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MST209 Mathematical methods
and models
Block 4

Contents
UNIT 13 Modelling with non-linear differential
equations 5
Introduction 6
1 Modelling populations of predators and prey 7
1.1 Exponential growth of a single population 7
1.2 A first model for populations of rabbits and foxes 9
1.3 A second model for populations of rabbits and foxes 12
1.4 Equilibrium populations 15
1.5 Stability of equilibrium points 16
1.6 Populations close to equilibrium 18
2 Classifying equilibrium points 21
2.1 Matrices with two real distinct eigenvalues 22
2.2 Matrices with complex eigenvalues 24
2.3 Matrices with repeated eigenvalues 27
2.4 Classifying the equilibrium point of a linear system of
differential equations 28
2.5 Classifying the equilibrium points of non-linear
systems 29
3 Modelling a pendulum 33
3.1 Pendulum equations 33
3.2 The phase plane for a pendulum 34
4 Computer activities 37
Outcomes 39
Solutions to the exercises 40
UNIT 14 Modelling motion in two and three
dimensions 47
Introduction 48
1 Bumpy rides 49
1.1 Acceleration on slopes 49
1.2 Crossing a hump-backed bridge 50
1.3 Complicated paths 53
2 Projectiles without air resistance 56
2.1 Horizontal launch 56
2.2 Launches at an angle 64
2.3 The trajectory of a projectile 68
2.4 Energy and projectile motion 74
3 Resisted projectiles 76
Outcomes 78

UNIT 15 Modelling heat transfer 85
Introduction 86
1 Heat and temperature 87
1.1 The relationship between heat and temperature 87
1.2 The three modes of heat transfer 89
2 Conduction 92
2.1 Heat loss through a wall: initial assumptions 92
2.2 Fourier’s law 94
2.3 Heat loss through a wall: a ﬁrst model 95
2.4 Conduction through a pipe 97
3 Convection and U -values 100
3.1 A simple model of convection 100
3.2 Heat loss through a wall: a second model 103
3.3 Convection and conduction through a pipe 107
4 Radiation 110
4.1 Modelling radiation 110
4.2 Combined modes of heat transfer and U-values 113
Outcomes 114
UNIT 16 Interpretation of mathematical models 123
Introduction 124
1 An overview of mathematical modelling 125
1.1 Specify the purpose 127
1.2 Create the model 128
1.3 Do the mathematics 132
1.4 Interpret the results 132
1.5 Evaluate the outcomes 134
2 Dimensional consistency and change of units 137
2.1 Dimensional consistency 137
2.2 Change of units 141
3 The method of dimensional analysis 144
4 A case study of modelling: Transient heat transfer 1
— Creating the model 149
5 A case study of modelling: Transient heat transfer 2
— Evaluating and revising the model 156
Outcomes 159

Index 169
The Open University, Walton Hall, Milton Keynes, MK7 6AA.
First published 2005. Second edition 2008.
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1.1

UNIT 13 Modelling with non-linear
differential equations
Study guide for Unit 13
This is the last unit in this course devoted to the analytical solution of
PC
2
3
4
2
3
4
1
1
ordinary differential equations. We considered the solution of first-order
differential equations in Unit 2, then in Unit 3 we looked at linear constant-
coefficient second-order differential equations, and in Unit 11 we studied
simultaneous systems of both first-order and second-order linear differential
equations. In this unit, we shall be looking again at systems of first-order
differential equations, but we shall be concentrating on systems which are
non-linear.
You will need to be familiar with the techniques developed in Unit 10 for
finding eigenvalues and eigenvectors, and in Unit 11 for the solution of
systems of first-order linear differential equations.
We recommend that you study one section per study session. However,
the main ideas of this unit are developed in the first two sections. You
could sensibly decide to spend two study sessions on Section 1, spending
one study session on Subsections 1.1 to 1.3, and the second study session
on Subsections 1.4 to 1.6. If you are short of time, you should devote more
time to the first two sections than to the later sections. You will need to use
your computer for Section 4, which is entirely devoted to computer activities
using the computer algebra package for the course.
The material discussed here is introduced in the context of the interaction of
animal populations, but it is applicable to many areas involving systems of
non-linear differential equations. The motion of pendulums is also discussed.
5

Unit 13 Modelling with non-linear differential equations
Introduction
In this unit we study the mathematical models associated with two physical
systems:
• the growth of two interacting populations, one a predator and the other
its prey;
• the motion of a rigid pendulum.
At first sight these systems appear to be unrelated, but each can be sensibly
modelled by a non-linear differential equation or a system of such equations.
• The interaction between two populations is modelled by the Lotka–
Volterra equations
ẋ = kx
2
1 −
y
Y
6
, ẏ = −hy
2
1 −
x
X
6
,
The equations are non-linear
because of the xy terms.
where h, k, X and Y are known constants, and x = x(t) and y = y(t)
represent the two population sizes at time t.
• The unresisted motion of a rigid pendulum is modelled by the second-
order differential equation
ẍ + ω2
sin x = 0,
The equation is non-linear
because of the sin x term.
where ω is a constant and x = x(t) is the angle the pendulum makes
with the downwards vertical at time t.
When you studied linear differential equations in Units 2, 3 and 11, the em-
phasis was on finding an explicit equation for the solution. For non-linear
equations this is rarely possible; even when it is possible, the solution is
usually difficult to interpret. As we shall see in Unit 26, it is often possible
to find a solution using a numerical method, but such a method may give
a solution that is equally difficult to interpret — and this will also be true
for any graphs that the solution produces. For these reasons, when study-
ing a system of non-linear differential equations, we often concentrate on
a qualitative approach; that is, we try to obtain useful information about
the solution, rather than trying to find the solution itself. This information
may be in the form of a diagram. For the Lotka–Volterra equations, we look
at the paths defined by (x(t), y(t)) in the (x, y)-plane. In the case of the We use the notations
x or x(t), ẋ or ẋ(t), etc.,
interchangeably to suit the
context.
pendulum, we again discuss paths in the plane, but while one axis is still
used to represent x(t), the second axis is used for ẋ(t), so that paths are
defined by (x(t), ẋ(t)).
We shall see that a common feature of these two models is the relevance of a
constant solution, which, in each case, describes an equilibrium state of the
system. For example, a constant solution x(t) = X, y(t) = Y to the Lotka–
Volterra equations describes a situation where the two populations are in
equilibrium. Near such a solution, we shall see that some useful information
can be obtained by replacing the original non-linear equations by linear
approximations to the differential equations. The equilibrium states and
the behaviour of the system when it is nearly in equilibrium play a major
part in obtaining a qualitative overview of the behaviour of the model.
We begin in Section 1 with the Lotka–Volterra equations, which apply to
a pair of interacting populations, and see how these equations can be lin-
earized near an equilibrium state. The resulting systems of linear differential
equations were discussed in Unit 11, but here, in Section 2, we concentrate
on the graphical representation of the solutions. Section 3 looks at models
for the motion of a pendulum: second-order differential equations are trans-
formed to systems of first-order equations, and the techniques developed
earlier in the unit are applied to find and interpret graphical solutions.
6

Section 1 Modelling populations of predators and prey
One feature of all the differential equation models that appear in this unit
is that the independent variable, usually t, does not appear explicitly in the
differential equation, and the system takes the form The important point is that
the functions on the
right-hand sides of these
differential equations are
independent of t.
ẋ = u(x, y),
ẏ = v(x, y).
Systems of this form, where t does not appear explicitly, are said to be
autonomous.
1 Modelling populations of predators and
prey
In this section we shall develop a model for the first type of system listed
in the Introduction: populations of a predator and its prey. A predator
population depends for its survival on there being sufficient prey to provide
it with food. Intuition suggests that when the number of predators is low,
the prey population may increase quickly, and that this will result in turn in
an increase in the predator population. On the other hand, a large number
of predators may diminish the prey population until it is very small, and this
in turn will lead to a collapse in the predator population. Our mathematical
model will need to reflect this behaviour.
It is not possible to find algebraic solutions to all such models, so we shall
introduce you to a geometric approach, based on the notion that a point
(x, y) = (x(t), y(t)) in the plane may be used to represent two populations
x = x(t) and y = y(t) at time t. As t increases, the point (x, y) = (x(t), y(t))
will trace a path that represents the variation of both populations with time.
In particular, we shall discuss the equilibrium values for such a system, and
a linear approximation to the system near these equilibrium values.
1.1 Exponential growth of a single population
Before we consider a system of two interacting populations, we shall first
develop a simple continuous model of the growth of a single population,
which is called the exponential model. This will allow us to develop some of
the concepts of population models in a simpler context.
Here we model the population size x, which we shall usually simply refer
to as the population x (omitting the word ‘size’), as a function of time t.
This function cannot take negative values (since there are no negative pop-
ulations), but we shall allow it to take the value zero. We shall normally
assume that t is measured in years. We deal with a continuous model, rather A population x can take only
integer values, so we say (as
in Unit 1) that x is a discrete
variable. It is often
convenient to approximate a
discrete variable by a variable
that can take any real value,
referred to as a continuous
variable.
than a discrete one, so the derivative ẋ represents the rate of increase of the
population, which we shall often refer to as the growth rate (even when,
if ẋ < 0, it actually represents a decay rate — compare the use in mechan-
ics of ‘acceleration’ to cover both of the everyday terms ‘acceleration’ and
‘deceleration’).
In the exponential model, we make the assumption that the growth rate ẋ
is proportional to the current population x. (This means that if the growth
7

rate is 20 per year when the population is 100, then the growth rate will be
40 per year when the population is 200, the growth rate will be 60 per year
when the population is 300, and so on.) This assumption leads directly to
the differential equation
ẋ = kx (x ≥ 0), (1.1)
where k is a constant. If k is positive, then x is an increasing function of
time, while if k is negative, then x is decreasing.
Exercise 1.1
Under what circumstances is it reasonable to assume that the growth rate
of a population is proportional to the current population?
In Unit 2 you saw that a differential equation like Equation (1.1) can be
described by a direction field. The graph of a solution is a curve whose tan-
gent at any point has a slope that is equal to the value of the direction field
at that point (see Figure 1.1). However, Equation (1.1) can be solved ex-
Figure 1.1 Direction field for
Equation (1.1), with k > 0
plicitly. Choosing a value for the population x(t) at time t = 0, for example,
x(0) = x0, gives the solution
x(t) = x0ekt
(x ≥ 0), (1.2)
which is an exponential function. If the constant k is positive, the population
is increasing. If k is negative, the population is decreasing. This difference is
demonstrated by comparing the graphs of the functions et and e−t, for t ≥ 0,
as shown in Figure 1.2. We may describe the two situations as exponential
Figure 1.2
growth and exponential decay, although we could say that the latter case
is merely exhibiting ‘negative growth’. For this reason we refer to Equa-
tion (1.1) as the exponential differential equation, or, when applied to
a population, the exponential model.
The proportionate growth rate ẋ/x represents the rate of increase of
the population per unit of the current population, and may be considered
as the difference between the birth and death rates per head of population.
It may be positive (for a growing population in which the birth rate exceeds
the death rate), negative (for a declining population in which the death rate
exceeds the birth rate) or zero (for a static population in which the birth
and death rates are equal). In the case of the exponential model (1.1), we
have ẋ/x = k, so the proportionate growth rate is constant. (We previously
assumed that x ≥ 0. While talking about proportionate growth rate, we
exclude the possibility that x takes the value zero.)
Exercise 1.2
Can you suggest any reason why the assumption that the proportionate
growth rate ẋ/x is constant is unrealistic for real populations?
The exponential model is fairly accurate for many populations in a state
of rapid increase, but it can be reasonable only over a restricted domain of
validity. As a model of the behaviour of increasing populations over longer
periods of time, the exponential model is clearly unsatisfactory because it
predicts unbounded growth. However, in this unit we shall not revise our
first (simple) model to give a more realistic description of the growth of a
single population.
8

