Differential equations as models in science and engineering gregory baker - The ebook with all chapt
Differential Equations as Models in Science and Engineering Gregory Baker
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Theory of Differential Equations in Engineering and Mechanics 1st Edition Kam Tim Chau
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Engineering students have strongly influenced this book. By paying attention to their questions and discussing their frustrations, I realized what they wanted: to see and appreciate how mathematics is beneficial in science and engineering. They are attracted to the power of mathematics rather than its beauty. This book, then, is dedicated to them.
Preface
Change occurs around us all the time. Indeed, our very notion of time emanates from our daily experience of life. There is always, before, now and next. Much of human endeavor is directed to gaining better and better understanding of swirling changes around us, presumably because that understanding will aid our survival
Some change is very gradual, some very rapid and others take about the time of a few heart beats. The universal unit of change is the second, and has been so since the dawn of civilization, perhaps because it is a measure of a heart beat. Connected to the idea of time is that of motion, the movement of things. Some things move very fast (the speed of light), some at a moderate pace (walking people), and others extremely slowly (the rock in the garden) There appears to be no limit to the possible ranges in time. But our sense of time depends very much on the capabilities of our eyes. The changes we observe seem for the most part to be a continuous variation of events. What really happens is that our brains receive a very fast sequence of snapshots that is interpreted as continuous. That is why we can go to the theatre and watch a movie made from a fast sequence of images.
Some changes are predictable, for example, falling objects, the ticking of a clock, sunrise and sunset. Others occur regularly but with some randomness, like the crashing of waves at the beach. In some cases, we can predict for a while but not for too long, like the tracking of hurricanes. In our search to provide valuable pre- dictions, we have come to rely more and more on mathematical models that can describe the patterns underlying change; and for continuous behavior the mathe- matical models are primarily differential equations, equations that connect change (time derivatives) with the current state of affairs.
Much of the current research and design activity of scientists and engineers is geared around the solution to differential equations. In the not too distant past, the only tools to treat differential equations were those of analysis and approximation. Many clever techniques have been developed, especially for those equations that are of fundamental importance to science and engineering The solutions found by these techniques provide the insight that allows us to conceive what happens under much more complicated circumstances. Today we can add computer algorithms to our tool box and the power of modern computers is opening up doors in almost all subjects, but certainly in the solution to differential equations. Yet we all start the journey to understanding differential equations with simple, but important examples from science and engineering It is only with a basic understanding of differential equations and the properties of their solutions that the journey can continue into the more complex nature found in modern science and engineering.
Given that differential equations started their central position in science and engineering in the middle of the 17th century, it is not surprising that a large body of knowledge has been created about the nature of differential equations and their solutions and the techniques for constructing them More recently (since the 20th century), more abstract interpretations have
arisen that extend the type of possible solutions to cases where the functions are not even continuous. All this progress has enriched our understanding of differential equations, but it also means that it takes some considerable time to learn all the complexities of differential equations, the huge variety of solution techniques, and the importance of properly stated additional conditions, in particular, initial and boundary conditions. On the other hand, there are fundamental properties of differential equations that are extremely important in science and engineering, and these fundamental properties are what students need to learn first.
Some organization of the material is necessary for the purpose of guiding a student’s conceptual development of differential equations, and there are of course many possible themes For a science and engineering student, the conceptual view should be at the same time strongly connected to relevant applications. The hope is that students begin to incorporate mathematical activity, including mathematical reasoning, into the natural activities of science and engineering, and at the core of that activity is the ability to translate physical reality into mathematical models that allow analysis and construction of solutions that in turn lead to interpretation and understanding of physical phenomena. The notion is one of mathematical literacy for scientists and engineers: it is more than simply translation, but the ca- pability to identify and understand the role that mathematics plays in the scientific and engineering world. Literacy invokes the mantra “Learn to read so that you can read to learn.” Mathematical literacy suggests “Learn mathematics so that you can use mathematics.”
The material in this book reflects this perspective Common problems in science and engineering are introduced as the motivation for developing an understanding of differential equations. Students are exposed to the complete process:
• the idealization of a real-life situation, itself a partly mathematical process, for example, we record the location of a particle as a precise number when the precision is a convenient idealization but not completely scientifically justified;
• the conversion of this idealization into a mathematical framework, based on the decision of what is important and what is needed (usually reflected on what assumptions are made);
• the mathematical processes and solution strategies that result in meaningful conclusions;
• and the evaluation of the success of the model.
The purpose of the mathematical model should be clear: it should be able to shed light on behavior of the system. Does the solution establish the nature of the long term behavior, whether stationary or steady or unstable? How does the solution make a transition from some initial state to the long term behavior? Subordinate to this process is the identification of important length and time scales. What role do parameters play in the mathematical results? Are there possibilities for control of parameters to optimize design? All these properties are stated in essentially mathematical terms and must be incorporated into learning about differential equations. Mastery of these properties demand higher cognitive function and can only be learned through consistent use and practice throughout the material, but once mastered the ability is lost only through lack of use.
There is another very important aspect to the organization of the material in this book, cohesion. There is a central theme that connects all parts of the material. A good starting point is the nature of solutions to linear differential equations because they exploit many of the fundamental ideas behind linear operators, such as linear superposition of homogeneous
solutions. They also reveal several other important concepts that run through all differential equations, such as the role of initial and boundary conditions. The student needs to master these concepts before moving on to more advanced topics The central theme in this book is to introduce these concepts right from the start and then re-enforce them as the material moves from first-order ordinary differential equations, through second-order ordinary differential equations and on to the standard second-order partial differential equations. The fundamental concepts are:
• The solutions to differential equations are functions! While apparently obvious, it takes some adjustment in thinking to shift from solving algebra equations where a single quantity is determined to the idea that differential equations determine a function. Also, there is a shift from viewing differentiation as a process to produce a result (the slope) and integration as a process to obtain a result (the area) to a view where they are operators acting on functions to produce functions.
• The properties of the function that solves the differential equation must reflect the nature of the equation For example, if there is a jump in one of the terms, especially the forcing term, then an appropriate derivative must have the appropriate jump to ensure the balance of terms in the differential equation.
• Homogeneous linear equations have the property that if a solution is known, then any multiple of the solution will satisfy the differential equation. It is important to find all the independent homogeneous solutions because a linear combination of them is needed to solve initial and/or boundary conditions.
• Inhomogeneous equations arise from forcing terms and can be solved by seeking a particular solution, one that ensures a balance of the differential equation with the inhomogeneous or forcing term and/or a match with inhomogeneous or forcing effects at boundaries.
• A general solution is constructed by adding a particular solution, if needed, with the linear combination of all homogeneous solutions that are independent
• A specific solution is obtained when initial conditions, or boundary conditions, are used to evaluate the arbitrary coefficients in the linear combination of homogeneous solutions. In this setting, science and engineering students learn how initial and boundary conditions are important in the mathematical model to describe completely a physical situation.
This theme is driven home most easily when the differential equations have constant coefficients. This restriction does not weaken the conceptual understanding of differential equations but allows students to construct concrete solutions with important applications in science and engineering. Indeed, the engineering faculty complain that students who have taken differential equation courses are not able to solve even the simplest linear equations with constant coefficients Emphasizing these equations in this book helps students retain the necessary perspectives and skills needed in subsequent engineering courses without losing important mathematical concepts.
Course Outline
Homogeneous solutions to differential equations with constant coefficients are just exponential functions for the simple reason that any derivative of an exponential is an exponential, thus ensuring all terms in the homogeneous differential equation will balance. At
the same time, exponential functions describe the standard ideas behind growth and decay, in other words, the question of stability. For second-order differential equations, and higher-order, the arguments in the exponentials can be complex The connection between exponentials of complex arguments and trigonometric functions with growing or decaying oscillations extends the concepts of stability and introduces the possibility of oscillatory behavior that occurs frequently in science and engineering.
The next important concept is that particular solutions to inhomogeneous differential equations tend to follow the nature of the forcing term. Important examples arise when the forcing term is a combination of polynomials, exponential functions and cosines and/or sines A particular solution, then, is easily guessed and can be completely determined without “integration formulas.” Furthermore, this approach is intuitively appealing. If a system is forced periodically, its response is periodic: if the forcing effect is exponential, the response is exponential. It is closely associated with the method of undetermined coefficients, but has extensions also to partial differential equations Particular solutions rarely satisfy initial conditions, and so the need to compose a general solution with particular solutions and homogeneous solutions that contain unknown coefficients.
