In projective geometry, elements such as points, lines, and planes generally remain points, lines, and planes when projected. However, lengths, ratios of lengths, and angles may change under projection. In projective geometry, parallel lines in Euclidean geometry intersect at infinity in the projection.
SEE ALSO Euclid’s Elements (c. 300 BCE), Descartes’ La Géométrie (1637), Tesseract (1888).
Drawing by Jan Vredeman de Vries (1527–c. 1607), a Dutch Renaissance architect and engineer who experimented with the principles of perspective in his artwork. Projective geometry grew from the principles of perspective art established during the European Renaissance.
1654
Pascal’s Triangle • Clifford A. Pickover
Blaise Pascal (1623–1662), Omar Khayyam (1048–1131)
One of the most famous integer patterns in the history of mathematics is Pascal’s triangle. Blaise Pascal was the first to write a treatise about this progression in 1654, although the pattern had been known by the Persian poet and mathematician Omar Khayyam as far back as A.D. 1100, and even earlier to the mathematicians of India and ancient China. The first seven rows of Pascal’s triangle are depicted at upper right.
Each number in the triangle is the sum of the two above it. Mathematicians have discussed the role that Pascal’s triangle plays in probability theory, in the expansion of binomials of the form (x + y)n, and in various number theory applications for years. Mathematician Donald Knuth (b. 1938) once indicated that there are so many relations and patterns in Pascal’s triangle that when someone finds a new identity, there aren’t many people who get excited about it anymore, except the discoverer. Nonetheless, fascinating studies have revealed countless wonders, including special geometric patterns in the diagonals, the existence of perfect square patterns with various hexagonal properties, and an extension of the triangle and its patterns to negative integers and to higher dimensions.
When even numbers in the triangle are replaced by dots and odd numbers by gaps, the resulting pattern is a fractal, with intricate repeating patterns on different size scales. These fractal figures may have a practical importance in that they can provide models for materials scientists to help produce new structures with novel properties. For example, in 1986, researchers created wire gaskets on the micron size scale almost identical to Pascal’s triangle, with holes for the odd numbers. The area of their smallest triangle was about 1.38 microns squared, and the scientists investigated many unusual properties of their superconducting gasket in a magnetic field.
SEE ALSO Normal Distribution Curve (1733), Cellular Automata (1952), Fractals (1975).
LEFT: George W. Hart created this nylon model of Pascal’s pyramid using a physical process known as selective laser sintering. RIGHT: The fractal Pascal triangle discussed in the text. The number of cells in the central red triangles is always even (6, 28, 120, 496, 2,016. . . .) and includes all perfect numbers (numbers that are the sum of their proper positive divisors).
1660
Von Guericke’s Electrostatic Generator • Clifford A. Pickover
Otto von Guericke (1602–1686), Robert Jemison Van de Graaff (1901–1967)
Neurophysiologist Arnold Trehub writes, “The most important invention in the past two thousand years must be a seminal invention with the broadest and most significant consequences. In my opinion, it is the invention by Otto von Guericke of a machine that produced static electricity.” Although electrical phenomena were known by 1660, von Guericke appears to have produced the forerunner of the first machine for generating electricity. His electrostatic generator employed a globe made of sulfur that could be rotated and rubbed by hand. (Historians are not clear if his device was continuously rotated, a feature that would make it more easy to label his generator a machine.)
More generally, an electrostatic generator produces static electricity by transforming mechanical work into electric energy. Toward the end of the 1800s, electrostatic generators played a key role in research into the structure of matter. In 1929, an electrostatic generator known as the Van de Graaff generator (VG) was designed and built by American physicist Robert Van de Graaff, and it has been used extensively in nuclear physics research. Author William Gurstelle writes, “The biggest, brightest, angriest, and most fulgent electrical discharges don’t come from Wimshurst-style electrostatic machines [see Leyden Jar] . . . or Tesla coils either. They come from an auditorium-sized pair of tall cylindrical machines . . . called Van De Graaff generators, [which] produce cascades of sparks, electrical effluvia, and strong electric fields. . . .”