1.2 A first model for populations of rabbits and foxes
In the rest of this section we shall be concerned with developing models for
populations of rabbits (the prey) and foxes (the predators). Our purpose
is to determine how these populations evolve with time. At a particular
time t, we suppose that these populations are x(t) and y(t), respectively.
We represent this system by a point in the (x, y)-plane. The evolution of the
two populations can be represented as a path, as shown in Figure 1.3, where
the directions of the arrows on the path indicate the directions in which the
point (x(t), y(t)) moves along the path with increasing time. However, this
type of representation does not show how quickly or slowly the point moves
along the path. Figure 1.3
In our mathematical model we make the following assumptions.
• There is plenty of vegetation for the rabbits to eat.
• The rabbits are the only source of food for the foxes.
• An encounter between a fox and a rabbit contributes to the fox’s larder,
which leads directly to a decrease in the rabbit population and indirectly
to an increase in the number of foxes.
We may, for convenience, measure the populations in hundreds or thousands,
as appropriate, so that we are able to use quite small numbers in our models.
We begin by examining a very simple model which generalizes the exponen-
tial model that we used for a single population in the previous subsection.
This simple case has the advantage that we can determine easily an algebraic
solution, and we use it to investigate the geometric approach.
As a first model, we assume that the populations are evolving independently
(perhaps on separate islands). Because there are no interactions, the popu-
lations may be modelled by the pair of equations
ẋ = kx,
ẏ = −hy (x ≥ 0, y ≥ 0),
Each of these equations has
an exponential function as its
general solution.
(1.3)
where k and h are positive constants. The condition (x ≥ 0, y ≥ 0) applies
here, as well as in similar situations throughout the unit, to the first equation
ẋ = kx, as well as the second equation ẏ = −hy.
The first equation models a colony of rabbits not affected by the predation
of foxes, growing exponentially according to a rule of the form
x(t) = Cekt
,
where C is a positive constant representing the initial rabbit population. When t = 0, x = C.
*Exercise 1.3
Determine a formula for the population y(t) of foxes. How would you inter-
pret this solution for the population of foxes?
Equations (1.3) form a system of linear differential equations which you
met in Unit 11. Using vector notation, the pair of populations may be
represented by the vector x = [x y]T . The system of equations (1.3) then
becomes the vector equation
ẋ =
%
ẋ
ẏ
,
=
%
kx
−hy
,
. (1.4)
It is helpful now to introduce the notion of a vector field, which is simi- Vector fields are discussed in
more detail in Unit 23.
lar to a direction field. In a plane, a direction field associates a direction
9

f(x, y) with each point (x, y), whereas a vector field associates a vector
u(x, y) with each point (x, y). For example, if u(x, y) = [2x 3y + 2]T , then
u(1, 2) = [2 8]T = 2i + 8j. For our predator–prey model, we would asso-
ciate with Equation (1.4) the vector field
u(x, y) =
%
kx
−hy
,
, (1.5)
so that Equation (1.4) becomes
ẋ = u(x, y). Sometimes this will be
written as ẋ = u(x).
For a direction field, f(x, y) represents the slope of a particular solution
of the differential equation dy/dx = f(x, y) at the point (x, y). Similarly,
u(x, y) is the vector ẋi + ẏj that is tangential to a particular solution of
ẋ = u(x, y) at the point (x, y), because the slope of the tangent is
dy
dx
=
(dy/dt)
(dx/dt)
=
ẏ
ẋ
.
This suggests a geometric way of finding a particular solution to Equa-
tion (1.4): choose a particular starting point (x0, y0), then follow the direc-
tions of the tangent vectors. The essential difference between a direction field An exception, which we
discuss later, occurs when
ẋ = ẏ = 0 at (xe, ye), so
u(xe, ye) = 0 and there is no
tangent vector to follow.
and a vector field is that the latter consists of directed line segments (which
we indicate by arrows) whose lengths indicate the magnitude of u(x, y).
However, since the magnitudes of u(x, y) may vary considerably and so make
the diagram difficult to interpret, we often use arrows of a fixed length.
In many cases, it is the
direction of u(x, y) that is our
primary concern, rather than
its magnitude.
We now widen the discussion and look at similar systems that do not arise
from populations.
Example 1.1
(a) Using arrows of a fixed length, sketch the vector field given by
u(x, y) =
%
0.2x
0.3y
,
.
(b) Write down the system of differential equations ẋ = u(x) corresponding
to this vector field. Solve this system of equations, and hence find an
equation in terms of x and y for each path in the (x, y)-plane represented
by this solution.
(c) Sketch a sample of the paths in the (x, y)-plane which represent the
solutions.
Solution
(a) We use values of u(x, y) to construct Figure 1.4. The arrows represent For example, at the point
(1, 1), we have
u(1, 1) = [0.2 0.3]T
.
vectors of a fixed length parallel to u(x, y). Each arrow indicates the
direction of a path through a given point.
Figure 1.4
(b) We have
u(x, y) =
%
ẋ
ẏ
,
=
%
0.2x
0.3y
,
,
so the system of equations is
ẋ = 0.2x,
ẏ = 0.3y.
The general solution is
x(t) = Ce0.2t
, y(t) = De0.3t
. (1.6)
10

To obtain the equation for the paths in terms of x and y, we eliminate
t from x = Ce0.2t and y = De0.3t. Cubing the first equation and squar-
ing the second gives x3 = C3e0.6t and y2 = D2e0.6t, so x3/C3 = y2/D2.
Hence the equations of the paths are of the form
y = K |x|3/2
,
The modulus signs around x
ensure that we do not try to
obtain the square root of a
negative number. The
arbitrariness of K ensures
that the equation represents
all possible cases.
for some constant K.
(c) As we have been able to analytically find the equations of the paths in
part (b), namely y = K|x|3/2, we can use this to sketch the paths in
the (x, y)-plane. These are shown in Figure 1.5. It remains to put the
arrows on these paths to indicate the direction of increasing time. The
first of the differential equations is ẋ = 0.2x. So for positive x, ẋ > 0
so x is an increasing function of time. Therefore in the right half-plane,
the arrows on the paths point to the right. Similarly, for negative x,
ẋ < 0 so x is a decreasing function of time. Hence the arrows on the
paths point to the left in the left half-plane. These directions have
been added to the paths in Figure 1.5. (Consideration of the second
differential equation, ẏ = 0.3y, confirms the directions of the arrows on
the paths.) The arrows on the paths along the positive and negative
y-axes can be deduced from consideration of the differential equation
ẏ = 0.3y. Note that the origin is also a path, corresponding to K = 0,
but it has no time arrow associated with it.
The direction of the time arrow on a path is the same as the direction
of the vector field. This can be confirmed by looking at Figure 1.5. Figure 1.5
Alternatively, we could have drawn the paths using the techniques we
used for direction fields in Unit 2: the direction of the tangent to the
path at any point (x, y) is the same as the vector field u(x, y) at that
point.
A solution curve along which the coordinates x and y vary as t increases
is called a phase path or orbit. The (x, y)-plane containing the solution
curves is called the phase plane, and a diagram, such as Figure 1.5, showing
the phase paths is called a phase diagram.
Before we leave Example 1.1, you may have noticed in Figure 1.5 that the
paths radiate outwards from the origin in all directions. For this reason, we
refer to the origin as a source. A source can occur at a point
other than the origin.
We now look at the phase paths for a similar system, for which
u(x, y) =
%
−0.2x
−0.3y
,
.
This system behaves in a similar fashion to the system in Example 1.1. The
general solution is
x = Ce−0.2t
, y = De−0.3t
.
Eliminating t gives
y = K |x|3/2
,
as before. However, as t increases, there is a significant difference in the
motion along the phase paths between this case and the case in Exam-
ple 1.1. At any point (x, y), the vector field u(x, y) = [−0.2x −0.3y]T has
the same magnitude but the opposite direction to the vector field u(x, y) =
[0.2x 0.3y]T . So in this case, the phase diagram would be identical to
Figure 1.5 except that the directions would be pointing towards the origin.
11

Now the paths radiate inwards towards the origin and, for this reason, we
refer to the origin as a sink. A sink can occur at a point
other than the origin.
*Exercise 1.4
Write down the system of differential equations ẋ = u(x, y) given by the
vector field
u(x, y) =
%
x
−y
,
.
Apart from the restrictions
x ≥ 0, y ≥ 0, this is the
equation for our first model
for the predator–prey
populations (Equation (1.5))
with k = 1 and h = 1.
Solve this system of equations, and hence find an equation in terms of x and
y for the phase paths in the (x, y)-plane represented by this solution. Sketch
some of these paths. Is there a path through the origin?
The phase diagram for the vector field examined in Exercise 1.4 is shown in
Figure 1.6. You can see that the vast majority of paths do not radiate into or
out of the origin. On these paths, a point initially travels towards the origin,
but eventually travels away from it again. The only paths which actually
radiate inwards towards or outwards from the origin are those along the x-
and y-axes. In this case we call the origin a saddle (as you will understand
from Unit 12).
The behaviour of the populations of rabbits and foxes illustrated in the
quadrant x ≥ 0, y ≥ 0 of Figure 1.6 is what we would expect from our
first model. The population of rabbits increases without limit, as they are
isolated from their predators. On the other hand, the population of foxes
decreases to zero, as they have no access to their sole source of food. Figure 1.6
1.3 A second model for populations of rabbits and
foxes
In the previous subsection we looked at a model for rabbit and fox popu-
lations when there was no interaction between the two populations. This
may be a reasonable model when both species inhabit the same environment
but the populations are so low that the rabbits and foxes rarely meet. As
we saw, for initial positive populations, this first model predicts that rab-
bits will increase without limit and foxes will die out. However, this is not
what we expect for interacting populations, when encounters are unavoid-
able. We assume that the number of encounters between foxes and rabbits
is proportional both to the population x of rabbits and to the population y
of foxes.
In addition to our previous assumptions, stated in Subsection 1.2, our model
assumes that
• the number of encounters between foxes and rabbits is proportional to
the product xy.
In this subsection we look at a revised model, which allows for interaction
based on this assumption.
For a population x of rabbits in a fox-free environment, our first model for
population change is given by the equation ẋ = kx, where k is a positive
constant. This represents exponential growth. However, if there is a pop-
ulation y of predator foxes, then you would expect the growth rate ẋ of
rabbits to be reduced. A simple assumption is that
• the growth rate ẋ of rabbits decreases by a factor proportional to the
number of encounters between rabbits and foxes, i.e. by a factor propor-
tional to xy.
12

