Kepler’s First Law (The Law of Orbits, 1609) indicated that all of the planets in our Solar System move in elliptical orbits, with the Sun at one focus. His Second Law (The Law of Equal Areas, 1618) showed that when a planet is far from the Sun, the planet moves more slowly than when it is close to the Sun. An imaginary line that connects a planet to the Sun sweeps out equal areas in equal intervals of time. Given Kepler’s first two laws, planetary orbits and positions could now be easily calculated and with an accuracy that matched observations.
Kepler’s Third Law (The Law of Periods, 1618) showed that for any planet, the square of the period of its revolution about the Sun is proportional to the cube of the semi-major axis of its elliptical orbit. Thus, planets far from the Sun have very long years. Kepler’s Laws are among the earliest scientific laws to be established by humans, and, while unifying astronomy and physics, the laws provided a stimulus to subsequent scientists who attempted to express the behavior of reality in terms of simple formulas.
SEE ALSO Antikythera Mechanism (c. 125 BCE), Sun-Centered Universe (1543), Telescope (1608), Newton’s Laws of Motion and Gravitation (1687).
Artistic representation of the Solar System. Johannes Kepler was the German astronomer and theologian-cosmologist, famous for his laws that described the elliptical orbits of the Earth and the other planets around the sun.
1614
Logarithms • Clifford A. Pickover
John Napier (1550–1617)
Scottish mathematician John Napier is famous as the inventor and promoter of logarithms in his 1614 book A Description of the Marvelous Rule of Logarithms. This method has since contributed to countless advances in science and engineering by making difficult calculations possible. Before electronic calculators became widely available, logarithms and tables of logarithms were commonly used in surveying and navigation. Napier was also the inventor of Napier’s bones, rods carved with multiplication tables that could be arranged in patterns in order to aid in calculations.
A logarithm (to a base b) of a number x is expressed as logb(x) and equals the exponent y that satisfies x = by. For example, because 35 = 3 × 3 × 3 × 3 × 3 = 243, we say that the log of 243 (base 3) is 5, or log3(243) = 5. As another example, log10(100) = 2. For practical purposes, consider that a multiplication such as 8 × 16 = 128 can be rewritten as 23 × 24 = 27, thereby converting the calculations into ones involving the simple additions of the powers (3 + 4 = 7). Prior to calculators, in order to multiply two numbers, an engineer often looked up the logarithms of both numbers in a table, added them, and then looked up the result in the table to find the product. This could often be faster than multiplying by hand and is the principle on which slide rules are based.
Today, various quantities and scales in science are expressed as logarithms of other quantities. For example, the pH scale in chemistry, the bel unit of measurement in acoustics, and the Richter scale used for measuring earthquake intensity all involve a base-10 logarithmic scale. Interestingly, the discovery of logarithms just prior to the era of Isaac Newton had an impact on science comparable to the invention of the computer in the twentieth century.
SEE ALSO Slide Rule (1621), Euler’s Number, e (1727), ENIAC (1946).
John Napier, the discoverer of logarithms, created a calculation device known as Napier’s rods or bones. Napier’s rotatable rods reduced multiplication to a sequence of simple additions.
1620
Scientific Method • Michael C. Gerald with Gloria E. Gerald
Aristotle (384–322 BCE), Francis Bacon (1561–1626), Galileo Galilei (1564–1642), Claude Bernard (1813–1878), Louis Pasteur (1822–1895)
Formulation and fine-tuning of the scientific method has evolved over the ages and is based upon the contributions of many early distinguished scholars including Aristotle, who introduced logical deduction, a “top-down” approach, that is, starting with a theory or hypothesis and then testing that theory; Francis Bacon, the father of the modern scientific method, who in 1620 wrote Novum Organum Scientiarum, which proposed inductive reasoning as the foundation for scientific reasoning, a “bottom-up” approach in which specific observations led to the formulation of a general theory or hypothesis; and Galileo, who advocated experimentation rather than metaphysical explanations. In the mid-nineteenth century, Louis Pasteur elegantly utilized the scientific method when he designed experiments to disprove the theory of spontaneous generation.
In 1865, Claude Bernard, one of the greatest of all scientists, wrote An Introduction to the Study of Experimental Medicine, which he personalized by using his own thoughts and experiments. In this classic book, he examined the importance of the scientist bringing forth new knowledge to society, and he then proceeded to critically analyze what constituted a good scientific theory, the importance of observation rather than reliance on historical authorities and sources, inductive and deductive reasoning, and cause and effect.
When some nonscientists think of theories, such as the theory of evolution, not infrequently they use the term “theory” disparagingly and assume or imply that it imputes an unproven notion or a mere guess or speculation. Scientists, by contrast, use the term “theory” to refer to an explanation, model, or general principle that has been tested and confirmed and that explains or predicts a natural event. The scientific method follows a number of sequential steps and is an approach used to investigate phenomena or acquire new knowledge. Using a series of steps, it is based upon developing and testing a hypothesis that explains a given observation, objectively evaluating the test results obtained, and then accepting, rejecting, or modifying that hypothesis. A theory is broader and more general than a hypothesis and is supported by experimental evidence based on a number of hypotheses that can be tested independently.