VGs employ an electronic power supply to charge a moving belt in order to accumulate high voltages, usually on a hollow metal sphere. To use VGs in a particle accelerator, a source of ions (charged particles) is accelerated by the voltage difference. The fact that the VG produces precisely controllable voltages allowed VGs to be used in studies of nuclear reactions during the designing of the atomic bomb.
Over the years, electrostatic accelerators have been used for cancer therapies, for semiconductor production (via ion implantation), for electron-microscope beams, for sterilizing food, and for accelerating protons in nuclear physics experiments.
SEE ALSO Coulomb’s Law of Electrostatics (1785), Battery (1800), Electron (1897), Little Boy Atomic Bomb (1945).
LEFT: Von Guericke invented perhaps the first electrostatic generator, a version of which is illustrated in Hubert-François Gravelot’s engraving (c. 1750). RIGHT: World’s largest air-insulated Van de Graaff generator, originally designed by Van de Graaff for early atomic energy experiments and currently operating in the Boston Museum of Science.
c. 1665
Development of Modern Calculus • Clifford A. Pickover
Isaac Newton (1642–1727), Gottfried Wilhelm Leibniz (1646–1716)
English mathematician Isaac Newton and German mathematician Gottfried Wilhelm Leibniz are usually credited with the invention of calculus, but various earlier mathematicians explored the concept of rates and limits, starting with the ancient Egyptians who developed rules for calculating the volume of pyramids and approximating the areas of circles.
In the 1600s, both Newton and Leibniz puzzled over problems of tangents, rates of change, minima, maxima, and infinitesimals (unimaginably tiny quantities that are almost but not quite zero). Both men understood that differentiation (finding the tangent to a curve at a point—that is, a straight line that “just touches” the curve at that point) and integration (finding the area under a curve) are inverse processes. Newton’s discovery (1665–1666) started with his interest in infinite sums; however, he was slow to publish his findings. Leibniz published his discovery of differential calculus in 1684 and of integral calculus in 1686. He said, “It is unworthy of excellent men, to lose hours like slaves in the labor of calculation. . . . My new calculus . . . offers truth by a kind of analysis and without any effort of imagination.” Newton was outraged. Debates raged for many years on how to divide the credit for the discovery of calculus, and, as a result, progress in calculus was delayed. Newton was the first to apply calculus to problems in physics, and Leibniz developed much of the notation seen in modern calculus books.
Today, calculus has invaded every field of scientific endeavor and plays invaluable roles in biology, physics, chemistry, economics, sociology, and engineering, and in any field where some quantity, like speed or temperature, changes. Calculus can be used to help explain the structure of a rainbow, teach us how to make more money in the stock market, guide a spacecraft, make weather forecasts, predict population growth, design buildings, and analyze the spread of diseases. Calculus has caused a revolution. It has changed the way we look at the world.
SEE ALSO Early Calculus (c. 1500), Newton’s Laws of Motion and Gravitation (1687), Fourier Series (1807), Laplace’s Théorie Analytique des Probabilités (1812).
William Blake’s Newton (1795). Blake, a poet and artist, portrays Isaac Newton as a kind of divine geometer, gazing at technical diagrams drawn on the grou
nd as he ponders mathematics and the cosmos.
1665
Micrographia • Clifford A. Pickover
Marcello Malpighi (1628–1694), Anton Philips van Leeuwenhoek (1632–1723), Robert Hooke (1635–1703), Georgios Nicholas Papanikolaou (1883–1962)
Although microscopes had been available since about the late 1500s, the use of the compound microscope (a microscope with more than one lens) by English scientist Robert Hooke represents a particularly notable milestone, and his instrument can be considered an important optical and mechanical forerunner of the modern microscope. For an optical microscope with two lenses, the overall magnification is the product of the powers of the ocular (eyepiece lens) and the objective lens, which is positioned closer to the specimen.
