text:
User:Egm6936.s09/Dimensional analysis tutorial

= References =

Web references:
 * google search for "dimensional analysis"
 * Wikipedia: Dimensional analysis Buckingham's Pi theorem Similitude
 * E. Buckingham, On Physically Similar Systems; Illustrations of the Use of Dimensional Equations, Physical Review, Vol.4, No.4, pp.345-376, 1914.
 * P.W. Bridgman (Nobel laureate), Dimensional analysis (free-access internet book), by Yale University Press, 1922.
 * H. Hanche-Olsen, Buckingham's Pi theorem, 2004.
 * The Buckingham Pi theorem in dimensional analysis, 2.25 Advanced Fluid Mechanics, MIT OpenCourseWare
 * Google for "Vaschy Buckingham dimensional analysis" EXCELLENT SEARCH !!
 * T.G. Dobre and J.G. Sanchez Marcano, Chemical engineering: Modelling, simulation, and similitude, Wiley-VCH, 2007. Google book, p.465.
 * Measurement theory and dimensional analysis: Methodological impact on the comparison and evaluation process, E. Coatanea, J. Vareille, B. Yannou.
 * Conceptual modelling of life cycle design: A modelling and evaluation method based on analogies and dimensionless numbers, E. Coatanea, doctoral dissertation, Helsinki University of Technology, 2005.
 * Dynamos, Section 4, p.16, by T. Alboussiere, edited by P. Cardin and Cugliandolo, Elsevier, 2008.
 * Talk slides: 1. History, Equations 2. Energetics 3. Dimensionless parameters 4. MHD (Magneto-Hydro-Dynamics) effects
 * L. I. Sedov, Similarity and dimensional methods in mechanics, Academic Press 1959. Google book.

Other references:

 * Nondimensionalization (French thesis) (Bucky wiki) Nondimensionalization (French thesis) (Bucky wiki)

= Motivation =

Dimensionless partial differential equations
Cairo and Feix, Similarity and rescaling, Extracta Mathematicae, Vol.13, No.1, pp.1-19, 1998.

Experiments in mechanics and physics
= Prelude to the Pi theorem =

This problem has been frequently used to illustrate the procedure to nondimensionalize relations describing physical phenomena.

Nondimensionalization procedure
There are essentially three steps ( Bridgman (1922), p.1 ):
 * list all quantities that a physical phenomenon depends on.
 * write the dimensions of these quantities.
 * combine these quantities into a functional relation that remains true regardless of the unit system used.

But the above is not a systematic procedure to obtain the dimensionless quantities. Bridgman (1922)

resorted to physical reasoning to find dimensionless quantities. Further below, a systematic mathematical method is presented to search for dimensionless quantities.

The swinging pendulum
Let $$\displaystyle F_i$$, with $$\displaystyle i=1,\ldots,m$$, denote the set of $$\displaystyle m$$ fundamental units; e.g., in the SI system, a set fundamental units could be: $$\displaystyle F_1 = T = second$$ for time, $$\displaystyle F_2 = L = meter$$ for length, $$\displaystyle F_3 = M = kilogram$$ for mass, as in the SI system ( see also Wikipedia ). Another set of fundamental units could be $$\displaystyle \hat F_1 = \hat T = second$$ for time, $$\displaystyle \hat F_2 = \hat L = centimeter$$ for length, $$\displaystyle \hat F_3 = \hat M = gram$$ for mass, as in the cgs system.

Next, let the $$\displaystyle n$$ physical quantities of a system be denoted by $$\displaystyle R_i$$, with $$\displaystyle i=1,\ldots,n$$, with the corresponding units denoted by $$\displaystyle [R_i]$$, with $$\displaystyle i=1,\ldots,n$$.

For the swinging pendulum, there are five physical quantities $$\displaystyle R_i$$, which are listed below together with their corresponding dimensional formula for the unit $$\displaystyle [R_i]$$ in terms of the fundamental units $$\displaystyle F_i$$ :

Nondimensionalization by reasioning
It is obvious that the period $$\displaystyle \overline{T} $$ depends on the following parameters:

-	The length of the pendulum $$\displaystyle l$$

-	The mass of the bob $$\displaystyle m$$

-	The gravity $$\displaystyle g$$

The dimensions of the quantities involved are as follows:

 * $$\displaystyle [\overline{T}]=T,\,\,\, [l]=L ,\,\,\, [m]=M,\,\,\, [g]=LT^{-2}$$.