During the development of the constructive procedure outlined above, two other aspects of the nature of the solutions to differential equations must be brought out: the mathematical properties of the solution and the need for initial conditions to determine a specific solution completely Important mathematical properties of the solution follow from the requirement that terms in the differential equation must balance the nature of the forcing term. For example, if the forcing effect is a linear ramp followed by a constant value, then the forcing term has a jump in its derivative, and the term containing the highest derivative in the differential equation must have the same mathematical property. Such considerations help remind students of the nature of differential equations as determining functions and their properties These properties under the appropriate conditions allow the differential equation to be differentiated and the Taylor series evaluated provided there is sufficient information from the initial conditions to ensure all the initial derivatives can be evaluated recursively. While students dislike the Taylor series, it does bring home the need for appropriate initial conditions to determine a specific solution completely.
Besides the mathematical concepts underlying linear differential equations, the applications chosen to illustrate them also introduce the student to subsidiary ideas important in differential equations as models for phenomena in science and engineering. These include the notion of evolutionary processes that are continuous in time and lead naturally to time-dependent differential equations. Certain scientific principles, such as the conservation of quantities, are fundamental to understanding what governs the processes and leads to the derivation of differential equations. The differential equation is not just some mathematical expression but contains important information, and the student needs to be able to recognize that information. The solution to the differential equation will also reveal important information, such as the role of parameters in the model, the eventual state of the process, and the opportunity to design the process in some optimal way. It is the whole picture that counts, and if only solution techniques to differential equations are presented, the student is denied the opportunity to see the whole picture.
Once the student sees how differential equations arise in science and engineering and has grasped the ideas behind constructing particular solutions, homogeneous solutions and general
solutions, it is time to move on to forcing effects that have more complicated mathematical nature. The next obvious step is to consider forcing terms that are periodic, for example, as might be found in electric circuits This is a good time to introduce the Fourier series Without concepts from linear algebra, it is difficult to motivate the derivation of the Fourier series as a representation in terms of a basis or eigenfunction expansion. Instead, I appeal to the idea that certain averages must balance, for example, if the Fourier series is to represent a periodic function, then the averaged values of the function must match the averaged value of the Fourier series. The important point is that the periodic inhomogeneous or forcing term is replaced by a Fourier series with trigonometric functions that allow the ideas underlying the method of undetermined coefficients to be applied successfully in the construction of the particular solution which also takes the form of a Fourier series.
The journey through the material now takes on the challenge of developing these ideas for two-point boundary value problems. The motivation here is the need to find steady solutions to transport phenomena with spatial variation Previous ideas extend naturally: particular solutions will take care of inhomogeneous terms in the differential equation and the homogeneous solution will take care of the boundary values. But now the student will see for the first time that solutions do not always exist!
There is a new perspective that can be introduced at this point. Boundary value problems consist of both a differential equation and a set of boundary conditions. The problem is homogeneous if both the differential equation and the boundary conditions are homogeneous Generally, there is only the trivial solution to the homogeneous boundary value problem, except there might be times when the differential equation has oscillatory solutions that automatically satisfy the boundary conditions so that a unique solution is not determined. If there is a parameter in the differential equation, non-unique solutions may arise for special values of the parameter, akin to the role of eigenvalues
The stage is now set for the composition of the ideas developed for solutions to initial value problems, boundary value problems and the Fourier series to solve partial differential equations. Here the thread of the previous concepts can be brought together: the multiplicative nature of exponentials for homogeneous problems (both the partial differential equation and the boundary conditions) suggests the method of separation of variables. Separation of variables leads to homogeneous boundary value problems which need non-unique solutions (eigenvectors) for the construction of the solution to succeed. The Fourier series then completes the approach. If the problem is inhomogeneous, then a particular solution must be constructed separately and added with the solution to the homogeneous problem. The solutions to higher-dimensional, secondorder partial differential equations rest on the recursive construction of particular and homogeneous solutions, and this process is the most challenging conceptual part of the course
The last chapter introduces the ideas behind systems of first-order differential equations. It serves a dual purpose of presenting the construction of solutions to linear differential equations and presenting the geometric ideas underlying the interpretation of the global behavior of solutions, in other words, the phase diagram. To draw a phase diagram, equilibrium points must be located and their stability provides the information to start the trajectories in phase space. Stability of an equilibrium point is approached through linearization, so the dual purpose of this chapter is brought together through the importance of linear approximations even in nonlinear models. Here, linear algebra would be a great help. Instead, the systems are mainly restricted to two and three-dimensional where the calculations of the solutions are relatively straightforward.
During these conceptual developments in mathematics of differential equations, connections are continually made to: the origin of differential equations in science and engineering; the importance of initial and boundary conditions to complete the description of the physical models; the role of parameters; and the physical consequences of the solutions. In this way, the natural place of mathematics in the lives of scientists and engineers is placed at the center of this course.
Gregory Baker
A Note to the Student
This book, and the course that uses it, will most likely be very different from any mathematics course you have ever taken. The material is dedicated to showing you how mathematics is used in science and engineering. That means ideas in science and engineering must be expressed in mathematical terms and the ability to do so must be learnt along the way of learning how to solve differential equations. As a scientist and engineer, you will have to convert real-life situations into mathematical models and then employ mathematical thinking to find answers. This process is very different from that you embark on when learning mathematics in a traditional course, where problems are stated in mathematical terms and you simply have to follow the procedures given in class or the textbook to find answers
The usual strategy for a student to “survive” a mathematics class is:
• Attend lecture to gain some idea of what you will need to do to answer assigned problems.
• Attempt homework and, if stuck, scan lecture notes and the textbook to find a similar problem and copy how it is solved.
• Make sure your answer agrees with the one given in the back of the book
• Do many problems to be sure that you will find one or more of them on exams.
• Be sure to get as much partial credit as you can.
This strategy suffers from several disadvantages: it encourages low-level cognitive behavior such as pattern recognition and mimicking procedures but does not encourage understanding nor the development of a connected body of mathematical knowledge capable of being used whenever needed. It decreases the opportunity to struggle with mathematical reasoning that leads to learning and reduces the opportunity to gain confidence that what you do mathematically is correct.
Instead, a new approach will likely be very successful, not only in this course but in subsequent courses in science and engineering when mathematics is needed Try to:
• Go over the problems in class and this book in detail and understand how each step is done. The goal is that you can do these problems on your own.
• Ask yourself why did the procedure succeed in answering the problem. What is the “key” idea?
• Use the approach in similar problems to gain confidence in how to find answers
• Learn to “read” mathematics. Each mathematical statement should make sense to you.
• Check your work carefully. Not only must each mathematical statement make sense but also that subsequent statements follow logically and make sense.
• Pay attention to the units of measurement for all quantities. In particular, the arguments of standard functions, such as exp(x), cos(x), must have no units A mismatch of units indicates the presence of errors
• Repeat problems every few days to make sure that you remember what is important to solve the problems.
• When you believe that you have the correct answer, write it down clearly The very process of writing aids in learning because you must pay attention to what is expressed.
The ultimate goal is simple but requires dedication and hard work. If you can solve all the problems on your own and be sure that you are correct, then you will have mastered the material in this book. Just as an athlete must focus and practice, you too must focus and practice, but the payoff is great When you finally play the game, you will be prepared and confident and relish the thrill of succeeding.
4.4.2 Steady state solution as a particular solution
4.4.3 Electrostatic potential
4.4.4 Abstract view
4.4.5 Exercises
4 4 6 Heat transport in two dimensions – continued
5. Systems of Differential Equations
5.1 First-order equations
5.1.1 Population dynamics
5.1.2 Abstract view
5 1 3 Exercises
5.2 Homogeneous linear equations
5.2.1 Basic concepts
5.2.2 Chemical reactions
5.2.3 The LCR circuit
5.2.4 Abstract viewpoint
5 2 5 Exercises
5.2.6 Higher dimensional systems
5.3 Inhomogeneous linear equations
5.3.1 LCR circuit with constant applied voltage
5.3.2 LCR circuit with alternating applied voltage
5.3.3 Stability
5 3 4 Abstract view
5.3.5 Exercises
5.3.6 General forcing term
5.4 Nonlinear autonomous equations
5.4.1 Predator-prey model
5 4 2 Abstract view
5.4.3 Exercises
Appendix A The Exponential Function
A.1 A review of its properties
Appendix B The Taylor Series
B.1 A derivation of the Taylor series
B.2 Accuracy of the truncated Taylor series
B.3 Standard examples
Appendix C Systems of Linear Equations
C 1 Algebraic equations
C.2 Gaussian elimination
C.3 Matrix form
C.4 Eigenvalues and eigenvectors
Appendix D Complex Variables
D.1 Basic properties
D.2 Connections with trigonometric functions
Index
Linear Ordinary Differential Equations 1
1.1 Growth and decay
A good starting point is to see just how differential equations arise in science and engineering. An important way they arise is from our interest in trying to understand how quantities change in time. Our whole experience of the world rests on the continual changing patterns around us, and the changes can occur on vastly different scales of time. The changes in the universe take light years to be noticed, our heart beats on the scale of a second, and transportation is more like miles per hour. As scientists and engineers we are interested in how specific physical quantities change in time. Our hope is that there are repeatable patterns that suggest a deterministic process controls the situation, and if we can understand these processes we expect to be able to affect desirable changes or design products for our use.
