The Physics Book
DKUsing a bold, graphicsled approach, The Physics Book sets out more than 80 of the key concepts and discoveries that have defined the subject and influenced our technology since the beginning of time. With the focus firmly on unpacking the thought behind each theory—as well as exploring when and how each idea and breakthrough came about—five themed chapters examine the history and developments in specific areas such as Light, Sound, and Electricity.
Eureka moments abound: from Archimedes' bathtub discoveries about displacement and density, and Galileo's experiments with spheres falling from the Tower of Pisa, to Isaac Newton's apple and his conclusions about gravity and the laws of motion. You'll also learn about Albert Einstein's revelations about relativity; how the accidental discovery of...
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The Holy Bible was inspired by God and written by men across different ages. This men never knew one another. If it had been written by a single person, the person would have been able to force coherence on his writing.
The Bible is the only single book that that was written by many writers and yet maintain coherence from the beginning till the end.
Jesus Christ has been declared the Son of God even before he came to dwell within men as well as the Holy Spirit had been making appearances in the old times before he came to dwell in us.
All this was done by a God who loves us beyond our sins and wants us to be part of his family. He doesn't want us to be destroyed with the devil and he has given us a solution to that by giving us his son as a sacrifice for our eternal life. All we just need to do is to believe in our hearts and confess with our mouth that we believe in him and we are saved.


CONTENTS HOW TO USE THIS EBOOK INTRODUCTION MEASUREMENT AND MOTION • PHYSICS AND THE EVERYDAY WORLD Man is the measure of all things • Measuring distance A prudent question is one half of wisdom • The scientific method All is number • The language of physics Bodies suffer no resistance but from the air • Free falling A new machine for multiplying forces • Pressure Motion will persist • Momentum The most wonderful productions of the mechanical arts • Measuring time All action has a reaction • Laws of motion The frame of the system of the world • Laws of gravity Oscillation is everywhere • Harmonic motion There is no destruction of force • Kinetic energy and potential energy Energy can be neither created nor destroyed • The conservation of energy A new treatise on mechanics • Energy and motion We must look to the heavens for the measure of the Earth • SI units and physical constants ENERGY AND MATTER • MATERIALS AND HEAT The first principles of the universe • Models of matter As the extension, so the force • Stretching and squeezing The minute parts of matter are in rapid motion • Fluids Searching out the firesecret • Heat and transfers Elastical power in the air • The gas laws The energy of the universe is constant • Internal energy and the first law of thermodynamics Heat can be a cause of motion • Heat engines The entropy of the universe tends to a maximum • Entropy and the second law of thermodynamics The fluid and its vapor become one • Changes of state and making bonds Colliding billiard balls in a box • The development of statistical mechanics Fetching some gold from the sun • Thermal radiation ELECTRICITY AND MAGNETISM • TWO FORCES UNITE Wondrous forces • Magnetism The attraction of electricity • Electric charge Potential energy becomes palpable motion • Electric potential A tax on electrical energy • Electric current and resistance Each metal has a certain power • Making magnets Electricity in motion • The motor effect The dominion of magnetic forces • Indu; ction and the generator effect Light itself is an electromagnetic disturbance • Force fields and Maxwell’s equations Man will imprison the power of the sun • Generating electricity A small step in the control of nature • Electronics Animal electricity • Bioelectricity A totally unexpected scientific discovery • Storing data An encyclopedia on the head of a pin • Nanoelectronics A single pole, either north or south • Magnetic monopoles SOUND AND LIGHT • THE PROPERTIES OF WAVES There is geometry in the humming of the strings • Music Light follows the path of least time • Reflection and refraction A new visible world • Focusing light Light is a wave • Lumpy and wavelike light Light is never known to bend into the shadow • Diffraction and interference The north and south sides of the ray • Polarization The trumpeters and the wave train • The Doppler effect and redshift These mysterious waves we cannot see • Electromagnetic waves The language of spectra is a true music of the spheres • Light from the atom Seeing with sound • Piezoelectricity and ultrasound A large fluctuating echo • Seeing beyond light THE QUANTUM WORLD • OUR UNCERTAIN UNIVERSE The energy of light is distributed discontinuously in space • Energy quanta They do not behave like anything that you have ever seen • Particles and waves A new idea of reality • Quantum numbers All is waves • Matrices and waves The cat is both alive and dead • Heisenberg’s uncertainty principle Spooky action at a distance • Quantum entanglement The jewel of physics • Quantum field theory Collaboration between parallel universes • Quantum applications NUCLEAR AND PARTICLE PHYSICS • INSIDE THE ATOM Matter is not infinitely divisible • Atomic theory A veritable transformation of matter • Nuclear rays The constitution of matter • The nucleus The bricks of which atoms are built up • Subatomic particles Little wisps of cloud • Particles in the cloud chamber Opposites can explode • Antimatter In search of atomic glue • The strong force Dreadful amounts of energy • Nuclear bombs and power A window on creation • Particle accelerators The hunt for the quark • The particle zoo and quarks Identical nuclear particles do not always act alike • Force carriers Nature is absurd • Quantum electrodynamics The mystery of the missing neutrinos • Massive neutrinos I think we have it • The Higgs boson Where has all the antimatter gone? • Matter–antimatter asymmetry Stars get born and die • Nuclear fusion in stars RELATIVITY AND THE UNIVERSE • OUR PLACE IN THE COSMOS The windings of the heavenly bodies • The heavens Earth is not the center of the universe • Models of the universe No true times or true lengths • From classical to special relativity The sun as it was about eight minutes ago • The speed of light Does Oxford stop at this train? • Special relativity A union of space and time • Curving spacetime Gravity is equivalent to acceleration • The equivalence principle Why is the traveling twin younger? • Paradoxes of special relativity Evolution of the stars and life • Mass and energy Where spacetime simply ends • Black holes and wormholes The frontier of the known universe • Discovering other galaxies The future of the universe • The static or expanding universe The cosmic egg, exploding at the moment of creation • The Big Bang Visible matter alone is not enough • Dark matter An unknown ingredient dominates the universe • Dark energy Threads in a tapestry • String theory Ripples in spacetime • Gravitational waves DIRECTORY GLOSSARY CONTRIBUTORS QUOTATIONS ACKNOWLEDGMENTS COPYRIGHT How to use this eBook Preferred application settings For the best reading experience, the following application settings are recommended: Color theme: White background Font size: At the smallest point size Orientation: Landscape (for screen sizes over 9”/23cm), Portrait (for screen sizes below 9”/23cm) Scrolling view: [OFF] Text alignment: Autojustification [OFF] (if the eBook reader has this feature) Autohyphenation: [OFF] (if the eBook reader has this feature) Font style: Publisher default setting [ON] (if the eBook reader has this feature) Images: Double tap on the images to see them in full screen and be able to zoom in on them FOREWORD I fell in love with physics as a boy when I discovered that this was the subject that best provided answers to many of the questions I had about the world around me—questions like how magnets worked, whether space went on forever, why rainbows form, and how we know what the inside of an atom or the inside of a star looks like. I also realized that by studying physics I could get a better grip on some of the more profound questions swirling around in my head, such as: What is the nature of time? What is it like to fall into a black hole? How did the universe begin and how might it end? Now, decades later, I have answers to some of my questions, but I continue to search for answers to new ones. Physics, you see, is a living subject. Although there are many things we now know with confidence about the laws of nature, and we have used this knowledge to develop technologies that have transformed our world, there is still much more we do not yet know. That is what makes physics, for me, the most exciting area of knowledge of all. In fact, I sometimes wonder why everyone isn’t as in love with physics as I am. But to bring the subject alive—to convey that sense of wonder—requires much more than collecting together a mountain of dry facts. Explaining how our world works is about telling stories; it is about acknowledging how we have come to know what we know about the universe, and it is about sharing in the joy of discovery made by the many great scientists who first unlocked nature’s secrets. How we have come to our current understanding of physics can be as important and as joyful as the knowledge itself. This is why I have always had a fascination with the history of physics. I often think it a shame that we are not taught at school about how concepts and ideas in science first developed. We are expected to simply accept them unquestioningly. But physics, and indeed the whole of science, isn’t like that. We ask questions about how the world works and we develop theories and hypotheses. At the same time, we make observations and conduct experiments, revising and improving on what we know. Often, we take wrong turns or discover after many years that a particular description or theory is wrong, or only an approximation of reality. Sometimes, new discoveries are made that shock us and force us to revise our view entirely. One beautiful example of this that has happened in my lifetime was the discovery, in 1998, that the universe is expanding at an accelerating pace, leading to the idea of socalled dark energy. Until recently, this was regarded as a complete mystery. What was this invisible field that acted to stretch space against the pull of gravity? Gradually, we are learning that this is most likely something called the vacuum energy. You might wonder how changing the name of something (from “dark energy” to “vacuum energy”) can constitute an advance in our understanding. But the concept of vacuum energy is not new. Einstein had suggested it a hundred years ago, then changed his mind when he thought he’d made a mistake, calling it his “biggest blunder.” It is stories like this that, for me, make physics so joyous. This is also why The Physics Book is so enjoyable. Each topic is made more accessible and readable with the introduction of key figures, fascinating anecdotes, and the timeline of the development of the ideas. Not only is this a more honest account of the way science progresses, it is also a more effective way of bringing the subject alive. I hope you enjoy the book as much as I do. Jim AlKhalili INTRODUCTION We humans have a heightened sense of our surroundings. We evolved this way to outmaneuver stronger and faster predators. To achieve this, we have had to predict the behavior of both the living and the inanimate world. Knowledge gained from our experiences was passed down through generations via an everevolving system of language, and our cognitive prowess and ability to use tools took our species to the top of the food chain. We spread out of Africa from around 60,000 years ago, extending our abilities to survive in inhospitable locations through sheer ingenuity. Our ancestors developed techniques to allow them to grow plentiful food for their families, and settled into communities. Experimental methods Early societies drew meaning from unrelated events, saw patterns that did not exist, and spun mythologies. They also developed new tools and methods of working, which required advanced knowledge of the inner workings of the world—be it the seasons or the annual flooding of the Nile—in order to expand resources. In some regions, there were periods of relative peace and abundance. In these civilized societies, some people were free to wonder about our place in the universe. First the Greeks, then the Romans tried to make sense of the world through patterns they observed in