We revise our first model to include this extra term, so the differential
equation that models the population x of rabbits is now
ẋ = kx − Axy,
This equation comes from the
input–output principle, which
was introduced in Unit 2
Subsection 1.1. In a period of
time δt, the change in the
rabbit population is the
number kx δt of additional
rabbits born, taking into
account those dying from
natural causes, less the
number Axy δt of rabbits
eaten.
for some positive constant A. As we shall see later, it is convenient to write
A = k/Y , for some positive constant Y , giving
ẋ = kx
2
1 −
y
Y
6
. (1.7)
This is a non-linear equation, since the right-hand side contains an xy term.
Similarly, for a population y of foxes in a rabbit-free environment, our first
model for the population change is given by the equation ẏ = −hy, where h
is a positive constant. This represents exponential decay. However, if there
is a population x of rabbits for the foxes to eat, we should expect the growth
rate ẏ of foxes to increase. A simple assumption is that
• the growth rate ẏ of foxes increases by a factor proportional to the num-
ber of encounters between foxes and rabbits, i.e. by a factor proportional
to xy.
Our revised model for the foxes is given by
ẏ = −hy + Bxy
for some positive constant B. Again, it is convenient to write B = h/X, for
some positive constant X, so that this equation becomes
ẏ = −hy
2
1 −
x
X
6
.
Again, this equation is
non-linear because of the xy
term on the right-hand side.
(1.8)
Together, the differential equations (1.7) and (1.8) model the pair of inter-
acting populations.
*Exercise 1.5
Sketch the graph of the proportionate growth rate ẋ/x of rabbits as a func-
tion of the population y of foxes, and the graph of the proportionate growth
rate ẏ/y of foxes as a function of the population x of rabbits. Interpret these
graphs.
Lotka–Volterra equations This model was one of the
first successful applications of
mathematical models to
biological systems. It was
independently proposed in
1925 by the American
biophysicist Alfred Lotka and
in 1926 by the Italian
mathematician Vito Volterra.
The evolution of two interacting populations x and y can be modelled
As in Equations (1.3), the
condition (x ≥ 0, y ≥ 0)
applies to both differential
equations.
by the Lotka–Volterra equations
ẋ = kx
2
1 −
y
Y
6
,
ẏ = −hy
2
1 −
x
X
6
(x ≥ 0, y ≥ 0),
(1.9)
where x is the population of the prey and y is the population of the
predators, and k, h, X and Y are positive constants.
The Lotka–Volterra equations can be written as
ẋ = u(x, y),
where ẋ = [ẋ ẏ]T and the vector field u(x, y) is given by
u(x, y) =



kx
2
1 −
y
Y
6
−hy
2
1 −
x
X
6


 . (1.10)
13

Exercise 1.6
Suppose in Equations (1.9) that k = 0.05, h = 0.1, X = 1000 and Y = 100.
Find the values of the corresponding vector field u(x, y) at the following
points.
(a) (0, 0) (b) (0, 100) (c) (500, 100) (d) (1000, 0)
(e) (1000, 100) (f) (1500, 100) (g) (1000, 50) (h) (1000, 150)
Previously, we were able to solve the pairs of differential equations that arose
from our mathematical model, but, for Equations (1.9), no explicit formulae
for x(t) and y(t) are available. However, using the given parameters and
the values obtained in Exercise 1.6, we can begin to construct a vector
field, as in Figure 1.7, where we have used arrows of fixed length. As we Figure 1.7
add more data to this figure, we shall be able to draw some phase paths,
as in Figure 1.8. The path shown represents our guess at a solution to
Equations (1.9) Although it may seem possible that the phase path forms a
closed loop, this is far from certain.
In Figure 1.8 we have labelled a point A on the path, which can be taken
as the initial value for a particular solution. Some other points have been
marked to aid the following discussion. To interpret this guess at a solution,
we think about what happens to the values of the populations as we follow
the path.
Initially, at the point A, there are 1000 rabbits and 50 foxes. As we follow
the path, the rabbit population increases and so does the fox population
until, at the point B, we have reached a maximum rabbit population. As
the fox population continues to rise, the rabbit population goes into decline.
At C the fox population has reached its maximum, while the rabbits decline
further. After this point, there are not enough rabbits available to sustain Figure 1.8
the number of foxes, and the fox population also goes into decline. At D the
declining fox population gives some relief to the rabbit population, which
begins to pick up. Finally, at E, the decline of the fox population is halted
as the rabbit population continues to increase. We may even return exactly
to the point A, in which case the cycle will repeat indefinitely.
We have therefore modelled a system in which both the fox and rabbit
populations repeatedly increase and decline, each in response to the other.
*Exercise 1.7
Consider the system of differential equations defined in Exercise 1.6.
(a) For what values of x and y do the following hold?
(i) ẋ = 0 (ii) ẋ > 0 (iii) ẋ < 0
(b) For what values of x and y do the following hold?
(i) ẏ = 0 (ii) ẏ > 0 (iii) ẏ < 0
(c) Using your answers to parts (a) and (b), and Figure 1.7, sketch some
more phase paths representing solutions to the system of differential
equations.
Note that the differential equations are defined only in the quadrant x ≥ 0,
y ≥ 0.
14

1.4 Equilibrium populations
Usually there is one and only one phase path through a point in the phase
plane — this means that phase paths do not cross. One important excep-
tion to this is illustrated by the point (0, 0) in Figure 1.6 and the point
(1000, 100) in Exercise 1.7. These points correspond to a constant solution As you will see later, the
point (1000, 100) in
Exercise 1.7 is an isolated
point through which no phase
path passes.
x(t) = C, y(t) = D of the system of differential equations, and are called
the equilibrium points or fixed points of the system. These points generally
represent important physical properties of the system being modelled. For
example, the equilibrium point (1000, 100) in Exercise 1.7 corresponds to
the fact that a rabbit population of 1000 and a fox population of 100 can
co-exist in equilibrium, not changing with time.
More generally, if x(t) = C, y(t) = D is a constant solution of a system of
differential equations, it follows that ẋ(t) = 0, ẏ(t) = 0, and we can use this
property to find all the equilibrium points of the system.
Definition
An equilibrium point of a system of differential equations
ẋ = u(x, y)
is a point (xe, ye) such that x(t) = xe, y(t) = ye is a constant solution
of the system of differential equations, i.e. (xe, ye) is a point at which
ẋ(t) = 0 and ẏ(t) = 0.
Procedure 1.1 Finding equilibrium points
To find the equilibrium points of the system of differential equations
ẋ = u(x, y),
for some vector field u, solve the equation
u(x, y) = 0. Solving u(x, y) = 0 requires
the solution of two
simultaneous equations,
generally non-linear, for the
unknowns x and y. If the
variables x and y represent
populations, they must also
satisfy the conditions x ≥ 0
and y ≥ 0.
Example 1.2
Find the equilibrium points for the Lotka–Volterra equations (1.9) for the
rabbit and fox populations.
Solution
Using Procedure 1.1, we need to solve the equation u(x, y) = 0, which be-
comes



kx
2
1 −
y
Y
6
−hy
2
1 −
x
X
6


 =
%
0
0
,
.
This gives the simultaneous equations
kx
2
1 −
y
Y
6
= 0,
−hy
2
1 −
x
X
6
= 0.
From the first equation, we deduce that either x = 0 or y = Y . As stated earlier, h, k, X
and Y are positive constants.
If x = 0, the second equation reduces to −hy = 0, so y = 0 and hence (0, 0)
is an equilibrium point.
15

If y = Y , the second equation becomes −hY (1 − x/X) = 0, so x = X and
hence (X, Y ) is an equilibrium point.
Thus there are two possible equilibrium points for the pair of populations.
The first has both the rabbit and fox populations zero, i.e. the equilibrium
point is at (0, 0); there are no births or deaths — nothing happens. However,
the other equilibrium point occurs when there are X rabbits and Y foxes,
i.e. the equilibrium point is at (X, Y ), when the births and deaths exactly This explains our choice of
constants X and Y in
Subsection 1.3.
cancel out and both populations remain constant.
*Exercise 1.8
Suppose that the population x of a prey animal and the population y of a
predator animal evolve according to the system of differential equations
ẋ = 0.1x − 0.005xy,
ẏ = −0.2y + 0.0004xy (x ≥ 0, y ≥ 0).
Find the equilibrium points of the system. What does this tell you about
the populations? Put these equations in the standard form of the Lotka–
Volterra equations.
Exercise 1.9
Suppose that two interacting populations x and y evolve according to the
system of differential equations
ẋ = x(20 − y), These are not Lotka–Volterra
equations.
ẏ = y(10 − y)(10 − x) (x ≥ 0, y ≥ 0).
Find the equilibrium points of the system.
1.5 Stability of equilibrium points
In a real ecosystem it is unlikely that predator and prey populations are
in perfect harmony. What if equilibrium is disturbed by a small deviation
caused perhaps by a severe winter or hunting? If the number of rabbits
is reduced, there would be a decreased food supply for the foxes, and the
population of foxes could decrease to zero as a consequence. On the other Our investigations in
Exercise 1.7 suggest that
these speculations are not
correct.
hand, if the number of foxes is reduced, the birth rate for rabbits would then
exceed their death rate, and the number of rabbits could increase without
limit.
If a small change or perturbation in the populations of rabbits and foxes from
their equilibrium values, no matter what the cause, results in subsequent
populations which remain close to their equilibrium values, we say that the
equilibrium point is stable. On the other hand, if a perturbation results in
a catastrophic change, with, for example, the population of foxes or rabbits
collapsing to zero or increasing without limit, we say that the equilibrium
point is unstable.
16

If you look at the phase diagram Figure 1.9, where the origin is a sink, you
can see that any slight perturbation from the origin will result in a point
that returns to the origin as time t increases. So the point (0, 0) is a stable
equilibrium point. Similarly, the point (1000, 100) in Exercise 1.7 (page 41) is
a stable equilibrium point. In this case the perturbation from the equilibrium
point does not result in a point which returns to the equilibrium point as t
increases, but does result in a point which remains in the neighbourhood of
the equilibrium point.
Figure 1.9
On the other hand, the origin in the phase diagram shown in Figure 1.5
(page 11) is an unstable equilibrium point. Any perturbation of the point
(x, y) from the origin will result in the point travelling further and further
away from the origin with time. Similarly, the origin in the phase diagram
shown in Figure 1.6 (page 12) is an unstable equilibrium point. Apart from
increases or decreases in y with x unchanged, any perturbation will result
in a point which travels further and further away from the origin with time.
The stability of equilibrium points
Suppose that the system of diﬀerential equations
ẋ = u(x, y)
has an equilibrium point at x = xe, y = ye. The equilibrium point is
said to be:
• stable when all points in the neighbourhood of the equilibrium
point remain in the neighbourhood of the equilibrium point as time
increases;
• unstable otherwise.
*Exercise 1.10
Classify the equilibrium points (0, 0) shown in the following phase diagrams
as stable or unstable.
(a) (b) (c)
17