SEE ALSO Refuting Spontaneous Generation (1668), Darwin’s Theory of Natural Section (1859), Randomized Controlled Trials (1948), Placebo Effect (1955)
In Novum Organum (New Method), Bacon proposed a scientific method of inquiry based on inductive reasoning, in which a generalization is built based on incremental data collection. This method was intended to improve upon Aristotle’s deductive reasoning, which deduces specific facts from a generalization.
1621
Slide Rule • Clifford A. Pickover
William Oughtred (1574–1660)
Those of you who went to high school before the 1970s may recall that the slide rule once seemed to be as common as the typewriter. In just seconds, engineers could multiply, divide, find square roots, and do much more. The earliest version with sliding pieces was invented in 1621 by English mathematician and Anglican minister William Oughtred, based on the logarithms of Scottish mathematician John Napier. Oughtred may not have initially recognized the value of his work, because he did not quickly publish his findings. According to some accounts, one of his students stole the idea and published a pamphlet on the slide rule, which emphasized its portability, and raved that the device was “fit for use on horseback as on foot.” Oughtred was outraged by his student’s duplicity.
In 1850, a 19-year-old French artillery lieutenant modified the original design of the slide rule, and the French army used it to perform projectile calculations when fighting the Prussians. During World War II, American bombers often used specialized slide rules.
Slide-rule guru Cliff Stoll writes, “Consider the engineering achievements that owe their existence to rubbing two sticks together: the Empire State Building; the Hoover Dam; the curves of the Golden Gate Bridge; hydrodynamic automobile transmissions, transistor radios; the Boeing 707 airliner.” Wernher Von Braun, the designer of the German V-2 rocket, relied on slide rules made by the German company Nestler, as did Albert Einstein. Pickett slide rules were aboard Apollo space missions in case the computers failed!
In the twentieth century, 40 million slide rules were produced worldwide. Given the crucial role that this device played from the Industrial Revolution until modern times, the device deserves a place in this book. Literature from the Oughtred
Society states, “For a span of 3.5 centuries, it was used to perform design calculations for virtually all the major structures built on this earth.”
SEE ALSO Ishango Bone (c. 18,000 BCE), Antikythera Mechanism (c. 125 BCE) Logarithms (1614), ENIAC (1946).
The slide rule played a crucial role from the Industrial Revolution until modern times. In the twentieth century, 40 million slide rules were produced and were used by engineers for countless applications.
1628
Circulatory System • Clifford A. Pickover
Praxagoras (340 BCE–280 BCE), Ibn al-Nafis (1213–1288), Hieronymus Fabricius (1537–1619), William Harvey (1578–1657), Marcello Malpighi (1628–1694)
Science journalist Robert Adler writes, “Today, the basics of how blood circulates through the body seem trivial. . . . Grade-school children learn that the heart pumps oxygen-rich blood through the body via the arteries, that the veins return oxygen-depleted blood to the heart, and that tiny capillaries link the finest arteries and veins. Yet . . . the functioning of the heart and blood vessels remained a profound mystery from ancient times until the first quarter of the seventeenth century.”
English physician William Harvey was the first to correctly describe, in detail, the circulation of blood through the body. In his 1628 work, De motu cordis (fuller title in English: On the Motion of the Heart and Blood in Animals), Harvey traced the correct route of blood through his study of living animals, in which he could pinch various blood vessels near the heart (or could cut vessels) and note directions of flow. He also applied pressure to veins near the skin of human subjects and noted blood-flow direction by observing swelling, along with the parts of the arms that grew congested or pale. In contrast to physicians of the past, who conjectured that the liver produced blood that was continually absorbed by the body, Harvey showed that blood must be recycled. He also realized that the valves that exist in veins, discovered by his teacher Hieronymus Fabricius, facilitated one-way blood flow to the heart.
Harvey traced the blood through smaller and smaller arteries and veins but did not have a microscope and, thus, could only conjecture that connections must exist between arteries and veins. Just a few years after Harvey died, Italian physician Marcello Malpighi used a microscope to observe the tiny capillaries that provided the elusive connections.
Various related work in blood circulation predates Harvey. For example, the Greek physician Praxagoras discussed arteries and veins, but he suggested that arteries carried air. In 1242, the Arab Muslim physician Ibn al-Nafis elucidated the flow of blood between the heart and lungs.
SEE ALSO Morgagni’s “Cries of Suffering Organs” (1761), Blood Transfusion (1829), Heart Transplant (1967).
William Harvey correctly described, in detail, the circulation of blood through the body, including the path of oxygenated blood away from the heart and the return of deoxygenated blood back to the heart.
1637
Descartes’ La Géométrie • Clifford A. Pickover
René Descartes (1596–1650)
In 1637, French philosopher and mathematician René Descartes published La géométrie, which shows how geometrical shapes and figures can be analyzed using algebra. Descartes’ work influenced the evolution of analytical geometry, a field of mathematics that involves the representation of positions in a coordinate system and in which mathematicians algebraically analyze such positions. La géométrie also shows how to solve mathematical problems and discusses the representation of points of a plane through the use of real numbers, and the representation and classification of curves through the use of equations.