nature. Thales of Miletus, Socrates, Plato, Aristotle, and others began to reject supernatural explanations and produce rational answers in the quest to create absolute knowledge—they began to experiment. At the fall of the Roman Empire, so many of these ideas were lost to the Western world, which fell into a dark age of religious wars, but they continued to flourish in the Arab world and Asia. Scholars there continued to ask questions and conduct experiments. The language of mathematics was invented to document this newfound knowledge. Ibn alHaytham and Ibn Sahl were just two of the Arab scholars who kept the flame of scientific knowledge alive in the 10th and 11th centuries, yet their discoveries, particularly in the fields of optics and astronomy, were ignored for centuries outside the Islamic world. A new age of ideas With global trade and exploration came the exchange of ideas. Merchants and mariners carried books, stories, and technological marvels from east to west. Ideas from this wealth of culture drew Europe out of the dark ages and into a new age of enlightenment known as the Renaissance. A revolution of our world view began as ideas from ancient civilizations became updated or outmoded, replaced by new ideas of our place in the universe. A new generation of experimenters poked and prodded nature to extract her secrets. In Poland and Italy, Copernicus and Galileo challenged ideas that had been considered sacrosanct for two millennia—and they suffered harsh persecution as a result. Then, in England in the 17th century, Isaac Newton’s laws of motion established the basis of classical physics, which was to reign supreme for more than two centuries. Understanding motion allowed us to build new tools—machines—able to harness energy in many forms to do work. Steam engines and water mills were two of the most important of these—they ushered in the Industrial Revolution (1760–1840). Whosoever studies works of science must … examine tests and explanations with the greatest precision. Ibn alHaytham The evolution of physics In the 19th century, the results of experiments were tried and tested numerous times by a new international network of scientists. They shared their findings through papers, explaining the patterns they observed in the language of mathematics. Others built models from which they attempted to explain these empirical equations of correlation. Models simplified the complexities of nature into digestible chunks, easily described by simple geometries and relationships. These models made predictions about new behaviors in nature, which were tested by a new wave of pioneering experimentalists—if the predictions were proven true, the models were deemed laws which all of nature seemed to obey. The relationship of heat and energy was explored by French physicist Sadi Carnot and others, founding the new science of thermodynamics. British physicist James Clerk Maxwell produced equations to describe the close relationship of electricity and magnetism—electromagnetism. By 1900, it seemed that there were laws to cover all the great phenomena of the physical world. Then, in the first decade of the 20th century, a series of discoveries sent shock waves through the scientific community, challenging former “truths” and giving birth to modern physics. A German, Max Planck, uncovered the world of quantum physics. Then fellow countryman Albert Einstein revealed his theory of relativity. Others discovered the structure of the atom and uncovered the role of even smaller, subatomic particles. In so doing, they launched the study of particle physics. New discoveries weren’t confined to the microscopic—more advanced telescopes opened up the study of the universe. Within a few generations, humanity went from living at the center of the universe to residing on a speck of dust on the edge of one galaxy among billions. Not only had we seen inside the heart of matter and released the energy within, we had charted the seas of space with light that had been traveling since soon after the Big Bang. Physics has evolved over the years as a science, branching out and breaching new horizons as discoveries are made. Arguably, its main areas of concern now lie at the fringes of our physical world, at scales both larger than life and smaller than atoms. Modern physics has found applications in many other fields, including new technology, chemistry, biology, and astronomy. This book presents the biggest ideas in physics, beginning with the everyday and ancient, then moving through classical physics into the tiny atomic world, and ending with the vast expanse of space. One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvellous structure of reality. Albert Einstein INTRODUCTION Our survival instincts have made us creatures of comparison. Our ancient struggle to survive by ensuring that we found enough food for our family or reproduced with the correct mate has been supplanted. These primal instincts have evolved with our society into modern equivalents such as wealth and power. We cannot help but measure ourselves, others, and the world around us by metrics. Some of these measures are interpretive, focusing upon personality traits that we benchmark against our own feelings. Others, such as height, weight, or age, are absolutes. For many people in the ancient and modern world alike, a measure of success was wealth. To amass fortune, adventurers traded goods across the globe. Merchants would purchase plentiful goods cheaply in one location before transporting and selling them for a higher price in another location where that commodity was scarce. As trade in goods grew to become global, local leaders began taxing trade and imposing standard prices. To enforce this, they needed standard measures of physical things to allow them to make comparisons. Language of measurement Realizing that each person’s experience is relative, the ancient Egyptians devised systems that could be communicated without bias from one person to another. They developed the first system of metrics, a standard method for measuring the world around them. The Egyptian cubit allowed engineers to plan buildings that were unrivalled for millennia and devise farming systems to feed the burgeoning population. As trade with ancient Egypt became global, the idea of a common language of measurement spread around the world. The Scientific Revolution (1543–1700) brought about a new need for these metrics. For the scientist, metrics were to be used not for trading goods but as a tool with which nature could be understood. Distrusting their instincts, scientists developed controlled environments in which they tested connections between different behaviors—they experimented. Early experiments focused on the movement of everyday objects, which had a direct effect upon daily life. Scientists discovered patterns in linear, circular, and repetitive oscillating motion. These patterns became immortalized in the language of mathematics, a gift from ancient civilizations that had then been developed in the Islamic world for centuries. Mathematics offered an unambiguous way of sharing the outcomes of experiments and allowed scientists to make predictions and test these predictions with new experiments. With a common language and metrics, science marched forward. These pioneers discovered links between distance, time, and speed and set out their own repeatable and tested explanation of nature. Measuring motion Scientific theories progressed rapidly and with them the language of mathematics changed. Building on his laws of motion, English physicist Isaac Newton invented calculus, which brought a new ability to describe the change in systems over time, not just calculate single snapshots. To explain the acceleration of falling objects, and eventually the nature of heat, ideas of an unseen entity called energy began to emerge. Our world could no longer be defined by distance, time, and mass alone, and new metrics were needed to benchmark the measurement of energy. Scientists use metrics to convey the results of experiments. Metrics provide an unambiguous language that enables scientists to interpret the results of an experiment and repeat the experiment to check that their conclusions are correct. Today, scientists use the Système international (SI) collection of metrics to convey their results. The value of each of these SI metrics and their link to the world around us are defined and decided upon by an international group of scientists known as metrologists. This first chapter charts these early years of the science we today call physics, the way in which the science operates through experimentation, and how results from these tests are shared across the world. From the falling objects that Italian polymath Galileo Galilei used to study acceleration to the oscillating pendulums that paved the way to accurate timekeeping, this is the story of how scientists began to measure distance, time, energy, and motion, revolutionizing our understanding of what makes the world work. IN CONTEXT KEY CIVILIZATION Ancient Egypt BEFORE c. 4000 BCE Administrators use a system of measuring field sizes in ancient Mesopotamia. c. 3100 BCE Officials in ancient Egypt use knotted cords—prestretched ropes tied at regular intervals—to measure land and survey building foundations. AFTER 1585 In the Netherlands, Simon Stevin proposes a decimal system of numbers. 1799 The French government adopts the meter. 1875 Signed by 17 nations, the Meter Convention agrees a consistent length for the unit. 1960 The eleventh General Conference on Weights and Measures sets the metric system as the International System of Units (“SI,” from the French Système international). When people began to build structures on an organized scale, they needed a way to measure height and length. The earliest measuring devices are likely to have been primitive wooden sticks scored with notches, with no accepted consistency in unit length. The first widespread unit was the “cubit,” which emerged in the 4th and 3rd millennia BCE among the peoples of Egypt, Mesopotamia, and the Indus Valley. The term cubit derives from the Latin for elbow, cubitum, and was the distance from the elbow to the tip of the outstretched middle finger. Of course, not everyone has the same length of forearm and middle finger, so this “standard” was only approximate. The Egyptian royal cubit was based on the length of the forearm, measured from the elbow to the middle fingertip. Cubits were subdivided into 28 digits (each a finger’s breadth in length) and a series of intermediary units, such as palms and hands. Imperial measure As prodigious architects and builders of monuments on a grand scale, the ancient Egyptians needed a standard unit of distance. Fittingly, the royal cubit of the Old Kingdom of ancient Egypt is the first known standardized cubit measure in the world. In use since at least 2700 BCE, it was 20.6–20.8 in (523–529 mm) long and was divided into 28 equal digits, each based on a finger’s breadth. Archaeological excavations of pyramids have revealed cubit rods of wood, slate, basalt, and bronze, which would have been used as measures by craftsmen and architects. The Great Pyramid at Giza, where a cubit rod was found in the King’s Chamber, was built to be 280 cubits in height, with a base of 440 cubits squared. The Egyptians further subdivided cubits into palms (4 digits), hands (5 digits), small spans (12 digits), large spans (14 digits, or half a cubit), and t’sers (16 digits or 4 palms). The khet (100 cubits) was used to measure field boundaries and the ater (20,000 cubits) to define larger distances. Cubits of various length were used across the Middle East. The Assyrians used cubits in c. 700 BCE, while the Hebrew Bible contains plentiful references to cubits—particularly in the Book of Exodus’s account of the construction of the Tabernacle, the sacred tent that housed the Ten Commandments. The ancient Greeks developed their own 24unit cubit, as well as the stade (plural stadia), a new unit representing 300 cubits. In the 3rd century BCE, the Greek scholar Eratosthenes (c. 276 BCE–c. 194 BCE) estimated the circumference of Earth at 250,000 stadia, a figure he later refined to 252,000 stadia. The Romans also adopted the cubit, along with the inch—an adult male’s thumb—foot, and mile. The Roman mile was 1,000 paces, or mille passus, each of which was five Roman feet. Roman colonial expansion from the 3rd century BCE to the 3rd century CE introduced these units to much of western Asia and Europe, including England, where the mile was redefined as 5,280 feet in 1593 by Queen Elizabeth I. “You are to make upright frames of acacia wood for the Tabernacle. Each frame is to be ten cubits long and a cubit and a half wide.” Exodus 26:15–16 The Bible Cubit rods—such as