We are faced, then, with the challenge of developing a model for the phenomenon we are interested in and we quickly realize that we must introduce simplifications; otherwise the model will be hopelessly complicated and we will make no progress in understanding it. For example, if we wish to understand the trajectory of a particle, we must first decide that its location is a single precise number, even though the particle clearly has size. We cannot waste effort in deciding where in the particle its location is measured,1 especially if the size of the particle is not important in how it moves. Next, we assume that the particle moves continuously in time; it does not mysteriously jump from one place to another in an instant. Of course, our sense of continuity depends on the scale of time in which appreciable changes occur. There is always some uncertainty or lack of precision when we take a measurement in time Nevertheless, we employ the concepts of a function changing continuously in time (in the mathematical sense) as a useful approximation and we seek to understand the mechanism that governs its change.
In some cases, observations might suggest an underlying process that connects rate of change to the current state of affairs. The example used in this chapter is bacterial growth. In other cases, it is the underlying principle of conservation that determines how the rate of change of a quantity in a volume depends on how much enters or leaves the volume. Both examples, although simple, are quite generic in nature. They also illustrate the fundamental nature of growth and decay.
1.1.1 Bacterial growth
The simplest differential equation arises in models for growth and decay. As an example, consider some data recording the change in the population of bacteria grown under different temperatures. The data is recorded as a table of entries, one column for each temperature. Each row corresponds to the time of the measurement. The clock is set to zero when the bacteria is first placed into a source of food in a container and measurements are made every hour afterwards. The experimentalist has noted the physical dimensions of the food source. It occupies
a cylindrical disk of radius 5 cm and depth of 1 cm. The volume is therefore 78.54 cm3 . The population is measured in millions per cubic centimeter, and the results are displayed in Table 1.1.
Glancing at the table, it is clear that the populations increase in the first two columns but decay in the last Detailed comparison is difficult because they do not all start at the same density. That difficulty can be easily remedied by looking at the relative densities. Divide all the entries in each column by the initial density in the column. The results are displayed in Table 1.2 as the change in relative densities, a quantity without dimensions. Since we use the initial density as a yardstick, all the first entries are just 1, and now it is easy to see that the second column shows the fastest change when the temperature is 35 °C
But looking at data in a table has limitations. It does not show, for example, whether the changes appear regular or random. Is there a smooth gradual change? Instead, we prefer to “see” the data in a graph as displayed in Fig. 1.1. Now it is apparent that the changes appear systematic. The question is “what is the pattern in the data?” The data for temperature at 35 °C looks like it might be a quadratic One way forward then is to try to fit the data to a quadratic To proceed, we need to introduce some symbols to represent the data before we can test the fit to any choice of function.
An obvious symbol for the measurement of time is t, but there are measurements at several different times. We may designate each choice by introducing a subscript to t, tj where t0 = 0, t1 = 1, and so on. Note that the symbol t records a number but with some units in mind, in this case hours 2 We may, of course, use any symbol to represent time, but it makes good sense to pick one that will remind us what the quantity is. Now we need to pick a symbol for the relative density. My choice is ρ (no units). For each time measurement tj, we may associate a density measurement from one of the columns, for example, ρj (Temperature = 25 °C). Obviously, we have three choices for the temperature so we can introduce a subscript to a symbol, Tk say, for
Table 1 1 Population densities in millions per cubic centimeters
Table 1 2 Relative population densities
the temperature with T1 = 25 °C, T2 = 35 °C, and T3 = 45 °C. From the experimentalist’s point of view, tj is the independent variable (measurements are made to reveal how the density changes with time), ρj is the measurement, and Tk is a control variable held fixed while the bacteria grows. From a mathematical perspective, tj is the independent variable and ρj is the dependent variable, while Tk is a parameter (a fixed constant for each set of results). If we wish to emphasize the role of the temperature on the measurements, we can use ρj(Tk), but we generally take it for granted that we know which data we refer to and drop the reference to Tk.
Fig. 1.1 Relative population density of bacteria for three different temperatures.
Now we are in a position to ask how the increases in the relative density change in time. There are two obvious ways the change in relative density can be measured. One is to record the jump in value (the absolute increase) and the other is to record the relative increase (the ratio of the absolute change to the current value) At each time tj, we calculate the absolute increase by
and show its value in Table 1.3 for temperature T2. Also, we can calculate the relative increase,
and it is shown as a separate column in Table 1.3.
Table 1 3 Changes in the relative densities during an hour
From the data, it seems that the relative change during an hour remains very close to a constant. In other words, it appears as though
where C is a dimensionless constant 3 When we make this assumption, we are introducing a mathematical idealization. We are stating that there is a definite principle that guides the behavior of the growth of the bacteria, namely, that the relative change of the density is a constant. The advantage is that we can now use the power of mathematics to study the behavior of ρj and draw some conclusions.
Equation (1.1.1) is a simple example of a recursion relation. A solution is generated by applying the recursion sequentially
and so on It is very easy to make the guess that
But is this guess correct? Although it seems intuitive that it must be correct (how can it be wrong?), we should always verify our guess, in this case by direct substitution into Eq. (1.1.1). The right and left hand sides of Eq. (1.1.1) are
and both sides agree for any choice of j: thus Eq. (1.1.2) must be the solution for Eq. (1.1.1).
Sometimes it helps to see how a different guess for the solution fails Suppose we guess that ρj falls on a quadratic function,
where a is to be determined so that the equation is satisfied. The right and left hand sides of Eq. (1.1.1) are
and there is no value of a which will make the left and right hand sides match perfectly, for example, there is a term with j on the left hand side but there is no such term on the right hand side. Clearly, then, ρj = 1 + aj2 cannot be a solution to Eq. (1.1.1)
Informed choices for the solution of equations will prove to be a very valuable technique. The idea is that we make a certain guess for the solution, replace the unknown function in the equation by the guess and verify that the equation is always satisfied If this works then we are sure we have found the solution. For example, we made the guess Eq. (1.1.2) for the solution to Eq. (1.1.1) and verified it satisfies the equation for any choice of j.
The behavior of the solution in Eq. (1.1.2) is exponential. To see this clearly, use Eq. (A.1.1) to rewrite Eq. (1.1.2) as
and now it is clear that the density increases exponentially as j increases. A review of the properties of exponential functions is provided in Appendix A.
Reflection
Exponential growth or decay reflects a constant relative change.
1.1.2 From discrete to continuous
The growth of the bacteria at a temperature of 35 °C has proved to be exponential when measurements are made at each hour. Suppose we record the results more frequently, say every Δt units of time. If Δt is less than an hour, then obviously the relative change must be less. For example, if we now record results every half hour, then we might expect the relative change to be halved Let us make the assumption that the density recorded at tj = jΔt changes at a fixed relative rate that is proportional to Δt. In other words, we should replace C in Eq. (1.1.1) by λΔt. The parameter λ must have the units per hour because Δt must have the same units as t which is hours.
with the solution indicated by Eq. (1.1.3),
Fig. 1.2 General pattern in the discrete data.
Let us keep halving the measuring interval, and imagine more data points added to Fig. 1.1. Surely, the points will continue to fill in a continuous curve and eventually the relative density becomes ρ(t). Mathematically, the counter j becomes a continuous variable t. How this happens is illustrated in Fig 1 2 If we knew what the function ρ(t) was, then we could easily determine the data points ρj by simply evaluating
The challenge is to determine ρ(t) when all we know is ρj. Clearly, to connect ρj with ρ(t), we must pick a fixed time t = JΔt and state ρJ = ρ(JΔt). Note that as we decrease Δt, we must increase the number J of time increments to remain at the same time t. Indeed, J = t/Δt and
It may seem that we have constructed a solution ρ(t) that is a continuous function of t. Not so, because t still jumps in multiples of Δt and Δt still appears in Eq (1 1 5) We must take the limit of Δt → 0 so that t becomes continuous. Of course we must take this limit with t = JΔt kept fixed, which means J → ∞ (it takes infinitely many very small time increments to reach a fixed
time). By replacing J with t/(Δt) in Eq. (1.1.5) and then keeping t fixed, we avoid the problem of what to do with J.
In the limit Δt → 0,
One way to obtain this limit is by L’Hospital’s rule. Another is to use the Taylor series expansion for ln(1 + x) = x x2/2 + ··· and cancel Δt before taking the limit. Either way the result is,
The relative density grows continuously as an exponential function!
For the data in Table 1.2 to match with Eq. (1.1.6), we must have at t = 1, and the solution Eq. (1.1.6), along with the data is shown in Fig. 1.3. We have constructed a continuous approximation to the growth of the bacteria that matches the experimental observations!