1.6 Populations close to equilibrium
The predator–prey model developed in Subsection 1.3 resulted in the Lotka–
Volterra equations (1.9) which have equilibrium points at the origin (0, 0)
and the point (X, Y ). Our preliminary attempts at drawing a phase diagram,
in Figure 1.8 and Exercise 1.7, indicate that the behaviour of the populations
in the neighbourhood of the equilibrium point (X, Y ) could be cyclical,
but on the other hand could spiral outwards or inwards. To investigate
the behaviour of a non-linear system in the neighbourhood of equilibrium
points, we shall develop in this subsection linear approximations to the
system which are applicable close to the equilibrium points.
If (xe, ye) is an equilibrium point, consider small perturbations p and q giving Although a population x or y
cannot be negative, a
perturbation p or q can
(usually) be negative if the
population is less than the
equilibrium value.
new populations x and y defined by
x = xe + p, y = ye + q. (1.11)
We can find the time-development of the small perturbations p and q by
linearizing the differential equation ẋ = u(x, y). We shall make use of Taylor
polynomials to achieve this.
In order to do so, we must write each component of the vector u(x, y) as a
function of the two variables x and y:
u(x, y) =
%
u(x, y)
v(x, y)
,
.
At the equilibrium point (xe, ye), we have u(xe, ye) = 0, i.e.
u(xe, ye) = 0 and v(xe, ye) = 0.
Now, for small perturbations p and q, we can use the linear Taylor polyno- Taylor polynomials were
introduced in Unit 12.
mial for functions of two variables to approximate each of u(x, y) and v(x, y)
near the equilibrium point (xe, ye):
u(xe + p, ye + q) / u(xe, ye) + p
∂u
∂x
(xe, ye) + q
∂u
∂y
(xe, ye)
= p
∂u
∂x
(xe, ye) + q
∂u
∂y
(xe, ye),
since u(xe, ye) = 0, and
v(xe + p, ye + q) / v(xe, ye) + p
∂v
∂x
(xe, ye) + q
∂v
∂y
(xe, ye)
= p
∂v
∂x
(xe, ye) + q
∂v
∂y
(xe, ye),
since v(xe, ye) = 0.
The above two equations appear rather unwieldy, but are much more suc-
cinctly represented in matrix form:
%
u(x, y)
v(x, y)
,
=




∂u
∂x
(xe, ye)
∂u
∂y
(xe, ye)
∂v
∂x
(xe, ye)
∂v
∂y
(xe, ye)




%
p
q
,
.
Since x(t) = xe + p(t) and y(t) = ye + q(t), we also have
ẋ = ṗ, ẏ = q̇.
18

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CHAPTER XIX.
SETTLEMENT OF JEWS IN HOLLAND.—FEEBLE
ATTEMPTS AT ENFRANCHISEMENT.
Revival of Catholicism—Decay in European Culture—Ill-
treatment of Jews in Berlin—Emperor Rudolph II of
Austria—Diminution in the Numbers of Italian Jews—
Pope Gregory XIII—Confiscation of Copies of the Talmud
—Vigorous Attempts at the Conversion of Jews—Pope
Sixtus V—The Jewish Physician, David de Pomis—
Renewal of Persecution by Clement VIII—Expulsion from
Various Italian States—The Censors and the Talmud—The
Jews of Ferrara—Settlement of Jews in Holland—Samuel
Pallache—Jacob Tirado and the Marranos in Amsterdam
—Tolerant Treatment—The Poet, David Jesurun—Moses
Uri—Hebrew Printing in Amsterdam.
1593–1618 C.E.
The free spirit of the nations of Europe, which at the beginning
of the century had taken so bold a flight, had broken the ancient
bonds in which the church had long held minds captive, and cast the
blight of doubt on the hitherto sacred authority of the wearer of the
Roman purple—this spirit, which promised to bring the regeneration
of civilized humanity and political freedom, seemed in the second
half of the century to be utterly cast down. The papacy, or
Catholicism, had recovered from its first feeling of terror, and
collected itself. Extraordinarily strengthened by the council of Trent,
it forged new chains to which the nations that had remained faithful,

willingly submitted. The order of the Jesuits, restless and
indefatigable champions, who not only disarmed their opponents,
but even drew them over to their own ranks, had already
reconquered much lost ground by their widespread plots, and had
conceived new measures in order to win back with double interest
what they had lost. Italy, a great part of southern Germany and the
Austrian provinces, France—after long civil wars and convulsions,
after the blood-stained eve of St. Bartholomew, and the murder of
two kings—as also to a great extent Poland and Lithuania, had once
more become Catholic, as fanatically Catholic, too, as Spain and
Portugal, the blazing hells of the Inquisition. In Lutheran and
reformed Germany another papacy had gained the mastery, a
papacy of dry formulas of belief, and slavery to the letter of the law.
The Byzantine quarrel about shadowy dogmas and meaningless
words divided the evangelical communities into as many sects and
subsidiary sects as there were points of discussion, and had a
harmful influence upon political development. Classical philology, at
first liberalizing and suggestive, was neglected, owing to excessive
belief in the Bible by the one party and the sway of authority over
the other, and had degenerated into fanciful dilettanteism or learned
lumber. The study of the Hebrew language, which for a time had
kindled great enthusiasm, was similarly debased, or only carried on
superficially for the purposes of ecclesiastical wrangling. The
knowledge of Hebrew had always been considered, at any rate was
now thought, in orthodox Catholic society, to be actual heresy. And
the same was still truer of rabbinical literature. The learned Spanish
theologian, Arias Montano, published the first complete polyglot
Bible in Antwerp, at the expense of Philip II. He also compiled
grammars and dictionaries of the Hebrew and cognate languages, in
which regard was had to the older Jewish expositors. He, the
favorite of Philip II, who had himself drawn up a list of heretical
books, was accused by the Jesuits and the Inquisition of favoring
heresy, suspected of secret conversion to Judaism, and stigmatized
as a rabbi. Thus, Europe seemed to be actually making a retrograde
movement, only with this distinction—what had formerly been
cheery, naïve credulity now became sinister, aggressive fanaticism.

Refined ecclesiasticism, resulting in the tension which
subsequently relieved itself in the general destructiveness of the
Thirty Years' War, made the sojourn of Jews, both in Catholic and
Protestant countries, a continual torture. Luther's followers in
Germany forgot what Luther had so earnestly uttered in their favor,
only remembering the hateful things of which, in his bitterness, he
had accused them. The Jews of Berlin and the province of
Brandenburg, for instance, had the sad alternative put before them
of being baptized or expelled. A Jewish financier, the physician
Lippold, favorite of Elector Joachim II, and his right hand in his
corrupt, financial schemes, examined and tortured on the rack by
Joachim's successor, John George, admitted, though afterwards
recanting, that he had poisoned his benefactor. The Jews were
driven also out of Brunswick by Duke Henry Julius. Catholic nations
and princes had no cause to reproach their Protestant opponents
with toleration or humanity in regard to Jews.
It was, in some respects, fortunate for the Jews of Germany and
Austria, that the reigning emperor, Rudolph II, although a pupil of
the Jesuits, educated in a country where the fires of the stake were
always smoking, and a deadly enemy of the Protestants, was not
greatly prejudiced against Jews. Weak and vacillating, he was not
able to check the persecutions directed against them, but at least he
did not encourage them. He issued an edict to one bishop (of
Würzburg) that the Jews should not be deprived of their privileges,
and to another (of Passau) that they should not be tortured on the
rack. But, in order not to be decried by his contemporaries or by
posterity as a benefactor of Jews, he not only maintained the heavy
taxation of Jews in his crown land, Bohemia, but from time to time
increased it. He also ordered the Jews to be expelled from the
archduchy of Austria within six months.
In this position, robbed by Catholics and Lutherans alike,
trampled down or driven into misery, barely protected by the
emperor, but taxed under the pretense of enjoying this protection,
the ruin and degradation of German Jews reached ever lower
depths. They were so sorely troubled by the cares of the moment,

that they neglected the study of the Talmud, once their spiritual
food.
The Jews of Italy fared even worse at this time, and they, too,
sank into misery and decay. Italy was the principal seat of the
malicious and inexorable, ecclesiastical reaction, animated with the
thought to annihilate the opponents of Catholicism from the face of
the earth. The torch of civil war was hurled from the Vatican into
Germany, France, and the Netherlands. And as the Jews, from the
time of Paul IV and Pius V, had been upon the list of heretics, or foes
of the church, their lot was not to be envied. With the loss of their
independence, their numbers also decreased. There were no Jews
living in southern Italy. In northern Italy, the largest communities,
those of Venice and Rome, numbered only between 1,000 and 2,000
souls; the community in Mantua had only 1,844; and in the whole of
the district of Cremona, Lodi, Pavia, Alessandria, and Casalmaggiore,
there dwelt only 889 Jews. Pius V, by nature a sinister ecclesiastic
delighting in persecution, who treated Jews as the cursed children of
Ham, was succeeded by Gregory XIII (1572–1585), who had been
skillfully trained to fanaticism by the Jesuits and the Theatine monks.
As regards Jews, Gregory was a most consistent follower of the
cruelty of his predecessor. In spite of repeated warnings, there were
still many Christians in Italy, who, in their blindness, preferred
Jewish physicians of proved excellence, such as David de Pomis, or
Elias Montalto, to Christian charlatans. Gregory was desirous of
prohibiting their employment. He renewed the old canonical law that
Christian patients were not to be treated by Jewish physicians; not
only visiting Christians who transgressed this command with severe
penalties, but also punishing the Jewish physicians if they ventured
to prolong the life of a Christian patient, or even alleviate his
sufferings. His severity succeeded. Another of Gregory's edicts
referred not to one profession, but to the Jewish race in general. He
placed them under the Argus eye of the Inquisition. If any of them
maintained or taught what was heretical, i.e., obnoxious to the
church; if he held intercourse with a heretic or an apostate, helped
him or showed him sympathy, he was to be summoned by the

