescope, and a year later, the Italian astronomer Galileo Galilei constructed a telescope with about a three-fold magnification. He later made others with up to a 30-fold magnification. Although the early telescopes were designed to observe remote objects using visible light, modern telescopes are a range of devices capable of utilizing other regions of the electromagnetic spectrum. Refracting telescopes employ lenses to form an image, while reflecting telescopes use an arrangement of mirrors for this purpose. Catadioptric telescopes use mirrors and lenses.
Interestingly, many important astronomical discoveries with telescopes have been largely unanticipated. Astrophysicist Kenneth Lang writes in Science, “Galileo Galilei turned his newly constructed spyglass to the skies, and thus began astronomers’ use of novel telescopes to explore a universe that is invisible to the unaided eye. The search for the unseen has resulted in many important unexpected discoveries, including Jupiter’s four large moons, the planet Uranus, the first asteroid Ceres, the large recession velocities of spiral nebulae, radio emission from the Milky Way, cosmic X-ray sources, Gamma-Ray Bursts,
radio pulsars, the binary pulsar with its signature of gravitational radiation, and the Cosmic Microwave Background radiation. The observable universe is a modest part of a much vaster, undiscovered one that remains to be found, often in the least expected ways.”
SEE ALSO Sun-Centered Universe (1534), Newton’s Prism (1672), Hubble Telescope (1990).
LEFT: One antenna in the Very Large Array (VLA) used for studying signals from radio galaxies, quasars, pulsars, and more. RIGHT: Observatory staff astride the University of Pittsburgh’s Thaw 30-inch refractor just before its completion in 1913. A man sits atop counterweights needed to keep the massive telescope in balance.
1609
Kepler’s Laws of Planetary Motion • Clifford A. Pickover
Johannes Kepler (1571–1630)
“Although Kepler is remembered today chiefly for his three laws of planetary motion,” writes astronomer Owen Gingerich, “these were but three elements in his much broader search for cosmic harmonies. . . . He left [astronomy] with a unified and physically motivated heliocentric [Sun-centered] system nearly 100 times more accurate.”
Johannes Kepler was the German astronomer and theologian-cosmologist, famous for his laws that described the elliptical orbits of the Earth and other planets around the Sun. In order for Kepler to formulate his laws, he had to first abandon the prevailing notion that circles were the “perfect” curves for describing the cosmos and its planetary orbits. When Kepler first expressed his laws, he had no theoretical justification for them. They simply provided an elegant means by which to describe orbital paths obtained from experimental data. Roughly 70 years later, Newton showed that Kepler’s Laws were a consequence of Newton’s Law of Universal Gravitation.
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