We have found a continuous function Eq (1 1 6) as the limit of the solution Eq (1 1 5) of the recursion Eq (1 1 4), but what is the limit for the recursion? Pick j = J, and set t = JΔt Then the two sides of Eq. (1.1.4) become and Eq. (1.1.4) can be written as which has the limit,
How remarkable! The solution Eq. (1.1.6) solves Eq. (1.1.7). To be sure of this, we should check the solution by direct substitution.
Fig. 1.3 Continuous exponential growth compared to the data.
Note how this statement compares with Eq. (A.1.5)! It is equivalent to the property of differentiation of exponential functions.
Equation (1.1.7) is the simplest example of a differential equation. It is an equation that involves the derivative of the unknown function ρ(t). It is called first-order because only the first derivative is involved: more derivatives, higher order It is also linear That means the unknown function and its derivatives appear separately and not as the argument of some other function.
Reflection
The differential equation Eq. (1.1.7) states that the instantaneous rate of change of some quantity is proportional to that quantity. There are many examples of such a relationship and we will consider several of them in what follows
1.1.3 Conservation of quantity
Suppose we have a drum filled with contaminated water, and we want to flush it out. We plan to pour clean water in until the water reaches a desired level of purity. If we want to study this reallife situation we must be more clear about what is happening, and in bringing clarity we inevitably make assumptions. For example, what is the rate at which the clean water is injected into the drum? We will assume that it is a constant, and that the rate of contaminated water flowing out is the same. The consequence is that the volume of water in the drum remains constant as well. The flow of water in is clean while the flow of water out contains contamination. How much contaminant flows out? If the water in the tank is well stirred, we expect the concentration in the tank to be uniformly constant, and that must then be the concentration in the water flowing out of the drum
Fig. 1.4 A schematic of the drum cleansing.
We are now ready to develop a mathematical model. What do we need to know? It often helps to draw a “cartoon” of the situation and to consider what quantities are involved. In Fig. 1 4, the drum is represented as a rectangular shape with inlet and outlet pipes The capacity of the drum is important, and is assumed to be V = 100 gal. Also in the drum is the contaminant dissolved in the water. The amount of contaminant is represented by Q(lb), but we anticipate it will change in time, so we write it as a function of time Q(t) where t is measured in minutes. We
can set our timer at t = 0 min when we start to flush the drum, and we assume we know the amount of contaminant at this moment, Q(0) = Q0. The arrow at the mouth of the inlet pipe in Fig 1 4 indicates the flow of water into the drum at a rate of r = 3 gal/min and we assume that the outflow at the outlet pipe has the same rate, thus keeping the volume of water a constant value V. Finally, we assume that the water flowing out carries a concentration of the contaminant that is the same as the concentration in the drum ρ(t) = Q(t)/V (lb/gal). So what do we want to know? We would like to know how much contaminant is left at a later time t (min). This means we must know how the amount of contaminant changes in time Q(t) is an unknown function, t is the independent variable, and V, r and Q0 are parameters (remain fixed for each simulation of drum cleaning).
Now that we have established a mathematical description of the drum cleansing, we need an equation that determines Q(t). Let us consider a small time interval Δt (min) and imagine we have made a series of measurements Qj = Q(jΔt). How does the amount Qj change during the time interval Δt? The basic principle of conservation says that the amount of contaminant must change according to
The assumption we make is that the concentration flowing out is the same as the current concentration in the drum which is Qj/V (lb/gal). The volume of water flowing out during Δt (min) is rΔt (gal). The amount of contaminant in this water is the volume times the concentration Thus Eq (1 1 8) becomes
Pick t = jΔt and replace Qj = Q(t) and QJ+1 = Q(t + Δt). So the equation becomes
After dividing by Δt and taking the limit Δ → 0, we obtain the differential equation
Let us first take note of the nature of this equation. The unknown “variable” is the function Q(t) and the independent variable is t. There are two parameters r and V to be considered known for each time we solve the equation. We know how to solve this equation, since it is very similar to Eq. (1.1.7) and the solution must be similar to Eq. (1.1.6). Indeed, we just replace λ with r/V.
Great, but there is a problem! At t = 0, Q(0) = 1. That was true for ρ(t) in Eq. (1.1.6), but that is not right for Q(t) The initial amount is given by Q(0) = Q0 The parameter Q0 allows us to study the situation for any initial amount of contamination and we certainly do not want to restrict ourselves to the choice Q0 = 1.
Fortunately, the remedy is simple. We start afresh and assume that the solution has the form
The choice made reflects the expectation that the solution is an exponential but we do not know yet what λ to pick. We also multiply the solution by C because we will need to satisfy the initial
condition. Thus we have two unknown constants in the guess for the solution and we expect them to be determined by requiring the guess to solve the differential equation and the initial condition First, substitute the guess into Eq (1 1 9) Since
the two sides of Eq. (1.1.9) will balance perfectly with the choice λ = r/V.
Notice that the key to success is that the derivative of an exponential is proportional to the exponential. The constant C simply cancels on both sides and remains unknown. It is determined by the application of the initial condition, Q(0) = C = Q0 at t = 0, and finally the solution is completely determined,
Finally, we can answer how long we must wait until Q(t) < α, some tolerance factor. The solution states that the Q(t) decays exponentially, and so there will be a time T when Q(T) = α and that is the first moment when Q(t) begins to fall below the tolerance level
The factor V/r is a very important physical quantity. Note first that it has the units of time. For this example, V = 100 gal, r = 3 gal/min, so V/r ≈ 33 min. The result gives an estimate for the typical time it takes for Q to change appreciably. To be more specific, note that at time t = τ ≡ V/r,
and the quantity Q(τ) is reduced by the factor 1/e of its initial value For this reason τ is called the e-folding time. In conclusion, it will take about half an hour to reduce Q by the factor α/Q0 = 1/e ≈ 1/3.
Reflection
The instantaneous rate of change of a quantity must be related to how it is added or subtracted The presence of parameters usually governs the rate of change, and hence the time scales for change to occur. As scientists and engineers, we are very interested in just how long a process will take. Mathematically, we look at the argument of any function in the solution, in the example, the exponential function. The argument of this function can have no units! Imagine the difficulties in evaluating exp(3 min) = exp(180 s) since the results should be the same
1.1.4 Simple electric circuits
So far, two examples have shown ways in which differential equations arise: population growth/decay that led to a phenomenological model, and the conservation of quantity that led to a model for mixing. Another way that differential equations arise is in models for electric circuits. A simple electric circuit is shown in Fig. 1.5. It is composed of a resistor R (Ω) and a capacitor C (F) in series and may have an externally applied voltage E (V) Also shown is a
switch that can turn the applied voltage on or off As a start, suppose there is no applied voltage, but the capacitor is fully charged with charge Q0(C). The switch is closed and the capacitor simply discharges.
The governing principles for electric circuits are called Kirchoff’s laws. For the simple circuit in Fig. 1.5, Kirchoff’s laws state:
• The sum of all the voltage changes around a closed circuit must be zero.
• The current is the same at all parts of the circuit.
To apply Kirchoff’s laws it is useful to introduce values for the voltage at different parts of the circuit They are shown as Vj (V) in Fig 1 5 The usual assumption is adopted, namely, that there is no voltage change along the wire, but only across the resistor and capacitor. Also shown is the current and its direction. The charge on the capacitor is shown in a manner consistent with the direction of the current I (A). Let VR = V1 V2 be the change in voltage across the resistor and Vc = V2 V3 be the change in voltage across the capacitor. Then
Here we acknowledge that no external voltage is being applied. While there is no current through the capacitor, the rate of change of the charge Q(t) must match the current in the circuit.
The voltage drop across the resistor is given by VR = RI and across the capacitor by VC = Q/C Thus Eq. (1.1.13) becomes
To solve this equation, assume the solution has the form where A and λ are unknown constants.4 Substitute this guess into the differential equation,
Fig 1 5 RC circuit
When solving Eq. (1.1.10), we picked λ to ensure the balance of the terms on either side of the differential equation because the exponential was common. Here we present a different, but equivalent, point of view. We have factored A and the exponential function from both terms because they are common, and then we should pick
to ensure the differential equation is satisfied 5 Both approaches Eq (1 1 10) and Eq (1 1 17) are equivalent, and will lead to the same result for λ.
We have found a function that satisfies the equation,
with an unknown coefficient A. The last step is to invoke an initial condition to determine A. Initially, the capacitor has a charge Q0. Stated mathematically, Q(0) = Q0. The perspective here is that Q0 is some known number, a parameter that reflects different possible starting charges on the capacitor, but for each discharge, it is a fixed constant Apply this initial condition,
and A is now known. Thus the specific solution is
The solution Eq. (1.1.18) is similar to Eq. (1.1.11) except that the time scale is RC. In both cases, the concentration Eq. (1.1.11) and the charge on the capacitor Eq. (1.1.18) relax exponentially in time to the state where they become negligibly small.