this example from the 18th dynasty in ancient Egypt, c. 14th century BCE—were used widely in the ancient world to achieve consistent measurements. Going metric In his 1585 pamphlet De Thiende (The Art of Tenths), Flemish physicist Simon Stevin proposed a decimal system of measurement, forecasting that, in time, it would be widely accepted. More than two centuries later, work on the metric system was begun by a committee of the French Academy of Sciences, with the meter being defined as one tenmillionth of the distance from Earth’s equator to the North Pole. France became the first nation to adopt the measurement, in 1799. International recognition was not achieved until 1960, when the Système international (SI) set the meter as the base unit for distance. It was agreed that 1 meter (m) is equal to 1,000 millimeters (mm) or 100 centimeters (cm), and 1,000 m make up 1 kilometer (km). “A mile shall contain eight furlongs, every furlong forty poles, and every pole shall contain sixteen foot and a half.” Queen Elizabeth I Changing definitions In 1668, English clergyman John Wilkins followed Stevin’s proposal of a decimalbased unit of length with a novel definition: he suggested that 1 meter should be set as the distance of a twosecond pendulum swing. Dutch physicist Christiaan Huygens (1629–1695) calculated this to be 39.26 in (997 mm). In 1889, an alloy bar of platinum (90%) and iridium (10%) was cast to represent the definitive 1meter length, but because it expanded and contracted very slightly at different temperatures, it was accurate only at the melting point of ice. This bar is still kept at the International Bureau of Weights and Measures in Paris, France. When SI definitions were adopted in 1960, the meter was redefined in terms of the wavelength of electromagnetic emissions from a krypton atom. In 1983, yet another definition was adopted: the distance that light travels in a vacuum in 1/299,792,458 of a second. See also: Free falling • Measuring time • SI units and physical constants • Heat and transfers IN CONTEXT KEY FIGURE Aristotle (c. 384–322 BCE) BEFORE 585 BCE Thales of Miletus, a Greek mathematician and philosopher, analyzes movements of the sun and moon to forecast a solar eclipse. AFTER 1543 Nicolaus Copernicus’s De Revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) and Andreas Vesalius’s De humani corporis fabrica (On the Workings of the Human Body) rely on detailed observation, marking the beginning of the Scientific Revolution. 1620 Francis Bacon proposes the inductivist method, which involves making generalizations based on accurate observations. Careful observation and a questioning attitude to findings are central to the scientific method of investigation, which underpins physics and all the sciences. Since it is easy for prior knowledge and assumptions to distort the interpretation of data, the scientific method follows a set procedure. A hypothesis is drawn up on the basis of findings, and then tested experimentally. If this hypothesis fails, it can be revised and reexamined, but if it is robust, it is shared for peer review—independent evaluation by experts. People have always sought to understand the world around them, and the need to find food and understand changing weather were matters of life and death long before ideas were written down. In many societies, mythologies developed to explain natural phenomena; elsewhere, it was believed that everything was a gift from the gods and events were preordained. Early investigations The world’s first civilizations—ancient Mesopotamia, Egypt, Greece, and China—were sufficiently advanced to support “natural philosophers,” thinkers who sought to interpret the world and record their findings. One of the first to reject supernatural explanations of natural phenomena was the Greek thinker Thales of Miletus. Later, the philosophers Socrates and Plato introduced debate and argument as a method of advancing understanding, but it was Aristotle—a prolific investigator of physics, biology, and zoology—who began to develop a scientific method of inquiry, applying logical reasoning to observed phenomena. He was an empiricist, someone who believes that all knowledge is based on experience derived from the senses, and that reason alone is not enough to solve scientific problems—evidence is required. Traveling widely, Aristotle was the first to make detailed zoological observations, seeking evidence to group living things by behavior and anatomy. He went to sea with fishermen in order to collect and dissect fish and other marine organisms. After discovering that dolphins have lungs, he judged they should be classed with whales, not fish. He separated fourlegged animals that give birth to live young (mammals) from those that lay eggs (reptiles and amphibians). However, in other fields Aristotle was still influenced by traditional ideas that lacked a grounding in good science. He did not challenge the prevailing geocentric idea that the sun and stars rotate around Earth. In the 3rd century BCE, another Greek thinker, Aristarchus of Samos, argued that Earth and the known planets orbit the sun, that stars are very distant equivalents of “our” sun, and that Earth spins on its axis. Though correct, these ideas were dismissed because Aristotle and his student Ptolemy carried greater authority. In fact, the geocentric view of the universe was held to be true—due in part to its enforcement by the Catholic Church, which discouraged ideas that challenged its interpretation of the Bible—until it was superseded in the 17th century by the ideas of Copernicus, Galileo, and Newton. ARISTOTLE The son of the court physician of the Macedonian royal family, Aristotle was raised by a guardian after his parents died when he was young. At around the age of 17, he joined Plato’s Academy in Athens, the foremost center of learning in Greece. Over the next two decades, he studied and wrote about philosophy, astronomy, biology, chemistry, geology, and physics, as well as politics, poetry, and music. He also traveled to Lesvos, where he made groundbreaking observations of the island’s botany and zoology. In c. 343 BCE, Aristotle was invited by Philip II of Macedon to tutor his son, the future Alexander the Great. He established a school at the Lyceum in Athens in 335 BCE, where he wrote many of his most celebrated scientific treatises. Aristotle left Athens in 322 BCE and settled on the island of Euboea, where he died at the age of about 62. Key works Metaphysics On the Heavens Physics “All truths are easy to understand once they are discovered; the point is to discover them.” Galileo Galilei Testing and observation Arab scholar Ibn alHaytham (widely known as “Alhazen”) was an early proponent of the scientific method. Working in the 10th and 11th centuries CE, he developed his own method of experimentation to prove or disprove hypotheses. His most important work was in the field of optics, but he also made important contributions to astronomy and mathematics. AlHaytham experimented with sunlight, light reflected from artificial light sources, and refracted light. For example, he tested—and proved—the hypothesis that every point of a luminous object radiates light along every straight line and in every direction. Unfortunately, alHaytham’s methods were not adopted beyond the Islamic world, and it would be 500 years before a similar approach emerged independently in Europe, during the Scientific Revolution. But the idea that accepted theories may be challenged, and overthrown if proof of an alternative can be produced, was not the prevailing view in 16thcentury Europe. Church authorities rejected many scientific ideas, such as the work of Polish astronomer Nicolaus Copernicus. He made painstaking observations of the night sky with the naked eye, explaining the temporary retrograde (“backward”) motion of the planets, which geocentrism had never accounted for. Copernicus realized the phenomenon was due to Earth and the other planets moving around the sun on different orbits. Although Copernicus lacked the tools to prove heliocentrism, his use of rational argument to challenge accepted thinking set him apart as a true scientist. Around the same time, Flemish anatomist Andreas Vesalius transformed medical thinking with his multivolume work on the human body in 1543. Just as Copernicus based his theories on detailed observation, Vesalius analyzed what he found when dissecting human bodyparts. Copernicus’s heliocentric model, socalled because it made the sun (helios in Greek) the focus of planetary orbits, was endorsed by some scientists but outlawed by the Church. “If a man will begin with certainties, he shall end in doubts, but if he will be content to begin with doubts, he shall end in certainties.” Francis Bacon Anatomical drawings from 1543 reflect Vesalius’s mastery of dissection and set a new standard for study of the human body, unchanged since the Greek physician Galen (129–216 CE). Experimental approach For Italian polymath Galileo Galilei, experimentation was central to the scientific approach. He carefully recorded observations on matters as varied as the movement of the planets, the swing of pendulums, and the speed of falling bodies. He produced theories to explain them, then made more observations to test the theories. He used the new technology of telescopes to study four of the moons orbiting Jupiter, proving Copernicus’s heliocentric model—under geocentrism, all objects orbited Earth. In 1633 Galileo was tried by the Church’s Roman Inquisition, found guilty of heresy, and placed under housearrest for the last decade of his life. He continued to publish by smuggling papers to Holland, away from the censorship of the Church. Later in the 17th century, English philosopher Francis Bacon reinforced the importance of a methodical, skeptical approach to scientific inquiry. Bacon argued that the only means of building true knowledge was to base axioms and laws on observed facts, not relying (even if only partially) on unproven deductions and conjecture. The Baconian method involves making systematic observations to establish verifiable facts; generalizing from a series of facts to create axioms (a process known as “inductivism”), while being careful to avoid generalizing beyond what the facts tell us; then gathering further facts to produce an increasingly complex base of knowledge. Unproven science When scientific claims cannot be verified, they are not necessarily wrong. In 1997, scientists at the Gran Sasso laboratory in Italy claimed to have detected evidence of dark matter, which is believed to make up about 27 percent of the universe. The most likely source, they said, were weakly interacting massive particles (WIMPs). These should be detected as tiny flashes of light (scintillations) when a particle strikes the nucleus of a “target” atom. However, despite the best efforts of other research teams to replicate the experiment, no other evidence of dark matter has been found. It is possible that there is an unidentified explanation—or the scintillations could have been produced by helium atoms, which are present in the experiment’s photomultiplier tubes. The scientific method in practice Photo 51, taken by Franklin, is a 1952 Xray diffraction image of human DNA. The Xshape is due to DNA’s doublehelix structure. Deoxyribonucleic acid (DNA) was identified as the carrier of genetic information in the human body in 1944, and its chemical composition was shown to be four different molecules called nucleotides. However, it was unclear how genetic information was stored in DNA. Three scientists—Linus Pauling, Francis Crick, and James Watson—put forward the hypothesis that DNA possessed a helical structure, and realized from work done by other scientists that if that was the case, its Xray diffraction pattern would be Xshaped. British scientist Rosalind Franklin tested this theory by performing Xray diffraction on crystallized pure DNA, beginning in 1950. After refining the technique over a period of two years, her analysis revealed an Xshaped pattern (best seen in “Photo 51”), proving that DNA had a helical structure. The Pauling, Crick, Watson hypothesis was proven, forming the starting point for further studies on DNA. See also: Free falling • SI units and physical constants • Focusing light • Models of the universe • Dark matter IN CONTEXT KEY FIGURE Euclid of Alexandria (c. 325–c. 270 BCE) BEFORE 3000–300 BCE Ancient Mesopotamian and Egyptian civilizations develop number systems and techniques to solve mathematical problems. 