Now we suppose to change the system of units so that the unit of mass is decreased by a factor of $$\displaystyle M$$, the unit of length is decreased by a factor of $$\displaystyle L$$ and the unit of time is decreased by a factor of $$\displaystyle T$$. Therefore, the ratio $$\displaystyle l/g$$ increases by a factor of $$\displaystyle T^{2}$$ or the quantity $$\displaystyle (l/g)^{1/2}$$ increases by a factor of $$\displaystyle T$$. Hence the ratio

 * $$\displaystyle \Pi = \frac{\overline{T}}{\sqrt {l/g}}$$

is invariant under a change of units. $$\displaystyle \Pi $$ is called a dimensionless number. Since it doesn't depend upon the variables $$\displaystyle (m,l,g)$$, it is in fact a constant. Consequently, from dimensional considerations alone we find that:
 * $$\displaystyle \overline{T} = const \times \sqrt{\frac{l}{g}} $$

Nondimensionalization by physical argument
The goal now is to find the functional relation for the time of swing in terms of the other physical quantities such that this relation continues to hold when we change the current fundamental unit system to a different fundamental unit system, such as going from the SI system to the CGS system or to the Customary US system.

Assume that such functional relation can be written as

\displaystyle t  = f (l, m, g, \theta) \Longrightarrow t  - f (l, m, g, \theta) =:  \Phi (R_1, \ldots, R_5) =  0 $$

Define the physically meaningful function $$\displaystyle \Phi$$ ( Hanche-Olsen (2004), p.3

) of the physical quantities $$\displaystyle R_1, \ldots, R_5$$:

\displaystyle \Phi (R_1, \ldots, R_5) :=  t   - f (l, m, g, \theta) =  R_1 -  f (R_2, R_3, R_4, R_5) =  0 $$
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi (R_1, \ldots, R_5) :=  t   - f (l, m, g, \theta) =  R_1 -  f (R_2, R_3, R_4, R_5) =  0 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (1)
 * }

An explicit expression for $$\displaystyle \Phi$$ can be found in Pendulum (Wolfram) (with $$\displaystyle \theta$$ written as $$\displaystyle \theta_{max}$$ ). The angular amplitude of swing $$\displaystyle \theta$$ here is a finite (large) angle, and not restricted to "small" angles.

Let $$\displaystyle \overline{T}$$ be the period of oscillation, then

\displaystyle t   = \frac{\overline{T}}{4} $$ We have

 * {| style="width:100%" border="0"

$$  \displaystyle t  \sqrt{\frac{g}{l}} =  \int\limits_{\alpha = 0}^{\alpha = \theta} \frac {     d \alpha }  {      \sqrt{\cos \alpha - \cos \theta} }  =   \frac{\overline{T}}{4} \sqrt{\frac{g}{l}} \Longrightarrow \overline{T} =  4   \sqrt{\frac{l}{g}} \int\limits_{\alpha = 0}^{\alpha = \theta} \frac {     d \alpha }  {      \sqrt{\cos \alpha - \cos \theta} } $$ See Pendulum (Wolfram).
 * style="width:95%" |
 * style="width:95%" |
 * style= | (3)
 * }

Comparing Eq.(3) to the expression for $$\displaystyle t$$ in Pendulum (Wolfram), it can be seen that the mass $$\displaystyle m$$ had cancelled out in Eq.(3); see Bridgman (1922) , p.2,

on how he argued to remove the mass $$\displaystyle m$$ based on dimensional reasoning, without a need to have the explicit expression for $$\displaystyle t$$.