Hooke’s 1665 book Micrographia featured breathtaking microscopic observations and biological speculation on specimens that ranged from plants to fleas. The book also discussed planets, the wave theory of light, and the origin of fossils, while stimulating both public and scientific interest in the power of the microscope.
Hooke was first to discover biological cells and coined the word cell to describe the basic units of all living things. The word cell was motivated by his observations of plant cells that reminded him of cellulae, which were the quarters in which monks lived. About this magnificent work, the historian of science Richard Westfall writes, “Robert Hooke’s Micrographia remains one of the masterpieces of seventeenth century science, [presenting] a bouquet of observations with courses from the mineral, animal and vegetable kingdoms.”
In 1673, Dutch biologist Anton van Leeuwenhoek discovered living organisms in a drop of pond water, opening up the possibility of using the microscope in medical research. He later published pictures of red blood cells, bacteria, spermatozoa, muscle tissue, and capillaries, the last of which were also observed by Italian physician Marcello Malpighi. Through the years, the microscope has become essential in research into the causes of diseases, such as bubonic plague, malaria, and sleeping sickness. The device also plays a crucial role in the study of cells, such as when used in the Pap smear test (invented by Greek physician Georgios Papanikolaou) to detect premalignant and malignant (cancerous) cervical cells. Before this test became widely used around 1943, cervical cancer was the leading cause of death in American women.
SEE ALSO Eyeglasses (1284), Discovery of Sperm (1678), Zoo Within Us (1683), Cell Division (1855), Germ Theory of Disease (1862).
Flea, from Robert Hooke’s Micrographia (1665).
1668
Refuting Spontaneous Generation • Michael C. Gerald with Gloria E. Gerald
Aristotle (384–322 BCE), Francesco Redi (1626–1697), Lazzaro Spallanzani (1729–1799), Louis Pasteur (1822–1895)
In his book, The History of Animals, written over 2,000 years ago, Aristotle proclaimed that while some organisms arise from similar organisms, others, such as insects, arise spontaneously from putrefying earth or vegetable matter. Each spring, the ancients observed that the Nile overflowed its banks leaving behind muddy soil and frogs that were not present in dry times. In Shakespeare’s Antony and Cleopatra, we learn that crocodiles and snakes are formed in the mud of the Nile. The concept that some living beings could arise from nonliving inanimate matter, which Aristotle called spontaneous generation, remained essentially unchallenged until the seventeenth century. After all, it was commonly observed that maggots appeared to arise from decaying flesh.
In 1668, the Italian physician and poet Francesco Redi devised an experiment that questioned the validity of spontaneous generation and the origin of maggots from rotting meat. Redi placed meat in three widemouthed jars, which he set aside for several days. In the one that was open, flies reached the meat and laid their eggs. In another jar that was sealed, no flies or maggots were found. The mouth of the third was covered with gauze, preventing flies from entering the jar containing meat but upon which they laid their eggs that hatched into maggots.
One century later, Lazzaro Spallanzani, an Italian priest and biologist, boiled broth in a sealed container and permitted the air to escape. While he did not see the growth of any living organism, the question persisted as to whether air was an essential factor required for spontaneous generation to occur.
In 1859 the French Academy of Sciences sponsored a contest for the best experiment that would conclusively prove or disprove the validity of spontaneous generation. In his winning entry, Louis Pasteur placed boiled meat broth in a swan-necked flask, with the neck curved downward. This allowed the free flow of air into the flask, while preventing the entrance of airborne microbes. The broth-containing flask remained free of growth, and the concept of spontaneous generation was believed to be relegated to history.
SEE ALSO Darwin’s Theory of Natural Section (1859), Germ Theory of Disease (1862), Miller-Urey Experiment (1952).
Louis Pasteur, a French microbiologist and chemist, made significant discoveries relating to the germ causes of disease, vaccinations, fermentation, and pasteurization.