600–300 BCE Greek scholars, including Pythagoras and Thales, formalize mathematics using logic and proofs. AFTER c. 630 CE Indian mathematician Brahmagupta uses zero and negative numbers in arithmetic. c. 820 CE Persian scholar alKhwarizmi sets down the principles of algebra. c. 1670 Gottfried Leibniz and Isaac Newton each develop calculus, the mathematical study of continuous change. Physics seeks to understand the universe through observation, experiment, and building models and theories. All of these are intimately entwined with mathematics. Mathematics is the language of physics—whether used in measurement and data analysis in experimental science, or to provide rigorous expression for theories, or to describe the fundamental “frame of reference” in which all matter exists and events take place. The investigation of space, time, matter, and energy is only made possible through a prior understanding of dimension, shape, symmetry, and change. Driven by practical needs The history of mathematics is one of increasing abstraction. Early ideas about number and shape developed over time into the most general and precise language. In prehistoric times, before the advent of writing, herding animals and trading goods undoubtedly prompted the earliest attempts at tallying and counting. As complex cultures emerged in the Middle East and Mesoamerica, demands for greater precision and prediction increased. Power was tied to knowledge of astronomical cycles and seasonal patterns, such as flooding. Agriculture and architecture required accurate calendars and land surveys. The earliest place value number systems (where a digit’s position in a number indicates its value) and methods for solving equations date back more than 3,500 years to civilizations in Mesopotamia, Egypt, and (later) Mesoamerica. “Number is the ruler of forms and ideas, and the cause of gods and daemons.” Pythagoras Adding logic and analysis The rise of ancient Greece brought about a fundamental change in focus. Number systems and measurement were no longer simply practical tools; Greek scholars also studied them for their own sake, together with shape and change. Although they inherited much specific mathematical knowledge from earlier cultures, such as elements of Pythagoras’s theorem, the Greeks introduced the rigor of logical argument and an approach rooted in philosophy; the ancient Greek word philosophia means “love of wisdom.” The ideas of a theorem (a general statement that is true everywhere and for all time) and proof (a formal argument using the laws of logic) are first seen in the geometry of the Greek philosopher Thales of Miletus in the early 6th century BCE. Around the same time, Pythagoras and his followers elevated numbers to be the building blocks of the universe. For the Pythagoreans, numbers had to be “commensurable”—measurable in terms of ratios or fractions—to preserve the link with nature. This world view was shattered with the discovery of irrational numbers (such as √2, which cannot be exactly expressed as one whole number divided by another) by the Pythagorean philosopher Hippasus; according to legend, he was murdered by scandalized colleagues. The dichotomy paradox is one of Zeno’s paradoxes that show motion to be logically impossible. Before walking a certain distance a person must walk half that distance, before walking half the distance he must walk a quarter of the distance, and so on. Walking any distance will therefore entail an infinite number of stages that take an infinite amount of time to complete. Titans of mathematics In the 5th century BCE, the Greek philosopher Zeno of Elea devised paradoxes about motion, such as Achilles and the tortoise. This was the idea that, in any race where the pursued has a head start, the pursuer is always catching up—eventually by an infinitesimal amount. Such puzzles, which were logical—if simple to disprove in practice—would worry generations of mathematicians. They were resolved, at least partially, in the 17th century by the development of calculus, a branch of mathematics that deals with continuously changed quantities. Central to calculus is the idea of calculating infinitesimals (infinitely small quantities), which was anticipated by Archimedes of Syracuse, who lived in the 3rd century BCE. To calculate the approximate volume of a sphere, for instance, he halved it, enclosed the hemisphere in a cylinder, then imagined slicing it horizontally, from the top of the hemisphere, where the radius is infinitesimally small, downward. He knew that the thinner he made his slices, the more accurate the volume would be. Reputed to have shouted “Eureka!” on discovering that the upward buoyant force of an object immersed in water is equal to the weight of the fluid it displaces, Archimedes is notable for applying math to mechanics and other branches of physics in order to solve problems involving levers, screws, pulleys, and pumps. Archimedes studied in Alexandria, at a school established by Euclid, often known as the “Father of Geometry.” It was by analyzing geometry itself that Euclid established the template for mathematical argument for the next 2,000 years. His 13book treatise, Elements, introduced the “axiomatic method” for geometry. He defined terms, such as “point,” and outlined five axioms (also known as postulates, or selfevident truths), such as “a line segment can be drawn between any two points.” From these axioms, he used the laws of logic to deduce theorems. By today’s standards, Euclid’s axioms are lacking; there are numerous assumptions that a mathematician would now expect to be stated formally. Elements remains, however, a prodigious work, covering not only plane geometry and threedimensional geometry, but also ratio and proportion, number theory, and the “incommensurables” that Pythagoreans had rejected. Greek philosophers drew in the sand when teaching geometry, as shown here. Archimedes is said to have been drawing circles in the sand when he was killed by a Roman soldier. Language and symbols In ancient Greece and earlier, scholars described and solved algebraic problems (determining unknown quantities given certain known quantities and relationships) in everyday language and by using geometry. The highlyabbreviated, precise, symbolic language of modern mathematics—which is significantly more effective for analyzing problems and universally understood—is relatively recent. Around 250 CE, however, the Greek mathematician Diophantus of Alexandria introduced the partial use of symbols to solve algebraic problems in his principal work Arithmetica, which influenced the development of Arabic algebra after the fall of the Roman Empire. The study of algebra flourished in the East during the Golden Age of Islam (from the 8th century to the 14th century). Baghdad became the principal seat of learning. Here, at an academic center called the House of Wisdom, mathematicians could study translations of Greek texts on geometry and number theory or Indian works discussing the decimal placevalue system. In the early 9th century, Muhammad ibn Musa alKhwarizmi (from whose name comes the word “algorithm”) compiled methods for balancing and solving equations in his book al Jabr (the root of the word “algebra”). He popularized the use of Hindu numerals, which evolved into Arabic numerals, but still described his algebraic problems in words. French mathematician François Viète finally pioneered the use of symbols in equations in his 1591 book, Introduction to the Analytic Arts. The language was not yet standard, but mathematicians could now write complicated expressions in a compact form, without resorting to diagrams. In 1637, French philosopher and mathematician René Descartes reunited algebra and geometry by devising the coordinate system. Islamic scholars gather in one of Baghdad’s great libraries in this 1237 image by the painter Yahya alWasiti. Scholars came to the city from all points of the Islamic Empire, including Persia, Egypt, Arabia, and even Iberia (Spain). “Imaginary numbers are a fine and wonderful refuge of the divine spirit … almost an amphibian between being and nonbeing.” Gottfried Leibniz More abstract numbers Over millennia, in attempts to solve different problems, mathematicians have extended the number system, expanding the counting numbers 1, 2, 3 … to include fractions and irrational numbers. The addition of zero and negative numbers indicated increasing abstraction. In ancient number systems, zero had been used as a placeholder—a way to tell 10 from 100, for instance. By around the 7th century CE, negative numbers were used for representing debts. In 628 CE, the Indian mathematician Brahmagupta was the first to treat negative integers (whole numbers) just like the positive integers for arithmetic. Yet, even 1,000 years later, many European scholars still considered negative numbers unacceptable as formal solutions to equations. The 16thcentury Italian polymath Gerolamo Cardano not only used negative numbers, but, in Ars Magna, introduced the idea of complex numbers (combining a real and imaginary number) to solve cubic equations (those with at least one variable to the power of three, such as x3, but no higher). Complex numbers take the form a + bi, where a and b are real numbers and i is the imaginary unit, usually expressed as . The unit is termed “imaginary” because when squared it is negative, and squaring any real number, whether it is positive or negative, produces a positive number. Although Cardano’s contemporary Rafael Bombelli set down the first rules for using complex and imaginary numbers, it took a further 200 years before Swiss mathematician Leonhard Euler introduced the symbol i to denote the imaginary unit. Like negative numbers, complex numbers were met with resistance, right up until the 18th century. Yet they represented a significant advance in mathematics. Not only do they enable the solution of cubic equations but, unlike real numbers, they can be used to solve all higherorder polynomial equations (those involving two or more terms added together and higher powers of a variable x, such as x4 or x5). Complex numbers emerge naturally in many branches of physics, such as quantum mechanics and electromagnetism. “A new, a vast, and a powerful language is developed for the future use of analysis, in which to wield its truths so that these may become of more speedy and accurate practical application for the purposes of mankind.” Ada Lovelace British computer scientist Differential calculus examines the rate of change over time, shown geometrically here as the rate of change of a curve. Integral calculus examines the areas, volumes, or displacement bounded by curves. Infinitesimal calculus From the 14th century to the 17th century, together with the increasing use of symbols, many new methods and techniques emerged. One of the most significant for physics, was the development of “infinitesimal” methods in order to study curves and change. The ancient Greek method of exhaustion—finding the area of a shape by filling it with smaller polygons—was refined in order to compute areas bounded by curves. It finally evolved into a branch of mathematics called integral calculus. In the 17th century, French lawyer Pierre de Fermat’s study of tangents to curves inspired the development of differential calculus—the calculation of rates of change. Around 1670, English physicist Isaac Newton and German philosopher Gottfried Leibniz independently worked out a theory that united integral and differential calculus into infinitesimal calculus. The underlying idea is of approximating a curve (a changing quantity) by considering that it is made up of many straight lines (a series of different, fixed quantities). At the theoretical limit, the curve is identical to an infinite number of infinitesimal approximations. During the 18th and 19th centuries, applications of calculus in physics exploded. Physicists could now precisely model dynamic (changing) systems, from vibrating strings to the diffusion of heat. The work of 19thcentury Scottish physicist James Clerk Maxwell greatly influenced the development of vector calculus, which models change in phenomena that have both quantity and direction. Maxwell also pioneered the use of statistical techniques for the study of large numbers of particles. NonEuclidean geometries The fifth axiom, or postulate, on geometry that Euclid set out in his Elements, is also known as the parallel postulate. This was controversial, even in ancient times, as it appears less selfevident than the others, although many theorems depend on it. It states that, given a line and a point that is not on that line, exactly one line can be drawn through the given point and parallel to the given line. Throughout history, various mathematicians, such as Proclus of Athens in the 5th century or the Arabic mathematician alHaytham, have attempted in vain to show that the parallel postulate can be derived from the other postulates. In the early 1800s, Hungarian mathematician János Bolyai and Russian mathematician Nicolai Lobachevsky