Inquisition, and according to its verdict was to be condemned to
confiscation of his property, the punishment of the galleys, or even
sentenced to death. If, then, a refugee Marrano from Spain or
Portugal was caught in Italy, and it was proved that a brother Jew
had given him food or shelter, both might expect to be seized by the
inexorable arm of the Inquisition of Italy. The anger of Pope Gregory
XIII was poured forth also against the Talmud. The Jews were once
more admonished to deliver up the Talmud and other works
suspected of being hostile to the church. The Inquisitors and other
spiritual authorities were appointed to institute search for these
books everywhere. Anyone subsequently found in possession of
them, even after declaring that the offending passages had been
expunged, was rendered liable to severe punishment. Pope Gregory
XIII's most zealous effort was directed to the conversion of Jews.
This pope, who most heartily encouraged the Jesuits and their
proselytizing school of thought, endowed a propagandist seminary of
all nations—the curriculum included twenty-five languages—called
the "Collegium Germanicum," issued a decree that on Sabbaths and
holy days Christian preachers should deliver discourses upon
Christian doctrine in the synagogues, if possible in Hebrew, and that
Jews of both sexes, over twelve years of age, at least a third of the
community, must attend these sermons. The Catholic princes were
exhorted to support this vigorous attempt at conversion. Thus an
ordinance of a half-mad, schismatic pope, Benedict XIII, issued in a
moment of passionate excitement, was sanctioned, and even
exaggerated in cold blood by the head of the united Catholic church,
thereby exercising religious compulsion not very different from the
act of Antiochus Epiphanes in dedicating the Temple of the one true
God to Jupiter. It is characteristic of the views then prevailing, that
the Jews were to provide salaries for the preachers, in return for the
violence done their consciences! Like his predecessor, Pius V,
Gregory spared no means to win over the Jews. Many allowed
themselves to be converted either from fear or for their advantage;
for Gregory's edicts did not remain a dead letter, but were carried
out with all strictness and severity. The consequence was that many
Jews left Rome.

The condition of the Jews in Rome was apparently altered under
Gregory's successor, Sixtus V (1585–1590), who rose from the
position of a swine-herd to the office of the shepherd of Catholic
Christendom, and whose dauntless energy in the government of the
Papal States stamped him as an original type of character. He
allowed Jews to be around him, and harbored Lopez, a Jewish
refugee from Portugal, who made various suggestions as to the
improvement of the finances. He went still further; he issued a bull
(October 22d, 1586), which did away with almost all the restrictions
made by his predecessors. Sixtus not merely granted Jews
permission to dwell in all the cities of the Papal States, but also
allowed them to have intercourse with Christians and employ them
as assistants in business. He protected their religious freedom by
special provisions, and extended to them an amnesty for past
offenses, i.e., for condemnations on account of the possession of
religious books. Moreover, he forbade the Knights of Malta to make
slaves of Jews traveling by sea from Europe to the Levant, or vice
versâ, a practice to which these consecrated champions of God had
hitherto been addicted. Pope Sixtus knew how to secure obedience
to his command when it became law, and the Jews previously
expelled now returned to the papal dominions. Under him the Jewish
community at Rome numbered two hundred members. Finally he
removed the prohibition which prevented Jewish physicians from
attending Christian patients. The compulsory services instituted by
his predecessor were the only ordinances that Sixtus V allowed to
remain.
The permission, so important at that time, for Jewish physicians
to have access to Christian patients, was probably gained for himself
and his colleagues, by the then celebrated physician, David de Pomis
(born 1525, died 1588). With medical knowledge he combined
linguistic acquirements, and familiarity with Hebrew and classical
literature, writing both Hebrew and Latin with elegance. In the
course of his life he felt keenly the changes in the papal policy. He
lost all his property through the hostile decrees of Paul IV, was kindly
treated by Pius IV, and allowed by way of exception to practice

among Christians in consequence of a splendid Latin discourse
delivered before the pope and the college of cardinals. But he was
again subjected to irritating restrictions by Pius V, and had to employ
his skill in the service of petty, capricious nobles. To dispel the
unconquerable prejudices against Jews, particularly against Jewish
physicians, De Pomis wrote a Latin work, entitled "The Hebrew
Physician," which affords favorable testimony to his noble mind and
extensive culture. With considerable eloquence De Pomis maintained
that the Jew was bound by his religion to love the Christian as his
brother, and that a Jewish physician, far from wishing to do harm to
his Christian patient, was wont to treat him with the utmost care and
solicitude. He enumerated various Hebrew physicians who had
attended princes of the church, cardinals and popes, had restored
them to health, and had received distinctions from them and from
cities. In conclusion, De Pomis adduced some proverbs from the
Talmud in a Latin translation, to show that this much-calumniated
book was not so harmful and corrupt as enemies of the Jews
asserted. This apology for Judaism and Jewish physicians, dedicated
to Prince Francesco Maria of Urbino, the elegant Latin style of which
was highly praised by an experienced critic of the time, appears to
have made an impression upon Pope Sixtus. De Pomis must certainly
have been intimate with him, as he was allowed to dedicate to him
his second important literary work, a dictionary of the Talmud in
three languages.
The pope severely punished a Christian Shylock, because he
claimed a pound of flesh from a Roman Jew as the result of a wager.
This Christian, named Seche, had wagered with a Jew, named
Ceneda, that St. Domingo would be conquered, and on winning his
bet he claimed the penalty. On hearing of this, Sixtus condemned
him to death, but afterwards mitigated the punishment to
banishment, and allotted the same fate to Ceneda for wagering his
body, the property of his sovereign.
The favorable attitude of Sixtus towards Jews encouraged them
in the hope—to them a matter of conscience, of life itself—that the
prohibition directed against the Talmud and the Hebrew Scriptures

would be removed forever. Under the last two popes no copies of
the Talmud had been allowed to appear without causing the
possessor to incur the dangers of the watchful Inquisition. Nor was
the possession of other perfectly harmless Hebrew works without
risk, for as the Inquisitors and clerical authorities did not in the least
understand them, they condemned all without exception as inimical
to the church, a category which afforded ample room for
denunciation. Whether the possessor of a Hebrew book should be
condemned to lose his property, or be sent to the galleys, depended,
in the last instance, upon the decision of baptized Jews acquainted
with rabbinical literature. To escape these annoyances the
communities of Mantua, Ferrara, and Milan addressed a request to
Sixtus V to allow the Jews to possess copies of the Talmud and other
books, provided these works were previously expurgated of the
passages objectionable to Christianity. They referred to the decision
of Pope Pius IV that the Talmud could not be entirely condemned,
but that it contained passages worthy of censure, which were to be
struck out by the censor's marks. A Jewish delegate, Bezalel
Masserano, had gone to Rome, provided with 2,000 scudi, in order
to lay the request of the Jews at the feet of his Holiness. It was
granted in the bull of October 22d, 1586. Sixtus allowed the
reprinting of the Talmud and other writings, though only after
censorship. For this purpose two commissions were appointed, in
which baptized Jews were naturally included as experts. The Italian
Jews began to rejoice at being allowed to possess even a mutilated
Talmud. But scarcely had the commission arranged the conditions of
the censorship (August 7th, 1590), when the wise pope died, and
the undertaking, just begun, of reprinting the mutilated Talmud was
at once discontinued.
The regard paid Jews by Sixtus V arose not from any sentiment
of justice, but from his passionate desire to amass treasure. "This
pope bled Christians from the throat," says his biographer, "but he
drew the blood of Jews from all their limbs." They often found
themselves compelled to pay immense sums into the papal treasury.

With Clement VIII, however (1592–1605), the system of
intolerance, practiced by Paul IV, Pius V, and Gregory XIII, once
more came into vogue. He repeated the edict of expulsion against
the Jews in the Papal States (February 25th, 1593), and allowed
them to dwell only in Rome, Ancona, and Avignon. If a Jew were
caught in any other papal city, he was to expiate his offense by the
loss of his property and the penalty of the galleys. Clement re-
imposed the old restrictions upon the Jews in the three cities
mentioned, forbidding them either to read or possess the Talmud
and other rabbinical writings. The Jews, expelled from the Papal
States, seem to have been received by Ferdinand, Duke of Tuscany,
who assigned Pisa to them as a dwelling-place (July, 1593). He
allowed them to possess books of every kind and of all languages,
including the Talmud, but the copies first had to be expurgated
according to the regulations of the commission instituted by Sixtus V.
So great was the fanaticism of the apostolic throne that even noble
princes, like Ferdinand de Medici, of Tuscany, and Vicenzo Gonzago,
of Mantua, did not venture to relax it. Even in places where, as a
favor, the Jews were allowed to possess expurgated books, they
were exposed to all kinds of annoyances and extortions. They had to
pay the censors, mostly baptized Jews, for the mutilation of these
writings, nor were they assured that even then their books would
not again be confiscated, and the owners punished, merely because
some obnoxious word or other had remained unobliterated. Woe to
those who rubbed out one of the censors' marks! To avoid being
exposed to vexation, Jews themselves laid hands upon their sacred
literature, and expunged not only everything that referred to
idolatry, but also everything that glorified the Jewish race, or made
mention of the Messiah and his future advent. As Italy, at that time,
was the chief market for printed Hebrew works, the Jews in other
countries received only mutilated copies, from which open or covert
protests against Rome were completely obliterated.
Expulsion of the Jews from all Italian cities was the order of the
day in the reign of this pope. Thus the Jews were expelled (in the
spring of 1597) from the Milan district, i.e., from the cities of

Cremona, Pavia, Lodi, and others, to the number of about a
thousand. They were forced to beg for shelter in Mantua, Modena,
Reggio, Verona, and Padua. During their migrations, they were
robbed by heartless Christians. The sword of the church hovered for
a time also over the Jews in Ferrara, a town that had always been a
safe refuge for them, and even for the new-Christians from Spain.
The ducal race of De Este, whose representatives vied with the
Medici in magnanimity and culture, had died out. The Jews of
Ferrara felt themselves so identified with the fortunes of this princely
house, that they offered public prayers in the synagogue on the
occasion of the severe illness of the thoughtful Princess Leonore,
whom two great poets have immortalized by placing her in the
glorified heaven of poetry. She herself was a benefactress of Jews,
and frequently protected them. But now the last representative of
the race, Alfonso II, had died without heirs (1597), and, in
opposition to his last wishes, Ferrara was incorporated into the Papal
States by Clement VIII. The Jewish community, consisting chiefly of
Marrano refugees, was prepared to endure banishment, as it could
expect no mercy from this pope. They only asked Aldobrandini, the
pope's relative, who had taken possession of Ferrara, to grant them
a respite that they might make preparations for departure. As
Aldobrandini saw that a great portion of the trade of the town was in
the hands of Jews, he had sufficient consideration not to injure it,
granted them permission to remain for five years, and had this
decree carried out in spite of the fanatical wishes of Clement VIII,
who had hoped to banish them. No fugitive new-Christian, however,
could now stop in Ferrara without falling into the clutches of the
bloody Inquisition. Thus the last refuge in Italy for this class of Jews
was destroyed, and there was no longer any place of safety for them
in all Christendom.
It seems providential that the Jewish race, which, at the end of
the sixteenth century, had no longer a footing, properly speaking, in
Europe or Asia, under Christianity or Islam, should have taken firm
root in the empire of their obstinate foe, Philip II, of Spain, and
should have been able from that vantage ground to gain a position