1672
Measuring the Solar System • Clifford A. Pickover
Giovanni Domenico Cassini (1625–1712)
Before astronomer Giovanni Cassini’s 1672 experiment to determine the size of the Solar System, there were some rather outlandish theories floating about. Aristarchus of Samos in 280 BCE had said that the Sun was a mere 20 times farther from the Earth than the Moon. Some scientists around Cassini’s time suggested that stars were only a few million miles away. While in Paris, Cassini sent astronomer Jean Richer to the city of Cayenne on the northeast coast of South America. Cassini and Richer made simultaneous measurements of the angular position of Mars against the distant stars. Using simple geometrical methods, and knowing the distance between Paris and Cayenne, Cassini determined the distance between the Earth and Mars. Once this distance was obtained, he employed Kepler’s Third Law to compute the distance between Mars and the Sun (see “Kepler’s Laws of Planetary Motion”). Using both pieces of information, Cassini determined that the distance between the Earth and the Sun was about 87 million miles (140 million kilometers), which is only seven percent less than the actual average distance. Author Kendall Haven writes, “Cassini’s discoveries of distance meant that the universe was millions of times bigger than anyone had dreamed.” Note that it would be difficult to make direct measurements of the Sun without risking his eyesight.
Cassini became famous for many other discoveries. For example, he discovered four moons of Saturn and discovered the major gap in the rings of Saturn, which, today, is called the Cassini Gap in his honor. Interestingly, he was among the earliest scientists to correctly suspect that light traveled at a finite speed, but he did not publish his evidence for this theory because, according to Kendall Haven, “He was a deeply religious man and believed that light was of God. Light therefore had to be perfect and infinite, and not limited by a finite speed of travel.”
Since the time of Cassini, our concept of the Solar System has grown, with the discovery, for example, of Uranus (1781), Neptune (1846), Pluto (1930), and Eris (2005).
SEE ALSO Eratosthenes Measures Earth (c. 240 BCE) Sun-Centered Universe (1534), Kepler’s Laws of Planetary Motion (1609), Michelson-Morley Experiment (1887).
Cassini calculated the distance from Earth to Mars, and then the distance from Earth to the Sun. Shown here is a size comparison between Mars and Earth; Mars has approximately half the radius of Earth.
1672
Newton’s Prism • Clifford A. Pickover
Isaac Newton (1642–1727)
“Our modern understanding of light and color begins with Isaac Newton,” writes educator Michael Douma, “and a series of experiments that he publishes in 1672. Newton is the first to understand the rainbow—he refracts white light with a prism, resolving it into its component colors: red, orange, yellow, green, blue and violet.”
When Newton was experimenting with lights and colors in the late 1660s, many contemporaries thought that colors were a mixture of light and darkn
ess, and that prisms colored light. Despite the prevailing view, he became convinced that white light was not the single entity that Aristotle believed it to be but rather a mixture of many different rays corresponding to different colors. The English physicist Robert Hooke criticized Newton’s work on the characteristics of light, which filled Newton with a rage that seemed out of proportion to the comments Hooke had made. As a result, Newton withheld publication of his monumental book Opticks until after Hooke’s death in 1703—so that Newton could have the last word on the subject of light and could avoid all arguments with Hooke. In 1704, Newton’s Opticks was finally published. In this work, Newton further discusses his investigations of colors and the diffraction of light.
Newton used triangular glass prisms in his experiments. Light enters one side of the prism and is refracted by the glass into various colors (since their degree of separation changes as a function of the wavelength of the color). Prisms work because light changes speed when it moves from air into the glass of the prism. Once the colors were separated, Newton used a second prism to refract them back together to form white light again. This experiment demonstrated that the prism was not simply adding colors to the light, as many believed. Newton also passed only the red color from one prism through a second prism and found the redness unchanged. This was further evidence that the prism did not create colors, but merely separated colors present in the original light beam.
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