independently developed a version of geometry (hyperbolic geometry) in which the fifth postulate is false and parallel lines never meet. In their geometry, the surface is not flat as in Euclid’s, but curves inward. By contrast, in elliptic geometry and spherical geometry, also described in the 19th century, there are no parallel lines; all lines intersect. German mathematician Bernhard Riemann and others formalized such nonEuclidean geometries. Einstein used Riemannian theory in his general theory of relativity—the most advanced explanation of gravity—in which mass “bends” spacetime, making it nonEuclidean, although space remains homogeneous (uniform, with the same properties at every point). “Out of nothing I have created a strange new universe. All that I have sent you previously is like a house of cards in comparison with a tower.” János Bolyai in a letter to his father Abstract algebra By the 19th century, algebra had undergone a seismic shift, to become a study of abstract symmetry. French mathematician Évariste Galois was responsible for a key development. In 1830, while investigating certain symmetries exhibited by the roots (solutions) of polynomial equations, he developed a theory of abstract mathematical objects, called groups, to encode different kinds of symmetries. For example, all squares exhibit the same reflectional and rotational symmetries, and so are associated with a particular group. From his research, Galois determined that, unlike for quadratic equations (with a variable to the power of two, such as x2, but no higher), there is no general formula to solve polynomial equations of degree five (with terms such as x5) or higher. This was a dramatic result; he had proved that there could be no such formula, no matter what future developments occurred in mathematics. Subsequently, algebra grew into the abstract study of groups and similar objects, and the symmetries they encoded. In the 20th century, groups and symmetry proved vital for describing natural phenomena at the deepest level. In 1915, German algebraist Emmy Noether connected symmetry in equations with conservation laws, such as the conservation of energy, in physics. In the 1950s and 1960s, physicists used group theory to develop the Standard Model of particle physics. Modeling reality Mathematics is the abstract study of numbers, quantities, and shapes, which physics employs to model reality, express theories, and predict future outcomes—often with astonishing accuracy. For example, the electron gfactor—a measure of its behavior in an electromagnetic field—is computed to be 2.002 319 304 361 6, while the experimentally determined value is 2.002 319 304 362 5 (differing by just one part in a trillion). Certain mathematical models have endured for centuries, requiring only minor adjustments. For example, German astronomer Johannes Kepler’s 1619 model of the solar system, with some refinements by Newton and Einstein, remains valid today. Physicists have applied ideas that mathematicians developed, sometimes much earlier, simply to investigate a pattern; for instance, the application of 19thcentury group theory to modern quantum physics. There are also many examples of mathematical structures driving insight into nature. When British physicist Paul Dirac found twice as many expressions as expected in his equations describing the behavior of electrons, consistent with relativity and quantum mechanics, he postulated the existence of an antielectron; it was duly discovered, years later. While physicists investigate what “is” in the universe, mathematicians are divided as to whether their study is about nature, or the human mind, or the abstract manipulation of symbols. In a strange historical twist, physicists researching string theory are now suggesting revolutionary advances in pure mathematics to geometers (mathematicians who study geometry). Just exactly how this illuminates the relationship between mathematics, physics, and “reality” is yet to be seen. Emmy Noether was a highly creative algebraist. She taught at the University of Göttingen in Germany, but as a Jew was forced to leave in 1933. She died in the US in 1935, aged 53. EUCLID Although his Elements were immensely influential, few details of Euclid’s life are known. He was born around 325 BCE, in the reign of Egyptian pharaoh Ptolemy I and probably died around 270 BCE. He lived mostly in Alexandria, then an important center of learning, but he may also have studied at Plato’s academy in Athens. In Commentary on Euclid, written in the 5th century ce, the Greek philosopher Proclus notes that Euclid arranged the theorems of Eudoxus, an earlier Greek mathematician, and brought “irrefutable demonstration” to the loose ideas of other scholars. Thus, the theorems of the 13 books of Euclid’s Elements are not original, but for two millennia they set the standard for mathematical exposition. The earliest surviving editions of the Elements date from the 15th century. Key works Elements Data Catoptrics Optics See also: Measuring distance • Measuring time • Laws of motion • SI units and physical constants • Antimatter • The particle zoo and quarks • Curving spacetime IN CONTEXT KEY FIGURE Galileo Galilei (1564–1642) BEFORE c. 350 BCE In Physics, Aristotle explains gravity as a force that moves bodies toward their “natural place,” down toward the center of Earth. 1576 Giuseppe Moletti writes that objects of different weights free fall at the same rate. AFTER 1651 Giovanni Riccioli and Francesco Grimaldi measure the time of descent of falling bodies, enabling calculation of their rate of acceleration. 1687 In Principia, Isaac Newton expounds gravitational theory in detail. 1971 David Scott shows that a hammer and a feather fall at the same speed on the moon. When gravity is the only force acting on a moving object, it is said to be in “free fall.” A skydiver falling from a plane is not quite in free fall—since air resistance is acting upon him—whereas planets orbiting the sun or another star are. The ancient Greek philosopher Aristotle believed that the downward motion of objects dropped from a height was due to their nature—they were moving toward the center of Earth, their natural place. From Aristotle’s time until the Middle Ages, it was accepted as fact that the speed of a freefalling object was proportional to its weight, and inversely proportional to the density of the medium it was falling through. So, if two objects of different weights are dropped at the same time, the heavier will fall faster and hit the ground before the lighter object. Aristotle also understood that the object’s shape and orientation were factors in how quickly it fell, so a piece of unfolded paper would fall more slowly than the same piece of paper rolled into a ball. Falling spheres At some time between 1589 and 1592, according to his student and biographer Vincenzo Viviani, Italian polymath Galileo Galilei dropped two spheres of different weight from the Tower of Pisa to test Aristotle’s theory. Although it was more likely to have been a thought experiment than a reallife event, Galileo was reportedly excited to discover that the lighter sphere fell to the ground as quickly as the heavier one. This contradicted the Aristotelian view that a heavier freefalling body will fall more quickly than a lighter one—a view that had recently been challenged by several other scientists. In 1576, Giuseppe Moletti, Galileo’s predecessor in the Chair of Mathematics at the University of Padua, had written that objects of different weights but made of the same material fell to the ground at the same speed. He also believed that bodies of the same volume but made of different materials fell at the same rate. Ten years later, Dutch scientists Simon Stevin and Jan Cornets de Groot climbed 33 ft (10 m) up a church tower in Delft to release two lead balls, one ten times bigger and heavier than the other. They witnessed them hit the ground at the same time. The ageold idea of heavier objects falling faster than lighter ones was gradually being debunked. Another of Aristotle’s beliefs—that a freefalling object descends at a constant speed—had been challenged earlier still. Around 1361, French mathematician Nicole Oresme had studied the movement of bodies. He discovered that if an object’s acceleration is increasing uniformly, its speed increases in direct proportion to time, and the distance it travels is proportional to the square of the time during which it is accelerating. It was perhaps surprising that Oresme should have challenged the established Aristotelian “truth,” which at the time was considered sacrosanct by the Catholic Church, in which Oresme served as a bishop. It is not known whether Oresme’s studies influenced the later work of Galileo. “Nature is inexorable and immutable; she never transgresses the laws imposed upon her.” Galileo Galilei GALILEO GALILEI The oldest of six siblings, Galileo was born in Pisa, Italy, in 1564. He enrolled to study medicine at the University of Pisa at the age of 16, but his interests quickly broadened and he was appointed Chair of Mathematics at the University of Padua in 1592. Galileo’s contributions to physics, mathematics, astronomy, and engineering single him out as one of the key figures of the Scientific Revolution in 16th and 17thcentury Europe. He created the first thermoscope (an early thermometer), defended the Copernican idea of a heliocentric solar system, and made important discoveries about gravity. Because some of his ideas challenged Church dogma, he was called before the Roman Inquisition in 1633, declared to be a heretic, and sentenced to house arrest until his death in 1642. Key works 1623 The Assayer 1632 Dialogue Concerning the Two Chief World Systems 1638 Discourses and Mathematical Demonstrations Relating to Two New Sciences Galileo showed that objects of different mass accelerate at a constant rate. By timing how long a ball took to travel a particular distance down a slope, he could figure out its acceleration. The distance fallen was always proportional to the square of the time taken to fall. Balls on ramps From 1603, Galileo set out to investigate the acceleration of freefalling objects. Unconvinced that they fell at a constant speed, he believed that they accelerated as they fell—but the problem was how to prove it. The technology to accurately record such speeds simply did not exist. Galileo’s ingenious solution was to slow down the motion to a measurable speed, by replacing a falling object with a ball rolling down a sloping ramp. He timed the experiment using both a water clock—a device that weighed the water spurting into an urn as the ball traveled—and his own pulse. If he doubled the period of time the ball rolled, he found the distance it traveled was four times as far. Leaving nothing to chance, Galileo repeated the experiment “a full hundred times” until he had achieved “an accuracy such that the deviation between two observations never exceeded onetenth of a pulse beat.” He also changed the incline of the ramp: as it became steeper, the acceleration increased uniformly. Since Galileo’s experiments were not carried out in a vacuum, they were imperfect—the moving balls were subject to air resistance and friction from the ramp. Nevertheless, Galileo concluded that in a vacuum, all objects—regardless of weight or shape—would accelerate at a uniform rate: the square of the elapsed time of the fall is proportional to the distance fallen. In this fresco by Giuseppe Bezzuoli, Galileo is shown demonstrating his rollingball experiment in the presence of the powerful Medici family in Florence. Quantifying gravitational acceleration In spite of Galileo’s work, the question of the acceleration of freefalling objects was still contentious in the mid17th century. From 1640 to 1650, Jesuit priests Giovanni Riccioli and Francesco Grimaldi conducted various investigations in Bologna. Key to their eventual success were Riccioli’s timekeeping pendulums—which were as accurate as any available at the time—and a very tall tower. The two priests and their assistants dropped heavy objects from various levels of the 321ft (98m) Asinelli Tower, timing their descents. The priests, who described their methodology in detail, repeated the experiments several times. Riccioli believed that freefalling objects accelerated exponentially, but the results showed him that he was wrong. A series of falling objects were timed by pendulums at the top and bottom of the tower. They fell 15 Roman feet (1 Roman foot = 11.6 in) in 1 second, 60 feet in 2 seconds, 135 feet in 3 seconds, and 240 feet in 4 seconds. The data, published in 1651, proved that the distance of descent was proportional to the square of the length of time the object was falling—confirming Galileo’s ramp experiments. And for the