Welcome, Pilgrim.  My name is Salviati.  I will be your guide on this journey through Calculus.  These lectures are all about gravitation. They tell a small part of the long story of how our ideas about gravity developed over the years. And in that story is the germ of the story of Science itself, because the simple but insistent questions we have always asked, both about the cosmos, and about an apple falling, have been the anvil on which our best understanding of Nature has been forged. 

           My teacher, Signore Professore Galileo Galilei, was one of those giants on whose shoulders young Isaac Newton stood when he first glimpsed the truth about gravity.  But my teacher did not know Calculus, nor even did he use algebra as we know it today.  He preferred the geometric and synthetic reasoning of the Greeks and followed the example of Archimedes in all things but Science.  For it can be plainly seen that of Science and of its Method, Galileo was the first true architect.

           And so, we shall begin our studies with Signore Galileo in the Pre-Calculus dawn of modern science.   The questions he asked and the answers he found will not require Calculus to understand, but they will illuminate the path before you, especially if you make them your own questions.  My Master, a Copernican to the end, died in the very year that young Isaac was born.  As he passed the torch, in his writings, to the new Prometheus, we hope that these very writings, his Two New Sciences among them, will serve to kindle a desire in you to know Calculus, and to continue with the later Lectures to follow the footsteps of Isaac Newton through that marvelous realm. 

           Too often, the laws of Nature, as we come to understand them, are presented to us as "received wisdom," knowledge that is somehow graven in stone.  But there was a time when those ideas were fresh and ripe with possibilities that were not yet imagined, not even by their discoverers.  My teacher, Galileo, came of age in such a time.  And he was skeptical of the "wisdom" that had been handed down for generations, that was held to be true knowledge, on the basis of authority alone.  Like Copernicus before him, and Descartes and Newton after, he chose to judge for himself what was of value in the classical scheme of thought, and to trust his senses and his imagination to fill the gaps. You can begin to imagine what that was like in two ways:  You may discover some of those laws for yourself, and that we will attempt to help you to do that in our labs, our exercises, and with our questions.  And you can read the words of the ones who found them.   Those words are a treasure that tell the adventure of these ideas in a special way. You had to be there.   

          You may find a little of what my teacher, Galileo, wrote in a book called Two New Sciences.  It will not be easy reading, because it comes from a different time, from a mind occupied with different questions from the ones you are familiar with. And you will find the thoughts of Isaac Newton, as he penned them in his Principia, his daring manifesto and joyous celebration of perhaps the greatest discovery ever made.  That book is well worth the trouble to explore.  

          Today, the wisdom of these observations, capturing as they do some of the best thoughts we have had about gravity and its effects, are somehow dry and almost uninteresting, like the dessicated and departed specimens on a museum tray.  But when they were born in human imagination, these ideas were both youthful and fragile, playful and mischievous.  They were the dreams of lofty minds that came to roost in a world that was hardly prepared to receive them. Hear the voices of those dreamers, the "sleepwalkers," who taught us the secrets of the sky.  I recommend them to you! 

            These lectures are also about a wonderful unification of what had seemed to be two separate questions.  How do objects fall? And how do the Moon and the Planets move?  These are really the same question.  And the history of that question is the subject of this book.  We begin to ask the first question in this introductory chapter.  The second question will lead us, following John Kepler and Isaac Newton, to the Calculus itself.  But you will use this book best if you read it with your own questions.  You will, in many cases, be able to test your answers for yourself in the laboratories.

             We begin this introduction to the ideas of the Calculus with the study of freely falling objects.  Our first Laboratory is called Freely Falling Objects, and in it you will discover an interesting fact about objects that fall freely.  We must abstract away of course the resistance of objects to the air and wind.  We are really talking about gravity itself, and about a simple fact.

            Gravity acts on the speed of a falling object directly, and not directly on its position.  Gravity does not care how heavy an object is, or how it is shaped.  Once an object begins to fall, gravity causes its downward speed to grow in a uniform way.  Over equal intervals of time, the speed changes by equal amounts (at least when this happens close to the Earth).  This is called uniform acceleration, and that acceleration can be measured.  We show in the first lab a natural way to measure it. 