1.1.5 Abstract viewpoint
There are three examples of differential equations in this section, Eq. (1.1.7), Eq. (1.1.9) and Eq. (1.1.15). They all have a similar form aside from the choice of symbols. Let y(t) be either ρ(t) or Q(t), then the equations can be written in a standard form,
where a is either λ, r/V or 1/(RC). The equation is linear. In addition, we have an initial condition
which in the first case is y0 = 1 and in the other cases y0 = Q0.
The solutions we found, Eq. (1.1.6), Eq. (1.1.11) and Eq. (1.1.18), suggest that we try a guess for the solution of the form
There is nothing wrong with making a guess for a solution, but we must verify that it is a solution. In the process, we expect to determine the unknown constant λ. 6 The way this is done is to invoke a simple principle.
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elected to Congress and was re-elected in 1804. In 1805, he was appointed United States District Judge for the new Territory of Louisiana, now the State of Missouri.
Dr. Felix Brunot arrived in Pittsburgh in 1797. He came from France with Lafayette and was a surgeon in the Revolutionary War and fought in many of its battles. His office was located on Liberty Street, although he owned and lived on Brunot Island. An émigré, the Chevalier Dubac, was a merchant.66 Dr. F. A. Michaux, the French naturalist and traveler, related of Dubac:67 “I frequently saw M. Le Chevalier Dubac, an old French officer who, compelled by the events of the Revolution to quit France, settled in Pittsburgh where he engaged in commerce. He possesses very correct knowledge of the Western country, and is perfectly acquainted with the navigation of the Ohio and Mississippi Rivers, having made several voyages to New Orleans.” Morgan Neville a son of Colonel Presley Neville, and a writer of acknowledged ability, drew a charming picture of Dubac’s life in Pittsburgh.68
Perhaps the best known Frenchman in Pittsburgh was John Marie, the proprietor of the tavern on Grant’s Hill. Grant’s Hill was the eminence which adjoined the town on the east, the ascent to the hill beginning a short distance west of Grant Street. The tavern was located just outside of the borough limits, at the northeast corner of Grant Street and the Braddocksfield Road, where it connected with Fourth Street. The inclosure contained more than six acres, and was called after the place of its location, “Grant’s Hill.” It overlooked Pittsburgh, and its graveled walks and cultivated grounds were the resort of the townspeople. For many years it was the leading tavern. Gallatin, who was in Pittsburgh, in 1787, while on the way from New Geneva to Maine, noted in his diary that he passed Christmas Day at Marie’s house, in company with Brackenridge and Peter Audrian,69 a well-known French merchant on Water Street. Marie’s French nationality naturally led him to become a Republican when the party was formed, and his tavern was long the headquarters of that party Numerous Republican plans for defeating their opponents originated in Marie’s house, and many Republican victories were celebrated in
his rooms. Also in this tavern the general meetings of the militia officers were held.70 Michaux has testified that Marie kept a good inn.71 The present court house, the combination court house and city hall now being erected, and a small part of the South School, the first public school in Pittsburgh, occupy the larger portion of the site of “Grant’s Hill.”
Marie’s name became well known over the State, several years after he retired to private life He was seventy-five years of age in 1802, when he discontinued tavern-keeping and sold “Grant’s Hill” to James Ross, United States Senator from Pennsylvania, who was a resident of Pittsburgh. Marie had been estranged from his wife for a number of years and by some means she obtained possession of “Grant’s Hill,” of which Ross had difficulty in dispossessing her. In 1808, Ross was a candidate for governor against Simon Snyder. Ross’s difference with Mrs. Marie, whose husband had by this time divorced her, came to the knowledge of William Duane in Philadelphia, the brilliant but unscrupulous editor of the Aurora since the discontinuance of the National Gazette, in 1793, the leading radical Republican newspaper in the country. The report was enlarged into a scandal of great proportions both in the Aurora and in a pamphlet prepared by Duane and circulated principally in Philadelphia. The title of the pamphlet was harrowing. It was called “The Case of Jane Marie, Exhibiting the Cruelty and Barbarous Conduct of James Ross to a Defenceless Woman, Written and Published by the Object of his Cruelty and Vengeance.” Although Marie was opposed to Ross politically, he defended his conduct toward Mrs. Marie as being perfectly honorable. Nevertheless, the pamphlet played an important part in obtaining for Snyder the majority of twenty-four thousand by which he defeated Ross.
Notwithstanding the high positions which some of the Frenchmen attained, they left no permanent impression in Pittsburgh. After prospering there for a few years, they went away and no descendants of theirs reside in the city unless it be some of the descendants of Dr. Brunot. Some went south to the Louisiana country, and others returned to France. Gallatin, himself, long after
he had shaken the dust of Western Pennsylvania from his feet, writing about his grandson, the son of his son James, said: “He is the only young male of my name, and I have hesitated whether, with a view to his happiness, I had not better take him to live and die quietly at Geneva, rather than to leave him to struggle in this most energetic country, where the strong in mind and character overset everybody else, and where consideration and respectability are not at all in proportion to virtue and modest merit.”72 And the grandson went to Geneva to live, and his children were born there and he died there.73
The United States Government was still in the formative stage. Until this time the men who had fought the Revolutionary War to a successful conclusion, held a tight rein on the governmental machinery. Now a new element was growing up, and, becoming dissatisfied with existing conditions, organized for a conflict with the men in power. The rise of the opposition to the Federal party was also the outcome of existing social conditions. Like the modern cry against consolidated wealth, the movement was a contest by the discontented elements in the population, of the men who had little against those who had more. Abuses committed by individuals and conditions common to new countries were magnified into errors of government. Also the people were influenced by the radicalism superinduced by the French Revolution and the subsequent happenings in France. “Liberty, fraternity, and equality” were enticing catchwords in the United States.
Thomas Jefferson, on his return from France, in 1789, after an absence of six years, where he had served as United States Minister, during the development of French radicalism, came home much strengthened in his ideas of liberty. They were in strong contrast with the more conservative notions of government entertained by Washington, Vice-President Adams, Hamilton, and the other members of the Cabinet. In March, 1790, Jefferson became Secretary of State in Washington’s first Cabinet, the appointment being held open for him since April 13th of the preceding year, when Washington entered on the duties of the Presidency. Jefferson’s views being made public, he immediately
became the deity of the radical element. At the close of 1793, the dissensions in the Cabinet had become so acute that on December 31st Jefferson resigned in order to be better able to lead the new party which was being formed. By this element the Federalists were termed “aristocrats,” and “tories.” They were charged with being traitors to their country, and were accused of being in league with England, and to be plotting for the establishment of a monarchy, and an aristocracy. The opposition party assumed the title of “Republican.” Later the word “Democratic” was prefixed and the party was called “Democratic Republican,”74 although in Pittsburgh for many years the words “Republican,” “Democratic Republican,” and “Democratic” were used interchangeably.