first time, due to relatively accurate timekeeping, it was possible to work out the value of acceleration due to gravity: 9.36 (±0.22) m/s2. This figure is only about 5 percent less than the range of figures accepted today: around 9.81 m/s2. The value of g (gravity) varies according to a number of factors: it is greater at Earth’s poles than at the equator, lower at high altitudes than at sea level, and it varies very slightly according to local geology, for example if there are particularly dense rocks near Earth’s surface. If the constant acceleration of an object in free fall near Earth’s surface is represented by g, the height at which it is released is z0 and time is t, then at any stage in its descent, the height of the body above the surface z = z0 – 1⁄2 gt2, where gt is the speed of the body and g its acceleration. A body of mass m at a height z0 above Earth’s surface possesses gravitational potential energy U, which can be calculated by the equation U = mgz0 (mass × acceleration × height above Earth’s surface). “When Galileo caused balls … to roll down an inclined plane, a light broke upon all students of nature.” Immanuel Kant German philosopher The hammer and the feather In 1971, American astronaut David Scott—commander of the Apollo 15 moon mission—performed a famous freefall experiment. The fourth NASA expedition to land on the moon, Apollo 15 was capable of a longer stay on the moon than previous expeditions, and its crew was the first to use a Lunar Roving Vehicle. Apollo 15 also featured a greater focus on science than earlier moon landings. At the end of the mission’s final lunar walk, Scott dropped a 3lb geological hammer and a 1oz falcon’s feather from a height of 5 ft. In the virtual vacuum conditions of the moon’s surface, with no air resistance, the ultralight feather fell to the ground at the same speed as the heavy hammer. The experiment was filmed, so this confirmation of Galileo’s theory that all objects accelerate at a uniform rate regardless of mass was witnessed by a television audience of millions. “In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.” Galileo Galilei See also: Measuring distance • Measuring time • Laws of motion • Laws of gravity • Kinetic energy and potential energy IN CONTEXT KEY FIGURE Blaise Pascal (1623–1662) BEFORE 1643 Italian physicist Evangelista Torricelli demonstrates the existence of a vacuum using mercury in a tube; his principle is later used to invent the barometer. AFTER 1738 In Hydrodynamica, Swiss mathematician Daniel Bernoulli argues that energy in a fluid is due to elevation, motion, and pressure. 1796 Joseph Bramah, a British inventor, uses Pascal’s law to patent the first hydraulic press. 1851 Scottish–American inventor Richard Dudgeon patents a hydraulic jack. 1906 An oil hydraulic system is installed to raise and lower the guns of the US warship Virginia. While investigating hydraulics (the mechanical properties of liquids), French mathematician and physicist Blaise Pascal made a discovery that would eventually revolutionize many industrial processes. Pascal’s law, as it became known, states that if pressure is applied to any part of a liquid in an enclosed space, that pressure is transmitted equally to every part of the fluid, and to the container walls. The impact of Pascal Pascal’s law means that pressure exerted on a piston at one end of a fluidfilled cylinder produces an equal increase in pressure on another piston at the other end of the cylinder. More significantly, if the crosssection of the second piston is twice that of the first, the force on it will be twice as great. So, a 2.2 lb (1 kg) load on the small piston will allow the large piston to lift 4.4 lb (2 kg); the larger the ratio of the crosssections, the more weight the large piston can raise. Pascal’s findings weren’t published until 1663, the year after his death, but they would be used by engineers to make the operation of machinery much easier. In 1796, Joseph Bramah applied the principle to construct a hydraulic press that flattened paper, cloth, and steel, doing so more efficiently and powerfully than previous wooden presses. Liquids cannot be compressed and are used to transmit forces in hydraulics systems such as car jacks. A small force applied over a long distance is turned into a larger force over a small distance, which can raise a heavy load. See also: Laws of motion • Stretching and squeezing • Fluids • The gas laws IN CONTEXT KEY FIGURE John Wallis (1616–1703) BEFORE 1518 French natural philosopher Jean Buridan describes “impetus,” the measure of which is later understood to be momentum. 1644 In his Principia Philosophiae (Principles of Philosophy), French scientist René Descartes describes momentum as the “amount of motion.” AFTER 1687 Isaac Newton describes his laws of motion in his threevolume work Principia. 1927 German theoretical physicist Werner Heisenberg argues that for a subatomic particle, such as an electron, the more precisely its position is known, the less precisely its momentum can be known, and vice versa. When objects collide, several things happen. They change velocity and direction, and the kinetic energy of motion may be converted to heat or sound. In 1666, the Royal Society of London challenged scientists to come up with a theory to explain what happens when objects collide. Two years later, three individuals published their theories: from England, John Wallis and Christopher Wren, and from Holland, Christiaan Huygens. All moving bodies have momentum (the product of their mass and velocity). Stationary bodies have no momentum because their velocity is zero. Wallis, Wren, and Huygens agreed that in an elastic collision (any collision in which no kinetic energy is lost through the creation of heat or noise), momentum is conserved as long as there are no other external forces at work. Truly elastic collisions are rare in nature; the nudging of one billiard ball by another comes close, but there is still some loss of kinetic energy. In The Geometrical Treatment of the Mechanics of Motion, John Wallis went further, correctly arguing that momentum is also conserved in inelastic collisions, where objects become attached after they collide, causing the loss of kinetic energy. One such example is that of a comet striking a planet. Nowadays, the principles of conservation of momentum have many practical applications, such as determining the speed of vehicles after traffic accidents. “A body in motion is apt to continue its motion.” John Wallis See also: Laws of motion • Kinetic energy and potential energy • The conservation of energy • Energy and motion IN CONTEXT KEY FIGURE Christiaan Huygens (1629–1695) BEFORE c. 1275 The first allmechanical clock is built. 1505 German clockmaker Peter Henlein uses the force from an uncoiling spring to make the first pocket watch. 1637 Galileo Galilei has the idea for a pendulum clock. AFTER c. 1670 The anchor escapement mechanism makes the pendulum clock more accurate. 1761 John Harrison’s fourth marine chronometer, H4, passes its sea trials. 1927 The first electronic clock, using quartz crystal, is built. 1955 British physicists Louis Essen and Jack Parry make the first atomic clock. Two inventions in the mid 1650s heralded the start of the era of precision timekeeping. In 1656, Dutch mathematician, physicist, and inventor Christiaan Huygens built the first pendulum clock. Soon after, the anchor escapement was invented, probably by English scientist Robert Hooke. By the 1670s, the accuracy of timekeeping devices had been revolutionized. The first entirely mechanical clocks had appeared in Europe in the 13th century, replacing clocks reliant on the movement of the sun, the flow of water, or the burning of a candle. These mechanical clocks relied on a “verge escapement mechanism,” which transmitted force from a suspended weight through the timepiece’s gear train, a series of toothed wheels. Over the next three centuries, there were incremental advances in the accuracy of these clocks, but they had to be wound regularly and still weren’t very accurate. In 1637, Galileo Galilei had realized the potential for pendulums to provide more accurate clocks. He found that a swinging pendulum was almost isochronous, meaning the time it took for the bob at its end to return to its starting point (its period) was roughly the same whatever the length of its swing. A pendulum’s swing could produce a more accurate way of keeping time than the existing mechanical clocks. However, he hadn’t managed to build one before his death in 1642. Huygens’ first pendulum clock had a swing of 80–100 degrees, which was too great for complete accuracy. The introduction of Hooke’s anchor escapement, which maintained the swing of the pendulum by giving it a small push each swing, enabled the use of a longer pendulum with a smaller swing of just 4–6 degrees, which gave much better accuracy. Before this, even the most advanced nonpendulum clocks lost 15 minutes a day; now that margin of error could be reduced to as little as 15 seconds. Christiaan Huygens’ pendulum clock dramatically improved the accuracy of timekeeping devices. This 17thcentury woodcut shows the inner workings of his clock, including toothed gears and pendulum. Quartz and atomic clocks Pendulum clocks remained the most accurate form of time measurement until the 1930s, when synchronous electric clocks became available. These counted the oscillations of alternating current coming from electric power supply; a certain number of oscillations translated into movements of the clock’s hands. The first quartz clock was built in 1927, taking advantage of the piezoelectric quality of crystalline quartz. When bent or squeezed, it generates a tiny electric voltage, or conversely, if it is subject to an electric voltage, it vibrates. A battery inside the clock emits the voltage, and the quartz chip vibrates, causing an LCD display to change or a tiny motor to move second, minute, and hour hands. The first accurate atomic clock, built in 1955, used the cesium133 isotope. Atomic clocks measure the frequency of regular electromagnetic signals that electrons emit as they change between two different energy levels when bombarded with microwaves. Electrons in an “excited” cesium atom oscillate, or vibrate, 9,192,631,770 times per second, making a clock calibrated on the basis of these oscillations extremely accurate. Harrison’s marine chronometer John Harrison’s prototype chronometer, H1, underwent sea trials from Britain to Portugal in 1736, losing just a few seconds on the entire voyage. In the early 18th century, even the most accurate pendulum clocks didn’t work at sea—a major problem for nautical navigation. With no visible landmarks, calculating a ship’s position depended on accurate latitude and longitude readings. While it was easy to gauge latitude (by viewing the position of the sun), longitude could be determined only by knowing the time relative to a fixed point, such as the Greenwich Meridian. Without clocks that worked at sea, this was impossible. Ships were lost and many men died, so, in 1714, the British government offered a prize to encourage the invention of a marine clock. British inventor John Harrison solved the problem in 1761. His marine chronometer used a fastbeating balance wheel and a temperaturecompensated spiral spring to achieve remarkably accurate timekeeping on transatlantic journeys. The device saved lives and revolutionized exploration and trade. See also: Free falling • Harmonic motion • SI units and physical constants • Subatomic particles IN CONTEXT KEY FIGURES Gottfried Leibniz (1646–1716), Isaac Newton (1642–1727) BEFORE c. 330 BCE In Physics, Aristotle expounds his theory that it takes force to produce motion. 1638 Galileo’s Dialogues Concerning Two New Sciences is published. It is later described by Albert Einstein as anticipating the work of Leibniz and Newton. 1644 René Descartes publishes Principles in Philosophy, which includes laws of motion. AFTER 1827–1833 William Rowan Hamilton establishes that objects tend to move along the path that requires the least energy. 