Interestingly, La géométrie does not actually use “Cartesian” coordinate axes or any other coordinate system. The book pays as much attention to representing algebra in geometric forms as vice versa. Descartes believed that algebraic steps in a proof should usually correspond to a geometrical representation.
Jan Gullberg writes, “La géométrie is the earliest mathematical text that a modern student of mathematics could read without stumbling over an abundance of obsolete notations. . . . Along with Newton’s Principia, it is one of the most influential scientific texts of the seventeenth century.” According to Carl Boyer, Descartes desired to “free geometry” from the use of diagrams through algebraic procedures and to give meaning to the operations of algebra through geometric interpretation.
More generally, Descartes was groundbreaking in his proposal to unite algebra and geometry into a single subject. Judith Grabiner writes, “Just as the history of Western philosophy has been viewed as a series of footnotes to Plato, so the past 350 years of mathematics can be viewed as a series of footnotes to Descartes’ Geometry . . . and the triumph of Descartes’ methods of problem solving.”
Boyer concludes, “In terms of mathematical ability, Descartes probably was the most able thinker of his day, but he was at heart not really a mathematician.” His geometry was only one facet of a full life that revolved around science, philosophy, and religion.
SEE ALSO Pythagorean Theorem and Triangles (c. 600 BCE), Euclid’s Elements (c. 300 BCE).
The Ancient of Days (1794), a watercolor etching by William Blake. European medieval scholars often associated geometry and the laws of nature with the divine. Through the centuries, geometry’s focus on compass and straightedge constructions became more abstract and analytical.
1638
Acceleration of Falling Objects • Clifford A. Pickover
Galileo Galilei (1564–1642)
“To appreciate the full nature of Galileo’s discoveries,” writes I. Bernard Cohen, “we must understand the importance of abstract thinking, of its use by Galileo as a tool that in its ultimate polish was a much more revolutionary instrument for science than even the telescope.” According to legend, Galileo dropped two balls of different weights from the Leaning Tower of Pisa to demonstrate that they would both hit the ground at the same time. Although this precise experiment probably did not take place, Galileo certainly performed experiments that had a profound effect on contemporary understanding of the laws of motions. Aristotle had taught that heavy objects fall faster than light ones. Galileo showed that this was only an artifact of differing air resistances of the objects, and he supported his claims by performing numerous experiments with balls rolling down inclined planes. Extrapolating from these experiments, he demonstrated that if objects could fall without air resistance, all objects accelerate at the same rate. More precisely, he showed that the distance traveled by a constantly accelerating body starting at zero velocity is proportional to the square of the time falling.
Galileo also proposed the principle of inertia, in which an object’s motion continues at the same speed and direction unless acted upon by another force. Aristotle had erroneously believed that a body could be kept in motion only by applying a force. Newton later incorporated Galileo’s principle into his Laws of Motion. If it is not apparent to you that a moving object does not “naturally” stop moving without an applied force, you can imagine an experiment in which the face of a penny is sliding along an infinite smooth horizontal table that is so well oiled that there is no friction. Here, the penny would continue sliding along such an imaginary surface forever.
SEE ALSO Archimedes Principle of Buoyancy (c. 250 BCE) Newton’s Laws of Motion and Gravitation (1687), Conservation of Energy (1843).
Imagine spheres, or any objects, of different masses released at the same height at the same time. Galileo showed that they must all fall together at the same speed, if we neglect any differences in air resistance.
1639
Projective Geometry • Clifford A. Pickover
Leon Battista Alberti (1404–1472), Gérard Desargues (1591–1661), Jean-Victor Poncelet (1788–1867)
Projective geometry generally concerns the relationships between shapes and their mappings, or “images,” that result from projecting the shapes onto a surface. Projections may often be visualized as the shadows cast by objects.
The Italian architect Leo
n Battista Alberti was one of the first individuals to experiment with projective geometry through his interest in perspective in art. More generally, Renaissance artists and architects were concerned with methods for representing three-dimensional objects in two-dimensional drawings. Alberti sometimes placed a glass screen between himself and the landscape, closed one eye, and marked on the glass certain points that appeared to be in the image. The resulting 2-D drawing gave a faithful impression of the 3-D scene.
French mathematician Gérard Desargues was the first professional mathematician to formalize projective geometry while searching for ways to extend Euclidean geometry. In 1636, Desargues published Exemple de l’une des manières universelles du S.G.D.L. touchant la pratique de la perspective (Example of a Universal Method by Sieur Girard Desargues Lyonnais Concerning the Practice of Perspective), in which he presented a geometric method for constructing perspective images of objects. Desargues also examined the properties of shapes that were preserved under perspective mappings. Painters and engravers made use of his approach.
Desargues’ most important work, Brouillon project d’une atteinte aux événements des rencontres d’un cône avec un plan (Rough Draft of Attaining the Outcome of Intersecting a Cone with a Plane), published in 1639, treats the theory of conic sections using projective geometry. In 1882, French mathematician and engineer Jean-Victor Poncelet (1788–1867) published a treatise that revitalized interest in projective geometry.
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