Heretofore Pennsylvania had been staunchly Federal. On the organization of the Republican party, Governor Thomas Mifflin, and Chief Justice Thomas McKean of the Supreme Court, the two most popular men in the State, left the Federal party and became Republicans. There was also a cause peculiar to Pennsylvania, for the rapid growth of the Republican party in the State. The constant increase in the backwoods population consisted largely of emigrants from Europe, chiefly from Ireland, who brought with them a bitter hatred of England and an intense admiration for France. They went almost solidly into the Republican camp. The arguments of the Republicans had a French revolutionary coloring mingled with which were complaints caused by failure to realize expected conditions. An address published in the organ of the Republican party in Pittsburgh is a fair example of the reasoning employed in advocacy of the Republican candidates: “Albert Gallatin, the friend of the people, the enemy of tyrants, is to be supported on Tuesday, the 14th of October next, for the Congress of the United States. Fellow citizens, ye who are opposed to speculators, land jobbers, public plunderers, high taxes, eight per cent. loans, and standing armies, vote for Mr. Gallatin!”75
In Pittsburgh the leader of the Republicans was Hugh Henry Brackenridge, the lawyer and dilettante in literature. In the fierce invective of the time, he and all the members of his party were styled
by their opponents “Jacobins,” after the revolutionary Jacobin Club of France, to which all the woes of the Terror were attributed. The Pittsburgh Gazette referred to Brackenridge as “Citizen Brackenridge,” and after the establishment of the Tree of Liberty, added “Jacobin printer of the Tree of Sedition, Blasphemy, and Slander.”76 But the Republicans gloried in titles borrowed from the French Revolution. The same year that Governor Mifflin and Chief Justice McKean went over to the Republicans, Brackenridge made a Fourth of July address in Pittsburgh, in which he advocated closer relations with France. This was republished in New York by the Republicans, in a pamphlet, along with a speech made by Maximilien Robespierre in the National Convention of France. In this pamphlet Brackenridge was styled “Citizen Brackenridge.”77 The Pittsburgh Gazette and the Tree of Liberty, contained numerous references to meetings and conferences held at the tavern of “Citizen” Marie. On March 4, 1802, the first anniversary of the inauguration of Jefferson as President, a dinner was given by the leading Republicans in the tavern of “Citizen” Jeremiah Sturgeon, at the “Sign of the Cross Keys,” at the northwest corner of Wood Street and Diamond Alley, at which toasts were drunk to “Citizen” Thomas Jefferson, “Citizen” Aaron Burr, “Citizen” James Madison, “Citizen” Albert Gallatin, and “Citizen” Thomas McKean.78
In 1799, the Republicans had as their candidate for governor Chief Justice McKean. Opposed to him was Senator James Ross. Ross was required to maintain a defensive campaign. The fact that he was a Federalist was alone sufficient to condemn him in the eyes of many of the electors. He was accused of being a follower of Thomas Paine, and was charged with “singing psalms over a card table.” It was said that he had “mimicked” the Rev. Dr. John McMillan, the pioneer preacher of Presbyterianism in Western Pennsylvania, and a politician of no mean influence; that he had “mocked” the Rev. Matthew Henderson, a prominent minister of the Associate Presbyterian Church.79 Although Allegheny County gave Ross a majority of over eleven hundred votes, he was defeated in the State by more than seventy-nine hundred.80 McKean took office
on December 17, 1799,81 and the next day he appointed Brackenridge a justice of the Supreme Court. All but one or two of the county offices were filled by appointment of the governor, who could remove the holders at pleasure. The idea of public offices being public trusts had not been formulated. The doctrine afterward attributed to Andrew Jackson, that “to the victors belong the spoils of office,” was already a dearly cherished principle of the Republicans, and Judge Brackenridge was not an exception to his party Hardly had he taken his seat on the Supreme Bench, when he induced Governor McKean to remove from office the Federalist prothonotary, James Brison, who had held the position since September 26, 1788, two days after the organization of the county.
Brison was very popular. As a young man, he had lived at Hannastown, and during the attack of the British and Indians on the place had been one of the men sent on the dangerous errand of reconnoitering the enemy. 82 He was now captain of the Pittsburgh Troop of Light Dragoons, the crack company in the Allegheny County brigade of militia, and was Secretary of the Board of Trustees of the Academy. He was a society leader and generally managed the larger social functions of the town. General Henry Lee, the Governor of Virginia, famous in the annals of the Revolutionary War, as “LightHorse Harry Lee,” commanded the expedition sent by President Washington to suppress the Whisky Insurrection, and was in Pittsburgh several weeks during that memorable campaign. On the eve of his departure a ball was given in his honor by the citizens. On that occasion Brison was master of ceremonies. A few months earlier Brackenridge had termed him “a puppy and a coxcomb.” Brackenridge credited Brison with retaliating for the epithet, by neglecting to provide his wife and himself with an invitation to the ball. This was an additional cause for his dismissal, and toward the close of January the office was given to John C. Gilkison. Gilkison who was a relative of Brackenridge, conducted the bookstore and library which he had opened the year before, and also followed the occupation of scrivener, preparing such legal papers as were demanded of him.83
REFERENCES
C III
58 Pittsburgh Gazette, January 23, 1801.
59 C R . An Abridgment of the Laws of Pennsylvania, Philadelphia, MDCCCI, pp. 264–269.
60 Pittsburgh Gazette, December 7, 1799.
61 N B. C . The Olden Time, Pittsburgh, 1848, vol. ii., pp. 354–355.
62 A Brief State of the Province of Pennsylvania, London, 1755, p. 12.
63 Tree of Liberty, December 27, 1800.
64 J A S . Albert Gallatin, Boston, 1895, p. 370.
65 M E D . Military Journal, Philadelphia, 1859, p. 21.
66 Pittsburgh Gazette, October 23, 1801.
67 D . F. A. M . Travels to the Westward of the Alleghany Mountains in the Year 1802, London, 1805, p. 36.
68 M N . In John F. Watson’s Annals of Philadelphia and Pennsylvania, Philadelphia, 1891, vol. ii., pp. 132–135.
69 H A . The Life of Albert Gallatin, Philadelphia, 1880, p. 68.
70 Tree of Liberty, November 7, 1800; Pittsburgh Gazette, February 20, 1801.
71 D . F A. M Travels to the Westward of the Alleghany Mountains in the Year 1802, London, 1805, p. 29.
72 H A . The Life of Albert Gallatin, Philadelphia, 1880, p. 650.
73 C D G . “A Diary of James Gallatin in Europe”; Scribner’s Magazine, New York, vol. lvi., September, 1914, pp. 350–351.
74 R H . The History of the United States of America, New York, vol. iv., p. 425.
75 Tree of Liberty, September 27, 1800.
76 Pittsburgh Gazette, February 6, 1801.
77 Political Miscellany, New York, 1793, pp. 27–31.
78 Tree of Liberty, March 13, 1802.
79 Tree of Liberty, September 19, 1801.
80 Pittsburgh Gazette, October 26, 1799.
81 W C. A . Lives of the Governors of Pennsylvania, Philadelphia, 1873, p. 289.
82 N B. C . The Olden Time, Pittsburgh, 1848, vol. ii., p. 355.
83 H. M. B . Recollections of Persons and Places in the West, Philadelphia, 1868, p. 68; Pittsburgh Gazette, December 29, 1798.
CHAPTER IV
LIFE AT THE BEGINNING OF THE NINETEENTH CENTURY
The Pittsburgh Gazette was devoted to the interests of the Federal party, and Brackenridge and the other leading Republicans felt the need of a newspaper of their own. The result was the establishment on August 16, 1800, of the Tree of Liberty, by John Israel, who was already publishing a newspaper, called the Herald of Liberty, in Washington, Pennsylvania. The title of the new paper was intended to typify its high mission. The significance of the name was further indicated in the conspicuously displayed motto, “And the leaves of the tree were for the healing of the nations.” The Federalists, and more especially their organ, the Pittsburgh Gazette, 84 charged Brackenridge with being the owner of the new paper, and with being responsible for its utterances. Brackenridge, however, has left a letter in which he refuted this statement, and alleged that originally he intended to establish a newspaper, but on hearing of Israel’s intention gave up the idea.85
The extent of the comforts and luxuries enjoyed in Pittsburgh was surprising. The houses, whether built of logs, or frame, or brick, were comfortable, even in winter. In the kitchens were large open fire-places, where wood was burned. The best coal fuel was plentiful. Although stoves were invented barely half a century earlier, and were in general use only in the larger cities, the houses in Pittsburgh could already boast of many There were cannon stoves, so called because of their upright cylindrical, cannon-like shape, and Franklin or open stoves, invented by Benjamin Franklin; the latter graced the
parlor Grates were giving out their cheerful blaze. They were also in use in some of the rooms of the new court house, and in the new jail.
The advertisements of the merchants told the story of what the people ate and drank, and of the materials of which their clothing was made. Articles of food were in great variety. In the stores were tea, coffee, red and sugar almonds, olives, chocolate, spices of all kinds, muscatel and keg raisins, dried peas, and a score of other luxuries, besides the ordinary articles of consumption. The gentry of England, as pictured in the pages of the old romances, did not have a greater variety of liquors to drink. There were Madeira, sherry, claret, Lisbon, port, and Teneriffe wines, French and Spanish brandies,86 Jamaica and antique spirits.87 Perrin DuLac, who visited Pittsburgh in 1802, said these liquors were the only articles sold in the town that were dear.88 But not all partook of the luxuries. Bread and meat, and such vegetables as were grown in the neighborhood, constituted the staple articles of food, and homemade whisky was the ordinary drink of the majority of the population. The native fruits were apples and pears, which had been successfully propagated since the early days of the English occupation.89
Materials for men’s and women’s clothing were endless in variety and design and consisted of cloths, serges, flannels, brocades, jeans, fustians, Irish linens, cambrics, lawns, nankeens, ginghams, muslins, calicos, and chintzes. Other articles were tamboured petticoats, tamboured cravats, silk and cotton shawls, wreaths and plumes, sunshades and parasols, black silk netting gloves, white and salmon-colored long and short gloves, kid and morocco shoes and slippers, men’s beaver, tanned, and silk gloves, men’s cotton and thread caps, and silk and cotton hose.