            If we choose a unit of time T (say 1 second) then there is a distance D associated with it by the Earth itself.  That distance D is the distance any object would fall from rest in the interval T of time.  Near the earth, it is roughly 4.9 meters, or a little over 16 feet.  It is a natural unit to use, because, according to Galileo, the distance (measured in units of D) that an object will fall in t (measured in units of T) time is then

            This is a beautiful formula.   Every gravitating mass (the Earth, Moon, Sun, etc.) associates a constant distance with a unit of time, at least near its surface.  Galileo did not say this because he restricted himself to what was observable.  But Newton did.

             Now a starting point for the study and description of motion is to analyze it into its "parts".  Galileo did this and discovered that ballistic projectiles (cannon balls or baseballs) have two independent components of motion.  One of them is vertical, and it is directly affected by gravity.  The other is horizontal, and it is, in principle, unaffected by gravity.  Our second and third Labs on Ballistics and Inclined Planes follow Galileo in this analysis. 

            The lab on ballistics associates with each trajectory, a parabola.  It is interesting that when we later follow Kepler and Newton in the study of planetary orbits using Calculus, we will discover that the latter orbits are ellipses.  It seems the 17^th Century had a fascination with conic sections!  

            Our Lab on inclined planes will go farther.  In fact, we will discover two great principles there.  The first is the Galilean principle of uniform motion.  Later articulated by Newton in the Principia, it states:  "Every body perseveres in its state of rest or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon."  Galileo sees this in the limiting case of inclined planes with very little slope, and in the measure of initial and final speeds.

            The second principle was known but was not well understood in Galileo's time.  It became available to Galileo through his postulate of the uniform acceleration due to gravity.  Even without Calculus, Galileo was able to apply the Archimedean style of synthetic integration to conclude what we now call the principle of reciprocity, or the principle of conservation of energy.  With the Calculus in the hands of Newton, this will become a powerful explanatory principle, as we shall see. 

            As we shall discover in the Inclined Planes Lab, Galileo's formula

obtained by a clever sort of "integration" leads directly to this principle.  It will imply that the increase in speed (due to gravity) of a ball rolling down the plane only depends on the vertical distance through which it rolls. 

            Along with the conservation of energy, Galileo's principle of uniform motion might have led him to the conservation of momentum.  We owe the expression of this idea to Newton however, and in our Energy and Momentum Conservation Lab, we introduce the vector notation that can help us understand it.  We also perform an amusing experiment with elastic collisions that illustrates these principles very nicely.  This is another preparation for the Calculus, where we will apply a new idea using vectors, the conservation of angular momentum in a central force field to understand Kepler's second law.

            The second law states that if an imaginary line is drawn from the sun to any planet, the line will sweep out equal areas in space in equal periods of time for all points in the orbit of that planet. 

            Next, our two labs on Pendulum Motion illustrate the principle of reciprocity in a new way.  We see with our frictionless pendulums how the continuous interchange of "kinetic" and "potential" energy can give rise to very regular and stable motions (in the workings of clocks, for example). 

            The drama of 17^th Century mathematics was centered on the Copernican view of the Sun and Planets.  In order for the Galilean view of terrestrial gravity to lead to an explanation of these phenomena that could justify this view, it was necessary for Newton to extend the latter.  This led to a new view of gravity that we shall explore in detail with the Calculus in later lectures.  Our Lab Two Views of Gravity explores some of the simple differences between these views and sets the stage for later developments using the grand conservation principles.

            The Labs Bernoulli's Shortest Time Problem and The Brachistochrone are a diversion that introduce polar coordinates and the principle of Least Time formulated also by Fermat to explain the refraction of light.  It is said that Newton invented the Calculus of Variations to solve the Bernoulli problem.  We not only show you the surprising solution in the Lab, but let you experiment with various ways to test it.

            Finally, we provide a general purpose symbolic and graphical calculator on the last page of the Microworld.  You may go there whenever you like to do calculations, draw pictures or test hypotheses.