of equality. Indeed, in the chain of causation it was the bloody
Inquisition itself which helped gain them freedom. Holland, a land
wrung from the sea, became for the hunted victims of a horrible,
refined fanaticism, a resting-place where they could settle down, and
develop their national characteristics. But what changes and
vicissitudes they had to undergo before this almost undreamed of
possibility could become reality! The northwest corner of Europe had
hitherto been inhabited by only a few Jews. They suffered, as did
their brethren, under the extravagances of excited fanaticism, were
hunted down, and massacred at the time of the crusades and the
Black Death, bearing all in silent obscurity and patience. When the
country, under the name of the Netherlands, beneath the far-
reaching scepter of Charles V, was united to Spain, the Spanish
principle of hostility to Jews was transferred to it. The emperor
issued command after command that the Jews in the cities of the
Netherlands, small though their numbers were, should be expelled.
Every citizen was required to make known to the royal officers the
presence of Jews contrary to law. In consequence of the introduction
of the Inquisition into Portugal, several Jewish families had betaken
themselves, with all their wealth, industry, and skill, to the
flourishing cities of the Netherlands, Brussels, Antwerp, and Ghent,
in order to lead a religious life secure from danger. The severe edict
of Charles V, and his repeated command not to allow their presence,
extended to them. The magistrates duly fulfilled the commands of
their ruler in this matter, because they feared that the presence of
new-Christians would cause the Inquisition to be introduced—an evil
which seemed to their anxious hearts to forebode great danger for
themselves.
The people of the Netherlands could not escape the Inquisition.
Although an appendage of Spain, were they not surrounded by
Lutheran heretics, and did not these dwell in their very midst? So
this institution was to be introduced among them also. This was one
of the main causes of the revolt of the Netherlands, and of that
long-continued war, so small in its beginnings, and so great in its
results, that rendered powerless the might of Spain, and raised the

tiny land of Holland to a power of almost the first rank. It seemed as
if from every head that Alva struck off in the Netherlands, hundreds
of others sprang, as from the Hydra of old. It was a matter of course
that in this sanguinary struggle which transformed the whole land
into an arena of battle, there was no place for Jews. Upon the advice
of Arnheim and Zütphen, Alva had issued an edict that if Jews were
found there, they were to be kept in custody until such time as he
should pass judgment upon them. It was well known what this
meant from his mouth.
The Portuguese Marranos, or new-Christians, who, even in the
third generation, could not forget, and would not repudiate, their
Jewish descent, turned their eyes towards the Netherlands, now
wrestling for freedom, the more as the Inquisition was raging more
furiously than ever, and dragging them to the dungeon or the stake.
Since the first symptom of the decline of Spain's fortunes, since the
collapse of the invincible Armada, by means of which Philip II had
thought to carry the chains of actual and spiritual bondage not only
to England, but, if possible, to the ends of the earth, there had
arisen in the hearts of the pseudo-Christians, under the iron rule of
this tyrant, an eager desire for freedom. As Italy was closed to them
by the persecuting policy of the reactionary popes, their only hope of
refuge was in the Netherlands.
An eminent Jew, Samuel Pallache, sent by the king of Morocco
as consul to the Netherlands (about 1591), proposed to the
magistrate of Middelburg, in the province of Zealand, to receive the
Portuguese Marranos, and allow them religious freedom. In return,
they would develop the city into a flourishing, commercial center by
means of their wealth. The wise city fathers would willingly have
agreed to this plan, but the war for religion and freedom, so
passionately waged against the two-fold despotism of Spain, had
made even the reformed preachers fanatical and intolerant. They
were opposed to the admission of Jews into Zealand.
But the Portuguese new-Christians did not abandon the idea of
seeking security in the provinces of the Netherlands already freed

from the Spanish yoke. They felt themselves drawn towards this
republic by mighty bonds; they shared its fierce hatred against Spain
with its thirst for human sacrifices, and against its fanatical king,
Philip II. The great Protector, William of Orange, the soul of the
struggle for independence, had uttered the idea of mutual toleration
and friendly intercourse between different religious parties, creeds,
and sects. Although this first germ of genuine humanity at first fell
to the ground, the Marranos clung to it as affording hope of release
from their daily torments. A courageous Marrano woman, Mayor
Rodrigues, appears to have formed the plan of seeking a refuge for
her family in Holland. She, her husband, Gaspar Lopes Homem, her
two sons and two daughters, and several other members of this rich
and respected family, were devotedly attached to Judaism, and
weary of the pretense of following Christian customs, a pretense,
after all, powerless to protect them from the horrors of the
Inquisition. When a ship sailed from Portugal with a load of fugitive
Marranos, under the leadership of one Jacob Tirado, Mayor
Rodrigues intrusted to this vessel her charming and beautiful
daughter, Maria Nuñes, and also her son. The mother appears to
have relied upon the magic of her daughter's charms; the
extraordinary beauty of Maria Nuñes was to serve as an ægis to
these wanderers, surrounded by dangers on all sides, and secure to
them a place of refuge. As a matter of fact, her beauty was
successful in averting the first danger that threatened the party of
refugees, consisting of ten persons, men, women and children. They
were captured by an English ship making raids upon vessels sailing
under the Spanish-Portuguese flag, and were taken to England.
Maria Nuñes so bewitched the captain, an English duke, that he
offered her his hand, thinking that she belonged to the rank of the
Portuguese grandees; but she refused this honorable offer, because
she wished to live as a Jewess. The beauty of the fair Portuguese
prisoner made so great a sensation in London, that the virgin queen,
Elizabeth, was curious to make the acquaintance of this celebrated
beauty, inaccessible even to the love of a duke. She invited her to an
audience, and drove with her in an open carriage through the streets
of the capital. Probably owing to the mediation of Maria Nuñes, the

fugitive Jews were allowed to leave England unharmed, and set sail
for Holland. After enduring a most stormy voyage, they were able to
make for the harbor of Emden, where, as in the rest of East
Friesland, some few German Jews lived.
As soon as the Marranos became aware, by Hebrew letters and
other signs, of the presence of brethren in this city, Jacob Tirado, the
most eminent among them repaired to Moses Uri Halevi, who had
the reputation of being a learned man, and on whose house Hebrew
characters had been noticed. He discovered to him his own and his
companions' intention to give up pseudo-Christianity, and to be
received fully and, if possible, immediately into Judaism. But Moses
Uri had scruples about taking such a decisive course, the apparent
conversion of Christians to Judaism, in a small town, where nothing
could long remain hidden. He, therefore, advised the fugitives to
betake themselves to Amsterdam, where greater toleration was
enjoyed, and promised to come to them with his whole family, to
remain with them, and instruct them in Jewish doctrines.
Accordingly, the Marranos, led by Tirado, arrived at Amsterdam
(April 22d, 1593), sought an abode which would allow of their
remaining together, and were received back into Judaism as soon as
Moses Uri and his family came to them.
Moses Uri and his son arranged a house of prayer for the
Marranos, and officiated as conductors of the services. Great zeal
was shown, not only by Jacob Tirado, but also by Samuel Pallache,
the consul, and a Marrano poet, Jacob Israel Belmonte, come thither
from Madeira, who depicted the tortures of the Inquisition in verse,
giving his poem the appropriate title of "Job." The youthful
community was strengthened in numbers and in standing by fresh
arrivals. An English fleet, which, under the Earl of Essex, surprised
the fortress of Cadiz, and inflicted serious injuries upon the
Spaniards (in the summer of 1596), conveyed several Marranos to
Holland, amongst them a man of great originality, not without
importance for posterity. Alonso de Herrera was descended from
Jewish and ancient Spanish families. His ancestor was the great
Gonsalvo de Cordova, the conqueror of Naples for Spain. He himself

was the Spanish resident in Cadiz, and on the capture of this city
was taken prisoner by the English. On being liberated he went to
Amsterdam, became a Jew, and adopted the name of Abraham de
Herrera (wrongly called Irira).
The Marranos in Amsterdam did not find the practice of their
religion altogether easy. When this first Portuguese community was
secretly celebrating its fourth Fast of Atonement (October, 1596),
their Christian neighbors were surprised at the secret meeting of
disguised figures in one house; they suspected treacherous
assemblies of Catholic conspirators, and denounced them to the
magistrates. Whilst the Jews were engaged in prayer, armed men
suddenly rushed into the house, and spread terror amongst the
assembled worshipers. As most of them, mindful of the cruelties of
the Inquisition, and fearing a similar fate in Amsterdam, tried to save
themselves by flight, the suspicions of the Amsterdam officials were
increased. The latter searched for crucifixes and wafers, and led
Moses Uri and his son, the leaders of the service, to prison.
However, Jacob Tirado, who was able to make himself understood in
the Latin language, succeeded in convincing the authorities that the
assembly was not one of papists, but of Jews who had fled from the
Moloch of the Inquisition. Moreover, that they had brought much
wealth with them, and finally that they would induce many co-
religionists to come from Spain and Portugal with their riches, and
thus give an impulse to the trade of Amsterdam. Tirado's speech
made a great impression. The prisoners were released, and the
terrified Portuguese Jews were able to conclude the service of the
Fast of Atonement. Now that their religion was made known, they
ventured upon the step of petitioning the magistrate to allow them
to build a synagogue in which to hold their religious services. After
much consideration the request was granted. Jacob Tirado bought a
site, and in 1598 built the first Jewish temple in the north of Europe,
called the "House of Jacob" (Beth Jacob). It was consecrated amid
the enthusiasm of the little community.
The favorable news about the Marrano colonists, carried secretly
to Spain and Portugal, afforded additional inducement to emigration.