Men were changing their dress along with their political opinions. One of the consequences in the United States of the French Revolution was to cause the effeminate and luxurious dress in general use to give way to simpler and less extravagant attire. The rise of the Republican party and the class distinctions which it was responsible for engendering, more than any other reason, caused the men of affairs—the merchants, the manufacturers, the lawyers,
the physicians, and the clergymen—to discard the old fashions and adopt new ones. Cocked hats gave way to soft or stiff hats, with low square crowns and straight brims. The fashionable hats were the beaver made of the fur of the beaver, the castor made of silk in imitation of the beaver, and the roram made of felt, with a facing of beaver fur felted in. Coats of blue, green, and buff, and waistcoats of crimson, white, or yellow, were superseded by garments of soberer colors. Coats continued to be as long as ever, but the tails were cut away in front. Knee-breeches were succeeded by tight-fitting trousers reaching to the ankles; low-buckled shoes, by high-laced leather shoes, or boots. Men discontinued wearing cues, and their hair was cut short, and evenly around the head. There were of course exceptions. Many men of conservative temperament still clung to the old fashions. A notable example in Pittsburgh was the Rev. Robert Steele, who always appeared in black satin kneebreeches, knee-buckles, silk stockings, and pumps.
90
The farmers on the plantations surrounding Pittsburgh and the mechanics in the borough were likewise affected by the movement for dress reform. Their apparel had always been less picturesque than that of the business and professional men. Now the ordinary dress of the farmers and mechanics consisted of short tight-fitting round-abouts, or sailor’s jackets, made in winter of cloth or linsey, and in summer of nankeen, dimity, gingham, or linen. Sometimes the jacket was without sleeves, the shirt being heavy enough to afford protection against inclement weather. The trousers were loose-fitting and long, and extended to the ankles, and were made of nankeen, tow, or cloth. Some men wore blanket-coats. Overalls, of dimity, nankeen, and cotton, were the especial badge of mechanics. The shirt was of tow or coarse linen, the vest of dimity. On their feet, farmers and mechanics alike wore coarse high-laced shoes, halfboots, or boots made of neat’s leather. The hats were soft, of fur or wool, and were low and round-crowned, or the crowns were high and square.
The inhabitants of Pittsburgh were pleasure-loving, and the time not devoted to business was given over to the enjoyments of life. Men and women alike played cards. Whisk, as whist was called, and
Boston were the ordinary games. 91 All classes and nationalities danced, and dancing was cultivated as an art. Dancing masters came to Pittsburgh to give instructions, and adults and children alike took lessons. In winter public balls and private assemblies were given. The dances were more pleasing to the senses than any ever seen in Pittsburgh, except the dances of the recent revival of the art. The cotillion was executed by an indefinite number of couples, who performed evolutions or figures as in the modern german. Other dances were the minuet, the menuet à la cour, and jigs. The country dance, generally performed by eight persons, four men and four women, comprised a variety of steps, and a surprising number of evolutions, of which liveliness was the characteristic.
The taverns had rooms set apart for dances. The “Sign of the Green Tree,”92 had an “Assembly Room”; the “Sign of General Butler”93 and the “Sign of the Waggon”94 each had a “Ball Room.” The small affairs were given in the homes of the host or hostess, and the large ones in the taverns, or in the grand-jury room of the new court house.
The dancing masters gave “Practicing Balls” at which the cotillion began at seven o’clock, and the ball concluded with the country dance, which was continued until twelve o’clock.95 Dancing became so popular and to such an extent were dancing masters in the eyes of the public that William Irwin christened his race horse “Dancing Master.”96 The ball given to General Lee was talked about for years after the occurrence. Its beauties were pictured by many fair lips. The ladies recalled the soldierly bearing of the guest of honor, the tall robust form of General Daniel Morgan, Lee’s second in command, and the commander of the Virginia troops, famous as the hero of Quebec and Saratoga, who had received the thanks of Congress for his victory at Cowpens. They dwelt on the varicolored uniforms of the soldiers, the bright colors worn by the civilians, their powdered hair, the brocades, and silks, and velvets of the ladies.
In winter evenings there were concerts and theatrical performances which were generally given in the new court house. A
unique concert was that promoted by Peter Declary It was heralded as a musical event of importance. Kotzwara’s The Battle of Prague, was performed on the “forte piano” by one of Declary’s pupils, advertised as being only eight years of age; President Jefferson’s march was another conspicuous feature. The exhibition concluded with a ball.97
Comedy predominated in the theatrical performances. The players were “the young gentlemen of the town.” At one of the entertainments they gave John O’Keefe’s comic opera The Poor Soldier, and a farce by Arthur Murphy called The Apprentice. 98 There were also performances of a more professional character. Bromley and Arnold, two professional actors, conducted a series of theatrical entertainments extending over a period of several weeks. The plays which they rendered are hardly known to-day. At a single performance99 they gave a comedy entitled Trick upon Trick, or The Vintner in the Suds; a farce called The Jealous Husband, or The Lawyer in the Sack; and a pantomime, The Sailor’s Landlady, or Jack in Distress. Another play in the series was Edward Moore’s tragedy, The Gamester. 100
Much of Grant’s Hill was unenclosed. Clumps of trees grew on its irregular surface, and there were level open spaces; and in summer the place was green with grass, and bushes grew in profusion. Farther in the background were great forest trees. The hill was the pleasure ground of the village. Judge Henry M. Brackenridge, a son of Judge Hugh Henry Brackenridge dwelling on the past, declared that “it was pleasing to see the line of welldressed ladies and gentlemen and children, ... repairing to the beautiful green eminence.”101 On this elevation “under a bower, on the margin of a wood, and near a delightful spring, with the town of Pittsburgh in prospect,” the Fourth of July celebrations were held.102 On August 2, 1794, the motley army of Insurgents from Braddocksfield rested there, after having marched through the town.
103
Here they were refreshed with food and whisky, in order that they might keep in good humor, and to prevent their burning the town.
Samuel Jones has left an intimate, if somewhat regretful account of the early social life of Pittsburgh. “The long winter evenings,” he wrote, “were passed by the humble villagers at each other’s homes, with merry tale and song, or in simple games; and the hours of night sped lightly onward with the unskilled, untiring youth, as they threaded the mazes of the dance, guided by the music of the violin, from which some good-humored rustic drew his Orphean sounds. In the jovial time of harvest and hay-making, the sprightly and active of the village participated in the rural labors and the hearty pastimes, which distinguished that happy season. The balls and merrymakings that were so frequent in the village were attended by all without any particular deference to rank or riches. No other etiquette than that which natural politeness prescribed was exacted or expected.... Young fellows might pay their devoirs to their female acquaintances; ride, walk, or talk with them, and pass hours in their society without being looked upon with suspicion by parents, or slandered by trolloping gossips.”
104
The event of autumn was the horse races, which lasted three days. They were held in the northeasterly extremity of the town between Liberty Street and the Allegheny River,105 and were conducted under the auspices of the Jockey Club which had been in existence for many years. Sportsmen came from all the surrounding country. The races were under the saddle, sulkies not having been invented. Racing proprieties were observed, and jockeys were required to be dressed in jockey habits.106 Purses were given. The horses compared favorably with race horses of a much later day. A prominent horse was “Young Messenger” who was sired by “Messenger,” the most famous trotting horse in America, which had been imported into Philadelphia from England in 1788, and was the progenitor of Rysdyk’s Hambletonian, Abdallah, Goldsmith Maid, and a score of other noted race horses.
A third of a century after the race course had been removed beyond the limits of the municipality, Judge Henry M. Brackenridge published his recollections of the entrancing sport. “It was then an affair of all-engrossing interest, and every business or pursuit was neglected.... The whole town was daily poured forth to witness the Olympian games.... The plain within the course and near it was filled with booths as at a fair, where everything was said, and done, and sold, and eaten or drunk, where every fifteen or twenty minutes there was a rush to some part, to witness a fisticuff—where dogs barked and bit, and horses trod on men’s toes, and booths fell down on people’s heads!”
107
The social instincts of the people found expression in another direction. The Revolutionary War, the troubles with the Indians, the more or less strained relations existing between France and England, had combined to inbreed a military spirit. Pennsylvania, with a population, in 1800, of 602,365, had enrolled in the militia 88,707 of its citizens. The militia was divided into light infantry, riflemen, grenadiers, cavalry, and artillery.108 Allegheny County had a brigade of militia, consisting of eight regiments.109 The commander was General Alexander Fowler, an old Englishman who had served in America, in the 18th, or Royal Irish, Regiment of Foot. On the breaking out of the Revolutionary War, he had resigned his commission on account of his sympathy with the Americans. Being unfit for active service, Congress appointed him Auditor of the Western Department at Pittsburgh.