1907–1915 Einstein proposes his theory of general relativity. Prior to the late 16th century, there was little understanding of why moving bodies accelerated or decelerated—most people believed that some indeterminate, innate quality made objects fall to the ground or float up to the sky. But this changed at the dawn of the Scientific Revolution, when scientists began to understand that several forces are responsible for changing a moving object’s velocity (a combined measure of its speed and direction), including friction, air resistance, and gravity. Early views For many centuries, the generally accepted views of motion were those of the ancient Greek philosopher Aristotle, who classified everything in the world according to its elemental composition: earth, water, air, fire, and quintessence, a fifth element that made up the “heavens.” For Aristotle, a rock falls to the ground because it has a similar composition to the ground (“earth”). Rain falls to the ground because water’s natural place is at Earth’s surface. Smoke rises because it is largely made of air. However, the circular movement of celestial objects was not considered to be governed by the elements—rather, they were thought to be guided by the hand of a deity. Aristotle believed that bodies move only if they are pushed, and once the pushing force is removed, they come to a stop. Some questioned why an arrow unleashed from a bow continues to fly through the air long after direct contact with the bow has ceased, but Aristotle’s views went largely unchallenged for more than two millennia. In 1543, Polish astronomer Nicolaus Copernicus published his theory that Earth was not the center of the universe, but that it and the other planets orbited the sun in a “heliocentric” system. Between 1609 and 1619, German astronomer Johannes Kepler developed his laws of planetary motion, which describe the shape and speed of the orbits of planets. Then, in the 1630s, Galileo challenged Aristotle’s views on falling objects, explained that a loosed arrow continues to fly because of inertia, and described the role of friction in bringing to a halt a book sliding across a table. These scientists laid the basis for French philosopher René Descartes and German polymath Gottfried Leibniz to formulate their own ideas about motion, and for English physicist Isaac Newton to draw all the threads together in Mathematical Principles of Natural Philosophy (Principia). “There is neither more nor less power in an effect than there is in its cause.” Gottfried Leibniz A new understanding In Principles in Philosophy, Descartes proposed his three laws of motion, which rejected Aristotle’s views of motion and a divinely guided universe, and explained motion in terms of forces, momentum, and collisions. In his 1687 Essay on Dynamics, Leibniz produced a critique of Descartes’ laws of motion. Realizing that many of Descartes’ criticisms of Aristotle were justified, Leibniz went on to develop his own theories on “dynamics,” his term for motion and impact, during the 1690s. Leibniz’s work remained unfinished, and he was possibly put off after reading Newton’s thorough laws of motion in Principia, which—like Dynamics—was also published in 1687. Newton respected Descartes’ rejection of Aristotelian ideas, but argued that the Cartesians (followers of Descartes) did not make enough use of the mathematical techniques of Galileo, nor the experimental methods of chemist Robert Boyle. However, Descartes’ first two laws of motion won the support of both Newton and Leibniz, and became the basis for Newton’s first law of motion. Newton’s three laws of motion clearly explained the forces acting on all bodies, revolutionizing the understanding of the mechanics of the physical world and laying the foundations for classical mechanics (the study of the motion of bodies). Not all of Newton’s views were accepted during his lifetime—one of those who raised criticisms was Leibniz himself—but after his death they were largely unchallenged until the early 20th century, just as Aristotle’s beliefs about motion had dominated scientific thinking for the best part of 2,000 years. However, some of Leibniz’s views on motion and criticisms of Newton were far ahead of their time, and were given credence by Albert Einstein’s general theory of relativity two centuries later. The bicycle is in motion due to the force supplied by the pedalling of the rider, until the external force of the rock acts upon it, causing it to stop. Law of inertia Newton’s first law of motion, which is sometimes called the law of inertia, explains that an object at rest stays at rest, and an object in motion remains in motion with the same velocity unless acted upon by an external force. For instance, if the front wheel of a bicycle being ridden at speed hits a large rock, the bike is acted upon by an external force, causing it to stop. Unfortunately for the cyclist, he or she will not have been acted upon by the same force and will continue in motion—over the handlebars. For the first time, Newton’s law enabled accurate predictions of motion to be made. Force is defined as a push or pull exerted on one object by another and is measured in Newtons (denoted N, where 1 N is the force required to give a 1 kg mass an acceleration of 1 m/s²). If the strength of all the forces on an object are known, it is possible to calculate the net external force—the combined total of the external forces—expressed as ∑ F (∑ stands for “sum of”). For example, if a ball has a force of 23 N pushing it left, and a force of 12 N pushing it right, ∑F = 11 N in a leftward direction. It is not quite as simple as this, since the downward force of gravity will also be acting on the ball, so horizontal and vertical net forces also need to be taken into account. There are other factors at play. Newton’s first law states that a moving object that is not acted upon by outside forces should continue to move in a straight line at a constant velocity. But when a ball is rolled across the floor, for example, why does it eventually stop? In fact, as the ball rolls it experiences an outside force: friction, which causes it to decelerate. According to Newton’s second law, an object will accelerate in the direction of the net force. Since the force of friction is opposite to the direction of travel, this acceleration causes the object to slow and eventually stop. In interstellar space, a spacecraft will continue to move at the same velocity because of an absence of friction and air resistance—unless it is accelerated by the gravitational field of a planet or star, for example. Two rockets with different masses but identical engines will accelerate at different rates. The smaller rocket will accelerate more quickly due to its lower mass. Change is proportional Newton’s second law is one of the most important in physics, and describes how much an object accelerates when a given net force is applied to it. It states that the rate of change of a body’s momentum—the product of its mass and velocity—is proportional to the force applied, and takes place in the direction of the applied force. This can be expressed as ∑F = ma, where F is the net force, a is the acceleration of the object in the direction of the net force, and m is its mass. If the force increases, so does acceleration. Also, the rate of change of momentum is inversely proportional to the mass of the object, so if the object’s mass increases, its acceleration decreases. This can be expressed as a = ∑F∕m. For example, as a rocket’s fuel propellant is burned during flight, its mass decreases and—assuming the thrust of its engines remains the same—it will accelerate at an everfaster rate. “The laws of motion… are the free decrees of God.” Gottfried Leibniz Equal action and reaction Newton’s third law states that for every action there is an equal and opposite reaction. Sitting down, a person exerts a downward force on the chair, and the chair exerts an equal upward force on the person’s body. One force is called the action, the other the reaction. A rifle recoils after it is fired due to the opposing forces of such an action–reaction. When the rifle’s trigger is pulled, a gunpowder explosion creates hot gases that expand outward, allowing the rifle to push forward on the bullet. But the bullet also pushes backward on the rifle. The force acting on the rifle is the same as the force that acts on the bullet, but because acceleration depends on force and mass (in accordance with Newton’s second law), the bullet accelerates much faster than the rifle due to its far smaller mass. Notions of time, distance, and acceleration are fundamental to an understanding of motion. Newton argued that space and time are entities in their own right, existing independently of matter. In 1715–1716, Leibniz argued in favor of a relationist alternative: in other words, that space and time are systems of relations between objects. While Newton believed that absolute time exists independently of any observer and progresses at a constant pace throughout the universe, Leibniz reasoned that time makes no sense except when understood as the relative movement of bodies. Newton argued that absolute space “remains always similar and immovable,” but his German critic argued that it only makes sense as the relative location of objects. From Leibniz to Einstein A conundrum raised by Irish bishop and philosopher George Berkeley around 1710 illustrated problems with Newton’s concepts of absolute time, space, and velocity. It concerned a spinning sphere: Berkeley questioned whether, if it was rotating in an otherwise empty universe, it could be said to have motion at all. Although Leibniz’s criticisms of Newton were generally dismissed at the time, Einstein’s general theory of relativity (1907–1915) made more sense of them two centuries later. While Newton’s laws of motion are generally true for macroscopic objects (objects that are visible to the naked eye) under everyday conditions, they break down at very high speeds, at very small scales, and in very strong gravitational fields. “Motion is really nothing more than change of place. So motion as we experience it is nothing but a relation.” Gottfried Leibniz Two Voyager spacecraft were launched in 1977. With no friction or air resistance in space, the craft are still moving through space today, due to Newton’s first law of motion. GOTTFRIED LEIBNIZ Born in Leipzig (now Germany) in 1646, Leibniz was a great philosopher, mathematician, and physicist. After studying philosophy at the University of Leipzig, he met Christiaan Huygens in Paris and determined to teach himself math and physics. He became a political adviser, historian, and librarian to the royal House of Brunswick in Hanover in 1676, a role that gave him the opportunity to work on a broad range of projects, including the development of infinitesimal calculus. However, he was also accused of having seen Newton’s unpublished ideas and passing them off as his own. Although it was later generally accepted that Leibniz had arrived at his ideas independently, he never managed to shake off the scandal during his lifetime. He died in Hanover in 1716. Key works 1684 “Nova methodus pro maximis et minimis” (“New method for maximums and minimums”) 1687 Essay on Dynamics See also: Free falling • Laws of gravity • Kinetic energy and potential energy • Energy and motion • The heavens • Models of the universe • From classical to special relativity IN CONTEXT KEY FIGURE Isaac Newton (1642–1727) BEFORE 1543 Nicolaus Copernicus challenges orthodox thought with a heliocentric model of the solar system. 1609 Johannes Kepler publishes his first two laws of planetary motion in Astronomia Nova (A New Astronomy), arguing that the planets move freely in elliptical orbits. AFTER 1859 French astronomer Urbain Le Verrier argues that Mercury’s precessionary orbit (the slight variance in its axial rotation) is incompatible with Newtonian mechanics. 1905 In his paper “On the Electrodynamics of Moving Bodies,” Einstein introduces his theory of special relativity. 