Mayor Rodrigues Homem, the first promoter of this course, also
found an opportunity of escaping from Portugal and joining her
beautiful daughter, Maria Nuñes (about 1598). She brought her
younger son and daughter with her; her husband had probably died
before this time. Simultaneously, barely escaping the Inquisition,
another eminent family arrived from Portugal, that of Franco
Mendes, including the parents and two sons, Francisco Mendes
Medeïros, a cultured literary man, who took the Hebrew name of
Isaac, and Christoval Mendes Franco, rich and benevolent, who
called himself Mordecai. Both played important parts in the
Amsterdam community, but subsequently caused a division.
Philip II lived to see the two races whom he had most savagely
hated and persecuted, the Netherlanders and the Jews, in a
measure join hands to destroy what he had created, for Holland
derived advantage from the Jewish settlers from Portugal. Previously
it had been one of the poorest states, and the bitter, destructive
wars had made the land still poorer. The capital brought by the
Marranos to Amsterdam was very acceptable, and benefited the
whole country. The Dutch were now enabled to lay the foundations
of their prosperity by taking the Indian trade out of the hands of the
Portuguese, who had been connected with Spain in an unprofitable
alliance. The capital of the fugitive Jews made it possible to found
great transmarine companies and fit out trading expeditions, in
which they participated. The connections, too, of the Portuguese
Jews with their secret co-religionists in the Portuguese possessions
in the Indies assisted the undertakings of Dutch merchants.
Philip II died in September, 1598, a terrible warning to obstinate,
unscrupulous despots. His body was covered with abscesses and
vermin, which made him such an object of horror that his trembling
servants approached him only with disgust. The great empire which
he bequeathed to his feeble son, Philip III, was likewise diseased. It
was succumbing to its infirmities, and no longer possessed influence
in the councils of Europe. The reins of government were loosened,
and thus the new-Christians found it still easier to escape the
clutches of the Inquisition. They now had a goal to which to direct

their steps. An extraordinary occurrence in Lisbon had excited the
most lukewarm apostate Jews to return to Judaism. A Franciscan
monk, Diogo de la Asumção, of an ancient Christian family, had
become convinced of the truth of Judaism and the falsity of
Christianity by reading the Bible—Bible reading has its dangers—and
had openly expressed his convictions to the other monks of his
order. For what purpose had the Inquisition been instituted, if it were
to let such crimes go unpunished? Diogo was thrown into a
dungeon; but it was not necessary to extort confession, for he
openly and without reservation admitted his offense, love for
Judaism. The tribunal needed to put him to the rack only to induce
him to denounce his accomplices, he having asserted that several of
his fellow-monks shared his convictions. Certain learned theologians
were charged to dissuade the apostate Franciscan from his belief,
and remove so dark a stain from the order and Christendom in
general; but in vain. Diogo remained true to his belief in the truth of
Judaism. After he had spent about two years in the dungeons of the
Inquisition, he was finally burnt alive at a solemn auto-da-fé in
Lisbon, in the presence of the regent (August, 1603).
The fact that a Christian by birth, a monk to boot, had suffered
for the sake of Judaism, and had died steadfast in faith, made a
powerful impression upon apostate Portuguese Jews, and impelled
them to return publicly to the faith of their fathers. The Inquisition
lost its terrors for them. They reverted to Judaism, without heeding
whether or not they were rushing upon death. David Jesurun, a
young poet, a favorite of the Muses since his childhood, on this
account called "the little poet" by his acquaintances, celebrated the
burning of the martyr, Diogo de la Asumção, in a fiery Portuguese
sonnet:

Thou wast the gold, buried in the dark vaults of the tribunal of blood;
And even as gold is purified from dross by flames,
So, too, by flames would'st thou be purified.
Thou wast as the phœnix, renewing his life,
Disdaining to remain the slave of death.
Thou wast consumed in the fire.
Only to rise again from thine ashes,
A burnt-offering
Brought to God in the flames.
n heaven dost thou laugh at those who tortured thee;
And no more art called Brother Diogo,
But Golden Phœnix, Angel, Sacrifice."
This eager young poet was fortunate enough to escape the
Inquisition, and hastened to Amsterdam. He composed a powerful
poem in Spanish on seeing this city, which seemed to him a new
Jerusalem. Another young Marrano poet also reverted to Judaism
through the tragic death of Diogo, the Franciscan. Paul de Pina, a man
of some poetic talent, was inclined to religious enthusiasm, and was on
the point of becoming a monk. This step caused great sorrow to his
relative, Diego Gomez Lobato, at heart faithful to Judaism, and he
wished to hinder him from apostasy. When he was about to make a
journey to Italy, Diego, therefore, gave him a letter, addressed to the
celebrated Jewish physician, Elias Montalto, known as Felix Montalto
when professing Christianity. The letter was as follows: "Our cousin,
Paul de Pina, is going to Rome to become a monk. Your Grace will do
me the favor to dissuade him."
If this letter had fallen into the hands of the Roman or Portuguese
Inquisition, it would have cost both the writer and his correspondent
their lives. Elias Montalto endeavored to dissuade young De Pina from
his purpose and win him back to the religion of his fathers. He seems
to have succeeded only in so far that De Pina abandoned his journey
to Rome, went off to Brazil, and then returned to Lisbon. The
martyrdom of Diogo de la Asumção appears to have finally decided
him against Christianity. He hastened to Amsterdam with the sad news
(1604), became an eager convert to Judaism, and adopted the Hebrew

name of Rohel Jesurun. He became a most enthusiastic Jew, an
ornament to the Amsterdam community.
The loyalty to Judaism manifested by the Portuguese Marranos
regardless of consequences naturally swelled the numbers of the
victims of the Inquisition. Not long afterwards, one hundred and fifty
of them were thrown into gloomy dungeons, tortured, and forced to
confess. Even the regent of Portugal hesitated to burn so large a
number. Moreover, the new-Christian capitalists had a certain amount
of power over the Spanish court, to which, since the union of the two
kingdoms, Portugal now belonged. The court owed them large sums
which it could not pay in consequence of the increasing poverty of
both countries. The Marranos offered to release Philip III from this
debt, and give in addition a present of 1,200,000 crusados (£120,000),
if the imprisoned Jews were pardoned. They also spent 150,000
crusados to persuade the councilors to make the king grant this favor.
Hence the court manifested an inclination to mercy, and applied to
Pope Clement VIII to empower the Inquisition to deal mildly with the
sinners on this occasion. The pope remembered, or was reminded,
that his predecessors, Clement VII and Paul III, had granted
absolution to Portuguese Marranos. He did the same, and issued a bull
pardoning the imprisoned Jews (August 23d, 1604). The Inquisition
contented itself with the hypocritical repentance of its prisoners.
Several hundred of them, clad in the garb of penitents, were led to the
auto-da-fé at Lisbon (January 10th, 1605), not, however, to mount the
stake, but to make public confession of their guilt, and be condemned
to deprivation of all civic rights. All, or a large proportion, of those set
free, repaired to their new place of refuge. Among them was Joseph
ben Israel, who had thrice suffered torture, and escaped with
shattered health and the loss of his property. He took with him his son
Manasseh—or whatever his name may have been as a pseudo-
Christian—then a child, subsequently destined to fill a distinguished
rôle in Jewish history.
Moses Uri (born 1544, died 1620) at different times received into
the Hebrew faith two hundred and forty-eight men, so greatly did the
numbers of the community at Amsterdam increase. They sent to
Salonica for a rabbi of Sephardic descent, by name Joseph Pardo, who

well understood the character of the semi-Catholic members of the
community. He put into their hands a book written in Spanish,
Christian rather than Jewish in tone. The synagogue Beth Jacob, built
by Tirado, no longer sufficed for the accommodation of its worshipers,
and a new one had to be built in 1608, called "Neve Shalom." It was
founded by Isaac Francisco Mendes Medeïros and his relatives. As the
discoverers of a new country regard every step they take in it, every
new path into which they strike out, and every person prominent in
the enterprise, as important and worthy of remembrance, so the
young Amsterdam community joyfully recorded everything that
occurred in their midst at the commencement of their career.
The arrival of Isaac Uziel (died in 1620) was a piece of good
fortune for this unique community. Apparently of a family of refugees,
this rabbi could thoroughly sympathize with his companions in
misfortune at Amsterdam. He was a poet, grammarian, and
mathematician, but, above all, a preacher of rare power and influence,
the first who dared arouse, with his mighty voice, the consciences of
his hearers, lulled to sleep by the practice of Catholic customs, and
warn them not to believe that they had purchased indulgence or
remission for their sins, follies, and vices, by religious observances
thoughtlessly practiced. Isaac Uziel did not spare even the most
respected and powerful in the community, although he thereby drew
upon himself their hatred, which went so far as to cause a split; on the
other hand, he gained devoted followers, who celebrated him in
spirited verse.
In this manner religious union was encouraged and faith
strengthened among the Portuguese fugitives, who had so
degenerated in religious matters. But as yet no arrangements had
been made for the proper burial of their dead. They were compelled to
bury them far away from the city, at Groede, in northern Holland. By
the endeavors of the leading members of the community, they
succeeded in obtaining a burial-ground, not too far from Amsterdam,
in Ouderkerk, near Muiderberg (in April, 1614), at which they rejoiced
greatly. The first person buried there was Manuel Pimentel, or, by his
Jewish name, Isaac Abenacar, called "king of players" by the French
king, Henry IV, who was in the habit of playing with him. Two years

later, the body of an eminent and noble man, Elias Felice Montalto,
was brought from far off to be buried in this peaceful spot. He had
formerly professed Christianity, but afterwards became a faithful Jew,
was a clever physician and elegant author, and lived in Livorno, Venice,
and finally in Paris as private physician to Queen Maria de Medici. He
died in Tours while on a journey with the French court, on February
16th, 1616. The queen caused his body to be embalmed, and taken to
the cemetery at Ouderkerk, accompanied by his son, his uncle, and his
disciple, Saul Morteira.
The Jews of Amsterdam were long compelled to pay a tax, for
every corpse, to the churches past which the body was carried. On the
whole, they were at first not tolerated officially, their presence was
only connived at. They were distrusted as Catholic spies in the service
of Spain, plotting treason disguised as Jews. Even when the authorities
and the population in general had become convinced of their genuine
hatred of Spain and Portugal, they were still far from being recognized
and tolerated as an independent, religious body. For a short time the
synagogues were closed, and public worship prohibited. Jewish
refugees from the Spanish peninsula, on arriving in Havre, were
thrown into prison. This intolerance in the country destined to be the
first where religious freedom was to raise its temple, was chiefly
caused by the passionate conflict between two parties of Reformers—
the Remonstrants and Contra-Remonstrants. The former were more
gentle in their exposition and practical application of Christianity than
their opponents, the gloomy Calvinists, Dutch Independents. In
Amsterdam the latter party predominated and persecuted their
opponents, considered secret, treacherous adherents of Spain.
Although the Remonstrants had cause to try to effect toleration for all
sects, it was they who came forward as the accusers of the Jews. They
complained to the chief magistrate of Amsterdam that all kinds of
sects, even Jews, were tolerated in the capital of Holland, they being
the sole exception.
The governor, Prince Maurice of Orange, was certainly favorable to
Jews, but he could do nothing against the spirit of intolerance, and the
independence of the cities and states. Consequently, even in Holland
the Jewish question came up for discussion, and a commission was

appointed for its settlement. Finally it was decided (March 17th, 1615)
that every city, as in the case of Amsterdam, could issue a special
regulation about Jews, either to tolerate them, or to expel them; but in
those cities where they were admitted, they were not to be forced to
wear a badge. Upon the repeated complaints of the Remonstrants, the
burgomaster, Reinier Pauw, laid before the council (October 15th,
1619) the question as to what was to be done in the case of the
numerous fugitive Portuguese Jews who had intermarried with the
daughters of the land, thereby causing great scandal and annoyance.
Hereupon it was decided (November 8th), that intercourse between
Jews and Christian women, even prostitutes, was to be strictly
forbidden. On the other hand, permission was granted to Jews freely
to acknowledge their religion.
As Amsterdam was not so wealthy as it afterwards became, it
could not do without Jews, who had transferred to it their riches and
their knowledge of affairs. The old-established prejudices against them
disappeared more and more upon closer acquaintance. The Jews from
Portugal betrayed neither by their cultured language, their demeanor,
nor their manners, that they belonged to a despised caste; on the
contrary, their carriage was that of people of rank, with whom it was
an honor for many a Christian burgher to be acquainted. They were,
therefore, treated with a certain amount of consideration. Their
number soon increased to four hundred families, with three hundred
houses in the city, and before long, a Hebrew printing press was set up
in Amsterdam, without fear of the Argus eye of the censor.
The prosperity of Amsterdam, caused by the influx of Portuguese
Jews, excited the envy of many Christian princes, and they invited the
Jews into their dominions. Christian IV, king of Denmark, addressed a
letter to the Jewish Council of Amsterdam (November 25th, 1622),
asking them to encourage some of their members to settle in his state.
He promised them freedom of worship, and other favorable privileges.
The Duke of Savoy invited Portuguese Jews to come to Nice, and the
Duke of Modena offered them the right of residence in Reggio, both
granting them extensive privileges. Thus, in the midst of the gloomy
persecution of Christendom, whose two religious factions were drawing
the sword against each other in the Thirty Years' War, the Jews found

pleasant little oases, as it were, from which they could recover their
lost liberty, and gradually raise themselves from their heavy bondage.