The militia had always been more or less permeated with partisan politics. During the Revolution the American officers wore a cockade with a black ground and a white relief, called the black cockade. This the Federalists had made their party emblem. The Republican party, soon after its organization, adopted as a badge of party distinction a cockade of red and blue on a white base, the colors of revolutionary France. The red and blue cockade thereafter became the distinguishing mark of the majority of the Pennsylvania militia, being adopted on the recommendation of no less a person than Governor McKean. General Fowler’s advocacy of the red and
blue cockade and his disparagement of the black cockade were incessant. He was an ardent Republican, and his effusions with their classic allusions filled many columns of the Tree of Liberty and the Pittsburgh Gazette. At a meeting of the Allegheny County militia held at Marie’s tavern, the red and blue cockade had been adopted. Fowler claimed that this was the result of public sentiment. He was fond of platitudes. “The voice of the people is the voice of God,” he quoted, crediting the proverb to an “English commentator,” and adding: “Says a celebrated historian, ‘individuals may err, but the voice of the people is infallible.’”110 A strong minority in Allegheny County remained steadfast to the Federal party, and the vote in favor of the adoption of the red and blue cockade was not unanimous. Two of the regiments, not to be engulfed in the growing wave of Republicanism, or overawed by the domineering disposition of General Fowler, opposed the adoption of the red and blue cockade, and chose the black cockade.111
The equipment furnished to the militia by the State was meagre, but the patriotism which had so lately won the country’s independence was still at flood tide, and each regiment was supplied with two silk standards. One was the national flag, the other the regimental colors. The national emblem differed somewhat from the regulation United States flag. The word “Pennsylvania” appeared on the union, with the number of the regiment, the whole being encircled by thirteen white stars. The fly of the regimental colors was dark blue; on this was painted an eagle with extended wings supporting the arms of the State. The union was similar to that of the national flag. The prescribed uniform which many of the men, however, did not possess, was a blue coat faced with red, with a lining of white or red. In Allegheny County a round hat with the cockade and buck’s tail, was worn. 112 The parade ground of the militia was the level part of Grant’s Hill which adjoined Marie’s tavern on the northeast. Here twice each year, in April and October, the militia received its training. Of no minor interest, was the social life enjoyed by officers and men alike, during the annual assemblages.
In the territory contiguous to Pittsburgh the uprising, for the right to manufacture whisky without paying the excise, had its inception. That taverns should abound in the town was a natural consequence. In 1808 the public could be accommodated at twenty-four different taverns.113 The annual license fee for taverns, including the clerk’s charges, was barely twenty dollars. Through some mental legerdemain of the lawmakers it had been enacted that if more than a quart was sold no license was required. Liquors, and particularly whisky, were sold in nearly every mercantile establishment. Also beer had been brewed in Pittsburgh since an early day, at the “Point Brewery,” which was purchased in 1795 by Smith and Shiras.114 Beer was likewise brewed in a small way by James Yeaman, two or three years later.115 In February, 1803, O’Hara and Coppinger, who had acquired the “Point Brewery,” began brewing beer on a larger scale.116
In the taverns men met to consummate their business, and to discuss their political and social affairs. Lodge No. 45 of Ancient York Masons met in the taverns for many years, as did the Mechanical Society. Even the Board of Trustees of the Academy held their meetings there.117 Religion itself, looked with a friendly eye on the taverns. In the autumn of 1785, the Rev. Wilson Lee, a Methodist missionary, appeared in Pittsburgh, and preached in John Ormsby’s tavern,118 on Water Street, at his ferry landing,119 at what is now the northeast corner of that street and Ferry Street. This was the same double log house which, while conducted by Samuel Semple, was in 1770 patronized by Colonel George Washington.120
Tavern keeping and liquor selling were of such respectability that many of the most esteemed citizens were, or had been tavernkeepers, or had sold liquors, or distilled whisky, or brewed beer. Jeremiah Sturgeon was a member of the session of the Presbyterian Church.121 John Reed, the proprietor of the “Sign of the Waggon,” in addition to being a leading member of the Jockey Club, and the owner of the race horse “Young Messenger,”122 was precentor in the
Presbyterian Church, and on Sundays “lined out the hymns” and led the singing.123 The pew of William Morrow is marked on the diagram of the ground-plan of the church as printed in its Centennial Volume. 124 The “Sign of the Cross Keys,” the emblem of Sturgeon’s tavern, was of religious origin and was much favored in England. Although used by a Presbyterian, it was the arms of the Papal See, and the emblem of St. Peter and his successors. That the way to salvation lay through the door of the tavern, would seem to have been intended to be indicated by the “Sign of the Cross Keys.” William Eichbaum, a pillar in the German church, after he left the employ of O’Hara and Craig, conducted a tavern on Front Street, near Market, at the “Sign of the Indian Queen.” The owners of the ferries kept taverns in connection with their ferries. Ephraim Jones conducted a tavern at his ferry landing on the south side of the Monongahela River; Robert Henderson had a tavern on Water Street at his ferry landing; Samuel Emmett kept a tavern at his landing on the south side of the Monongahela River; and James Robinson had a tavern on the Franklin Road at the northerly terminus of his ferry.125
Drinking was universal among both men and women. Judge James Veech declared that whisky “was the indispensable emblem of hospitality and the accompaniment of labor in every pursuit, the stimulant in joy and the solace in grief. It was kept on the counter of every store and in the corner cupboard of every well-to-do family The minister partook of it before going to church, and after he came back. At home and abroad, at marryings and buryings, at house raisings and log rollings, at harvestings and huskings, it was the omnipresent beverage of old and young, men and women; and he was a churl who stinted it. To deny it altogether required more grace or niggardliness than most men could command, at least for daily use.”126
A practical joke perpetrated by the Rev. Dr. John McMillan, on the Rev Joseph Patterson, another of the early ministers in this region, illustrates the custom of drinking among the clergy. On their way to attend a meeting of the Synod, the two men stopped at a
wayside inn and called for whisky, which was set before them. Mr Patterson asked a blessing which was rather lengthy. Dr. McMillan meanwhile drank the whisky, and to Mr. Patterson’s blank look remarked blandly, “You must watch as well as pray!”127
Families purchased whisky and laid it away in their cellars for future consumption, and that it might improve with age. Judge Hugh Henry Brackenridge declared that the visit of the “Whisky Boys”—as the Insurgents from Braddocksfield were called—to Pittsburgh cost him “four barrels of old whisky.”128 The statement caused Henry Adams, in his life of Albert Gallatin, to volunteer the assertion that it nowhere appeared “how much whisky the western gentleman usually kept in his house.”129
There was no legislation against selling liquors on Sundays. The only law on the subject was an old one under which persons found drinking and tippling in ale-houses, taverns, and other public houses on Sundays, were liable to be fined one shilling and sixpence; and the keepers of the houses upon conviction were required to pay ten shillings. The line of demarcation between proper and improper drinking being faint, the law proved ineffectual to prevent drinking on Sundays.
Religion had not kept pace with material progress. The people had been too much engrossed in secular affairs to attend to spiritual matters. They were withal generous, and practiced the Christian virtues; and never failed to help their unfortunate neighbors. This disposition was manifested in various ways. Losses by fire were of frequent occurrence and were apt to cause distress or ruin to those affected. In these cases the citizens always furnished relief. An instance where this was done was in the case of William Thorn. Thorn was a cabinet-maker on Market Street, and built windmills and Dutch fans.130 When the house which he occupied was burned to the ground and he lost all his tools and valuable ready-made furniture, a liberal subscription was made by the citizens, and he was enabled to again commence his business.131
But there was little outward observance of religious forms. The Germans had made some progress in that direction. The little log building where they worshipped had been succeeded by a brick church. The only English church was the Presbyterian Meeting House facing on Virgin Alley, now Oliver Avenue, erected in 1786. It was the same building of squared timbers in which the congregation had originally worshipped. From 1789 to 1793, the church had languished greatly. There was no regular pastor; services were held at irregular and widely separated intervals. Two of the men who served as supplies left the ministry and became lawyers.132 From 1793 to 1800, the church was all but dead. The house was deserted and falling into ruin. Only once, so far as there is any record, were Presbyterian services held in the building during this period. It was in 1799 that the Rev. Francis Herron, passing through Pittsburgh, was induced to deliver a sermon to a congregation consisting of fifteen or eighteen persons “much to the annoyance of the swallows,” as Herron ingenuously related, which had taken possession of the premises.133
A light had flashed momentarily in the darkness when John Wrenshall, the father of Methodism in Pittsburgh, settled in the town. Wrenshall was an Englishman who came to Pittsburgh in 1796 and established a mercantile business. He was converted to Wesleyanism in England and had been a local preacher there. As there was no minister or preaching of any kind in Pittsburgh, he commenced holding services in the Presbyterian Meeting House. His audiences increased, but after a few Sundays of active effort, a padlock was placed on the door of the church, and he was notified that the house was no longer at his disposal. The Presbyterians might not hold services themselves, but they would not permit the use of their building to adherents of the new sect of Methodists, “the offspring of the devil.”
A great religious revival swept over the Western country in the concluding years of the eighteenth century. In Kentucky it developed into hysteria,134 and in Western Pennsylvania the display of religious fervor was scarcely less intense.135 The effect was felt in Pittsburgh.