1915 Einstein’s theory of general relativity states that gravity affects time, light, and matter. Published in 1687, Newton’s law of universal gravitation remained—alongside his laws of motion—the unchallenged bedrock of “classical mechanics” for more than two centuries. It states that every particle attracts every other particle with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between their centers. Before the scientific age in which Newton’s ideas were formulated, the Western understanding of the natural world had been dominated by the writings of Aristotle. The ancient Greek philosopher had no concept of gravity, believing instead that heavy objects fell to Earth because that was their “natural place,” and celestial bodies moved around Earth in circles because they were perfect. Aristotle’s geocentric view remained largely unchallenged until the Renaissance, when Polish–Italian astronomer Nicolaus Copernicus argued for a heliocentric model of the solar system, with Earth and the planets orbiting the sun. According to him, “We revolve around the sun like any other planet.” His ideas, published in 1543, were based on detailed observations of Mercury, Venus, Mars, Jupiter, and Saturn made with the naked eye. “What hinders the fixed stars from falling upon one another?” Isaac Newton Astronomical evidence In 1609, Johannes Kepler published Astronomia Nova (A New Astronomy) which, as well as providing more support for heliocentrism, described the elliptical (rather than circular) orbits of the planets. Kepler also discovered that the orbital speed of each planet depends on its distance from the sun. Around the same time, Galileo Galilei was able to support Kepler’s view with detailed observations made with the aid of telescopes. When he focused a telescope on Jupiter and saw moons orbiting the giant planet, Galileo uncovered further proof that Aristotle had been wrong: if all things orbited Earth, Jupiter’s moons could not exist. Galileo also observed the phases of Venus, demonstrating that it orbits the sun. Galileo also challenged the idea that heavy objects fall to the ground more quickly than light objects. His contention was supported by Italian Jesuit priests Giovanni Battista Riccioli and Francesco Maria Grimaldi, who in the 1640s dropped objects from a Bologna tower and timed their descent to the street below. Their calculations provided reasonably accurate values for the rate of acceleration due to gravity, now known to be 32.15 ft/s2 (9.8 m/s2). Their experiment was recreated in 1971 by US astronaut David Scott, who dropped a hammer and a feather on NASA’s Apollo 15 mission. With no air resistance at the moon’s surface, the two objects hit the ground at the same time. The New Almagest, a 1651 work by Riccioli, illustrates the tussle between rival models of planetary motion: Tycho Brahe’s Earthcentric theory is shown outweighing heliocentrism. Newton’s apple While the story of an apple falling on Isaac Newton’s head is apocryphal, watching the fruit drop to the ground did spark his curiosity. By the time Newton started thinking seriously about gravity in the 1660s, much important groundwork had already been achieved. In his seminal work Principia, Newton credited the work of Italian physicist Giovanni Borelli (1608–1679) and French astronomer Ismael Bullialdus (1605–1694), who both described the sun’s gravity exerting an attractive force. Bullialdus believed incorrectly that the sun’s gravity attracted a planet at its aphelion (the point on its orbital curve when it is furthest from the sun) but repulsed it at the perihelion (when it is closest). The work of Johannes Kepler was probably the greatest influence on Newton’s ideas. The German astronomer’s third law of orbital motion states that there is an exact mathematical relationship between a planet’s distance from the sun and the length of time it takes to complete a full orbit. In 1670, English natural philosopher Robert Hooke argued that gravitation applies to all celestial bodies and that its power decreases with distance, and—in the absence of any other attractive forces—moves in straight lines. By 1679, he had concluded that the inverse square law applied, so gravity weakens in proportion to the square of the distance from a body. In other words, if the distance between the sun and another body is doubled, the force between them reduces to only one quarter of the original force. However, whether this rule could be applied close to the surface of a large planetary body such as Earth was unknown. “If the Earth should cease to attract its waters to itself all the waters of the sea would be raised and would flow to the body of the moon.” Johannes Kepler Grimaldi and Riccioli showed that gravity causes objects to fall at the same speed regardless of their mass. If air resistance is eliminated, objects accelerate at a constant rate of 32.15 ft/s (9.8 m/s) faster with each passing second. Universality of free fall The Asinelli tower in Bologna, Italy, was the chosen site for the freefall experiments of Riccioli and Grimaldi, in which Galileo’s theory was put to the test. The principle of universality of free fall was empirically discovered by Galileo and others, then mathematically proven by Newton. It states that all materials, heavy or light, fall at the same rate in a uniform gravitational field. Consider two falling bodies of different weight. Since Newton’s theory of gravity says that the greater the mass of an object, the greater the gravitational force, the heavy object should fall faster. However, his second law of motion tells us that a larger mass does not accelerate as quickly as a smaller one if the applied force is the same, so it will fall more slowly. In fact, the two cancel each other out, so the light and heavy objects will fall with the same acceleration as long as no other forces—such as air resistance—are present. Universal gravitation Newton published his own laws of motion and gravitation in Principia in 1687, asserting “Every particle attracts every other particle … with a force directly proportional to the product of their masses.” He explained how all matter exerts an attractive force—gravity—that pulls other matter toward its center. It is a universal force, the strength of which depends on the mass of the object. For instance, the sun has a greater gravitational strength than Earth, which in turn has more gravitational strength than the moon, which in turn has more gravitational strength than a ball dropped upon it. Gravitational strength can be expressed with the equation F = Gm1m2/r2, where F is the force, m1 and m2 are the masses of the two bodies, r is the distance between their centers, and G is the gravitational constant. Newton continued to refine his views long after the publication of Principia. His cannonball thoughtexperiment speculated on the trajectory of a ball fired from a cannon on top of a very high mountain in an environment in which there was no air resistance. If gravity was also absent, he argued that the cannonball would follow a straight line away from Earth in the direction of fire. Assuming gravity is present, if the speed of the cannonball was relatively slow it would fall back to Earth, but if it was fired at a much faster velocity it would continue to circle Earth in a circular orbit; this would be its orbital velocity. If its velocity was faster still, the ball would continue to travel around Earth in an elliptical orbit. If it reached a speed that was faster than 7 miles/s (11.2 km/s), it would leave Earth’s gravitational field and travel onward into outer space. More than three centuries later, modernday physics has put Newton’s theories into practice. The cannonball phenomenon can be seen when a satellite or spacecraft is launched into orbit. Instead of the gunpowder that blasted Newton’s imaginary projectile, powerful rocket engines lift the satellite from Earth’s surface and give it forward speed. When it reaches its orbital velocity, the spacecraft’s propulsion ceases and it falls all the way around Earth, never hitting the surface. The angle of the satellite’s path is determined by its initial angle and speed. The success of space exploration has relied heavily on Newton’s laws of gravitation. Newton argued that gravity is a universal, attractive force that applies to all matter, however large or small. Its strength varies according to how much mass the objects have, and how far they are from each other. “Nonsense will fall of its own weight, by a sort of intellectual law of gravitation. And a new truth will go into orbit.” Cecilia PayneGaposchkin British–American astronomer Understanding mass An object’s inertial mass is its inertial resistance to acceleration from any force, gravitational or not. It is defined by Newton’s second law of motion as F = ma, where F is the force applied, m is its inertial mass, and a is acceleration. If a known force is applied to the object, by measuring its acceleration the inertial mass is found to be F/a. In contrast, according to Newton’s law of universal gravitation, gravitational mass is the physical property of an object that causes it to interact with other objects through the gravitational force. Newton was troubled by the question: is the inertial mass of an object the same as its gravitational mass? Repeated experiments have shown that the two properties are the same, a fact that fascinated Albert Einstein, who used it as a basis for his theory of general relativity. Reinterpreting gravitation Newton’s ideas on universal gravitation and motion were unchallenged until 1905, when Einstein’s theory of special relativity was published. Whereas Newton’s theory depended on the assumption that mass, time, and distance remain constant, Einstein’s theory treats them as fluid entities that are defined by the observer’s frame of reference. A person standing on Earth as it spins on its axis orbits the sun—and moves through the universe in a different frame of reference to an astronaut flying through space in a spacecraft. Einstein’s theory of general relativity also states that gravity is not a force, but rather the effect of the distortion of spacetime by massive objects. Newton’s laws are adequate for most daytoday applications, but they cannot explain the differences in motion, mass, distance, and time that result when bodies are observed from two very different frames of reference. In this instance, scientists must rely on Einstein’s theories of relativity. Classical mechanics and Einstein’s theories of relativity agree as long as an object’s speed is low or the gravitational field it experiences is small. Newton correctly predicted that objects would orbit Earth if they were launched at the correct speed. If a satellite moves quickly enough, the curvature of its fall is less than that of Earth, so it stays in orbit, never returning to the ground. ISAAC NEWTON Born in the English village of Woolsthorpe on Christmas Day 1642, Newton went to school in Grantham and studied at Cambridge University. In Principia, he formulated the laws of universal gravitation and motion, which formed the basis of classical mechanics until the early 20th century when they were partially superseded by Einstein’s theories of relativity. Newton also made important contributions to mathematics and optics. Sometimes a controversial character, he had longrunning disputes with Gottfried Leibniz over who had discovered calculus, and Robert Hooke regarding the inverse square law. As well as being a keen scientist, Newton was very interested in alchemy and biblical chronology. He died in London in 1727. Key works 1684 On the Motion of Bodies in an Orbit 1687 Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) See also: Free falling • Laws of motion • The heavens • Models of the universe • Special relativity • The equivalence principle • Gravitational waves IN CONTEXT KEY FIGURE Leonhard Euler (1707–1783) BEFORE 1581 Galileo discovers the link between the length of a pendulum and its period of motion. 1656 Christiaan Huygens builds a pendulum clock that uses the periodic motion of a pendulum to regulate a timekeeping mechanism. AFTER 1807 Joseph Fourier, a French physicist, shows that any periodic process can be treated as the sum of simple harmonic oscillations overlaid on each other. 1909 German engineer Hermann Frahm develops a “dynamic vibration absorber,” a device that absorbs energy from oscillations and releases it out of sync in order to reduce vibration. Periodic motion—motion repeated in equal time intervals—is found in many natural and artificial phenomena. Studies of pendulums in the 16th and 17th centuries, for instance, helped lay the foundations for Isaac Newton’s laws of motion. But groundbreaking though these laws were, physicists still faced major barriers in applying them to realworld problems involving systems (interacting groups of items) that were more complex than Newton’s idealized, freelymoving bodies. Musical oscillations One particular area of interest was the vibration of musical strings—another form of periodic motion. In Newton’s day, the principle that strings vibrate at different frequencies to produce different sounds was well established, but the exact form of the vibrations was unclear. In 1732, Swiss physicist and mathematician Daniel Bernoulli found a means of applying Newton’s second law of motion to each segment of a vibrating string. He showed that the force on the string grew as it moved further from the center line (its stationary starting point), and always acted in the opposite direction to the displacement from the center. While tending to restore the string toward the center, it overshot to the other side, creating a repeating cycle. This type of movement, with a specific relationship between displacement and restoring force, is today known as simple harmonic motion. As well as vibrating strings, it encompasses phenomena such as a swinging pendulum and a weight bouncing on the end of a spring. Bernoulli also discovered that harmonic oscillations plotted on a graph form a sine wave—a mathematical function that is easily manipulated to find solutions to physical problems. Harmonic motion can also be applied in some more surprising places. For instance, both circula