CHAPTER XX.
THE DUTCH JERUSALEM AND THE THIRTY
YEARS' WAR.
The Amsterdam Jewish Community—Its Wealth, Culture, and
Honored Position—Zacuto Lusitano—Internal Dissensions—
The Talmud Torah School—Saul Morteira, Isaac Aboab, and
Manasseh ben Israel—The Portuguese Congregation in
Hamburg—The First Synagogue—Lutheran Intolerance—
John Miller—Jewish Colony in Brazil—The Chief
Communities in Germany—Persecution in Frankfort—Dr.
Chemnitz—The Vienna Congregation—Lipmann Heller—
Ferdinand II's Zeal for the Conversion of Jews—Influence
of the Thirty Years' War on the Fortunes of the Jews.
1618–1648 C.E.
The Jewish race during its dispersion of nearly two thousand years
may fitly be compared to a polyp. Though it was often wounded and
cut to pieces, the parts severed from the whole did not die, but began
an independent existence, developed organically, and formed a new
rootstock. Driven from their original Palestinian home, the scattered
members of this peculiar national organism assembled on the banks of
the Euphrates and Tigris and in the palm district of Arabia. Doomed to
ruin there, they emigrated to Spain with the Arabs, the most cultured
people of the Middle Ages, and became the teachers of Europe, then
plunged in barbarism. Expelled thence, weakened in heart and
numbers, they proceeded eastwards, and, as again they found no
resting-place, they settled in the north, always following advancing
civilization. The admission of Jews to Holland was the first quivering

dawn of a bright day after dense gloom. Amsterdam, the northern
Venice, in the beginning of the seventeenth century, had become a
new center for Jews; they rightly named it their new, great Jerusalem.
In time this city became an ark of refuge for the Jewish race in the
new deluge. With every trial conducted by the Inquisition in Spain and
Portugal on account of the Judaizing practices of the Marranos, with
every burning pile set ablaze for convicted or suspected persons, the
numbers of the Amsterdam community increased, as if the fanatics
aimed at depopulating and impoverishing the Catholic countries to
render the heretical states of the Netherlands populous and wealthy.
The Amsterdam Portuguese community, consisting of more than four
hundred members, already possessed three hundred stately houses
and palaces in this city, raised by them to a flourishing seat of
commerce. Their capital enabled them to carry on trade, for the most
part on a large scale, and they were interested in the East and West
India Companies, or conducted banking houses. But to usury, which
made the Jews of other countries so hated, they were sworn foes. The
synagogue dues imposed upon themselves give an approximate idea
of the extent of their capital and trade. For every pound of goods
exported or imported by them they were accustomed to pay a doit,
and these taxes, exclusive of those on the receipts of merchants
interested in trading companies, amounted to 12,000 francs annually.
Not on account of their wealth alone did they occupy a
distinguished position in the new Batavian seat of commerce. The
immigrant Marranos belonged for the most part to the educated class;
in Spain or Portugal, their unnatural mother country, they had occupied
positions as physicians, lawyers, government officials, officers, or
clergymen, and were familiar with the Latin language and literature no
less than with belles-lettres, and were accustomed to the usages of
society. In the Netherlands, then the most civilized part of Europe,
humanistic culture was in itself a recommendation. Hence, in Holland,
cultivated Jews had intercourse with educated Christians on terms of
equality, and obliterated the prejudices against the Jewish race. Some
of them obtained a European reputation, and were connected with
personages of high rank. Abraham Zacuto Lusitano (born 1576, died
1642), great-grandson of Zacuto, the historian and astronomer, was

one of the most celebrated physicians of his time. He corresponded
with Frederick, prince of the Palatinate, and his learned wife, the
unfortunate couple that occupied the throne of Bohemia for a brief
space, and was the cause of the Thirty Years' War. Zacuto's praise was
sounded in poetry and prose by Christian as well as Jewish
professional brethren. The Stadtholders of the Netherlands, princes of
the house of Orange-Nassau, Maurice, Henry, and William II, like the
founder of their race, William I, were well disposed towards Jews, and
treated them as citizens with full rights. Even the Spanish and
Portuguese kings, the persecutors of the Marranos in their own
countries, condescended to show respect to the descendants of their
hunted victims, to confer appointments upon them, and to intrust
them with consular functions for their states.
The attachment of the Amsterdam Jews to their re-adopted
religion, purchased with so many dangers, was deep, and was
renewed at every accession of fresh fugitives, and every report of the
martyrdom of their brethren on the burning pile of the Inquisition. This
devotedness was reflected in their conduct, and embodied in verses
composed in the language of their persecutors.
Paul de Pina, or Rëuel Jesurun, the poet, who had once been on
the point of becoming a monk, composed for a sacred festival part
songs in Portuguese, performed by seven youths to do honor to the
first synagogue (Beth-Jacob) in 1624. The mountains of the Holy Land,
Sinai, Hor, Nebo, Gerisim, Carmel, and Zethim (Mount of Olives), in
melodious verses celebrated the excellence of the Jewish religion, the
Jewish Law, and the Jewish people. They praised the thousand
merciful ways in which God had led His people from the earliest times
to the present. The unity of God, the holiness of the Law, and the
expectations of the Messianic age of grace, the more deeply felt by the
Sephardic Marranos because they were newly acquired and dearly
gained convictions—these were the inexhaustible themes of their
poetry. But in the background of the splendid picture there always
lowered the dreadful dungeon, the priests of Moloch, and the blazing
flames of the Inquisition.

In this mood, exalted by the recollection of sufferings and torture
endured, the members of the Amsterdam community, with full heart
and bountiful hand, founded benevolent institutions of every
description, orphan asylums, benevolent societies (brotherhoods), and
hospitals, such as were not in existence in any of the older
communities. They had the means and the disposition. Their piety was
shown in charity and generosity. But, exalted though their mood was,
they were men with passions, and dissensions arose in the young
community. Many members, born and brought up in Catholicism,
brought with them and retained their Catholic views and customs; they
thought that they could combine them with Judaism. "Can one carry
coals in his bosom without singeing his clothes?" From childhood the
Marranos had heard and seen that one is allowed to sin, if from time
to time he is reconciled with the church. Catholic priests of all ranks
were at hand to effect the reconciliation, and by ecclesiastical means
ward off future punishment from the sinner. In the eyes of most
Marranos, the rites and ceremonies of Judaism took the place of the
Catholic sacraments, and the rabbis of father-confessors. They
believed that he who conscientiously observes Jewish rites, and in
addition does a few other things, may yield to his desires without
forfeiting his soul's welfare. At any rate, the rabbis could give him
absolution. Hence the Marranos led a life far from perfect, especially in
point of chastity. The first two rabbis of the Amsterdam community,
Joseph Pardo and Judah Vega, in consideration of the circumstances
were indulgent to these weaknesses and shortcomings. But the third,
Isaac Uziel, did not restrain himself; with inexorable rigor he scourged
the evil habits of semi-Jews and semi-Catholics from the pulpit. This
severity wounded the attacked, but, instead of mending their ways,
they were angry with the preacher, and several left the community and
the synagogue, and combined to found a new one (the third) in 1618.
At the head of the seceders was David Osorio; possibly he felt most
deeply wounded by Uziel's severe sermons. For the new synagogue
(Beth Israel) which the seceders erected, they chose David Pardo, the
son of Joseph Pardo, as rabbi and preacher. He defended the
acceptance of this office in the new body, founded to some extent in
defiance of Isaac Uziel, by alleging that he wished to lessen
dissension. However, the tension lasted for twenty years (1618–1639).

Meanwhile German Jews, whom the ravages of the Thirty Years'
War had driven out of their Ghettos, sought the asylum of Amsterdam,
and were admitted to its shelter. If the Amsterdam Council had at first
merely connived at the immigration and settlement of Jews, at a later
period it decidedly furthered their admission, because it perceived the
important advantage which they brought the state. The immigrant
German Jews naturally could not unite closely with the Portuguese
community, because they differed, not only in language, but also in
demeanor and manners. A wide chasm divided the Portuguese and the
Germans of the same race and religion from each other. The former
haughtily looked down upon the latter as semi-barbarians, and the
latter did not regard the former as genuine Jews. As soon as a
sufficient number had assembled, the German Jews formed a
synagogue, with a rabbi of their own. Their first chief was Moses Weil.
The breach within the Portuguese community was painfully felt. Jacob
Curiel, a distinguished man, afterwards resident of the Portuguese
court in Hamburg, by the greatest exertions brought about a
reconciliation, and not till the union of the three synagogues in one
single corporate body, in April, 1639, did the Portuguese community,
by the harmonious co-operation of its powers, stand forth in all its
splendor, and surpass all its elder sisters in the three divisions of the
globe. The Amsterdam community in some points resembled the
ancient Alexandrian Jewish congregation. Like the latter, it possessed
great wealth, cultivation, and a certain distinction of character; but,
like it, suffered from insufficient knowledge of Jewish religious and
scientific literature. Nearly all Marrano members had to commence to
learn Hebrew in advanced age!
On uniting the three communities, for which statutes were passed,
the representatives took pains to obviate this ignorance of Hebrew.
They founded an institute (Talmud Torah) in which children and youths
might have instruction in the useful branches of Jewish theology. It
was, perhaps, the first graded institution of the kind among Jews. It
consisted, at first, of seven classes. Students could be conducted from
the lowest step, the Hebrew alphabet, to the highest rung of the
Talmud. It was at once an elementary school and a college for higher
studies. Thorough Hebrew philology, elocution, and modern Hebrew

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