In the case of small angles, the time of swing depends only on $$\displaystyle g$$ and $$\displaystyle l$$, i.e., the angular frequency $$\displaystyle \omega$$ and the period $$\displaystyle \overline{T}$$ of oscillation (not the dimensional formula $$\displaystyle T$$ ) are given by

 * {| style="width:100%" border="0"

$$  \displaystyle \omega =  \frac{2 \pi}{\overline{T}} =  \sqrt{\frac{g}{l}} \Longrightarrow \overline{T} =  2 \pi \sqrt{\frac{l}{g}} $$ See Pendulum small oscillations (Wolfram).
 * style="width:95%" |
 * style="width:95%" |
 * style= | (4)
 * }

Following Hanche-Olsen (2004) ,

let the numerical magnitude of a physical variable $$\displaystyle R_i$$ be denoted by $$\displaystyle \nu(R_i)$$, i.e., the function $$\displaystyle \nu$$ maps a physical quantity to its numerical value ( $$\displaystyle \nu$$ is Greek for "n", which is mnemonic for number ). Consider the time of swing $$\displaystyle R_1 = t$$; if the fundamental unit $$\displaystyle F_1$$ changes from hour to seconds, then the numerical value $$\displaystyle \nu(R_1) = \nu(t)$$ is multiplied by 3600. This example is a very special case such that the unit $$\displaystyle [R_1] = F_1$$. In general we can write
 * {| style="width:100%" border="0"

$$  \displaystyle R_i =  \nu(R_i) [R_i] =  \rho_i [R_i] \, \ {\rm with} \ \rho_i :=  \nu(R_i) $$ The notation $$\displaystyle \rho_i := \nu(R_i)$$ was introduced just to shorten the notation for the numerical value of $$\displaystyle R_i$$.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (5)
 * }

The numerical value $$\displaystyle \nu(t)$$ of the time of swing would not change if we changed the unit $$\displaystyle [m]$$ of mass or the unit $$\displaystyle [l]$$ of length; Bridman (1922) , p.2.

Note: Is this assertion an experimental fact or an assumption on the nature of the function $$\displaystyle \Phi$$?

Bridgman (1922) , p.2,

argued that if the function $$\displaystyle \Phi$$ in Eq.(1) is to remain true when we change the (fundamental) unit $$\displaystyle [m]$$ of mass, which in turn change the numerical value $$\displaystyle \nu(m)$$ of mass, then the mass $$\displaystyle m$$ should not enter into the arguments of $$\displaystyle \Phi$$, since otherwise the numerical value of $$\displaystyle \Phi$$ would change, as a result of the change in $$\displaystyle \nu(m)$$. Thus, Eq.(1) becomes
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi(t,l,g,\theta) = 0 $$ Thus, Bridgman (1922) , p.2,
 * style="width:95%" |
 * style="width:95%" |
 * style= | (6)
 * }

used dimensional analysis to remove the mass $$\displaystyle m$$ from the arguments of $$\displaystyle \Phi$$ without a need to know the explicit expression for $$\displaystyle \Phi$$.

Next, recall from Table 1 that

\displaystyle [l] = L = F_2 \, \    [g] = L T^{-2} = F_2 F_1^{-2} $$ Bridgman (1922) , p.2,

again argued that if the function $$\displaystyle \Phi$$ is to remain true when we change the fundamental unit $$\displaystyle [l]$$ of length and thus the numerical value $$\displaystyle \nu(l)$$ of length then the numerical value $$\displaystyle \nu(g)$$ of the acceleration of gravity should also change. As a result, $$\displaystyle l$$ and $$\displaystyle g$$ should appear together as the fraction $$\displaystyle l/g$$ or $$\displaystyle g/l$$ in the arguments of $$\displaystyle \Phi$$. Eq.(6), and thus Eq.(1), now becomes
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi(t,l/g,\theta) = 0 \ \, \      {\rm or} \ \Phi(t,g/l,\theta) = 0 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (7)
 * }

Indeed, Eqs.(3)-(4) were in the form of Eq.(7). On the other hand, Eq.(7) is not yet in dimensionless form.

Numerical value and unit of a physical quantity
Now let's use the above simple pendulum example to explain the notations and the corresponding meaning in Hanche-Olsen (2004) .

Bridgman (1922) , p.2,

wrote that "The dimensional formulas show how the various fundamental units determine the numerical magnitude of the variables".

The dimensional formula for the unit $$\displaystyle [R_i]$$ of a physical quantity $$\displaystyle R_i$$ indicates how the numerical value $$\displaystyle \nu(R_i)$$ of that quantity changes when the fundamental units in this dimensional formula change; Bridgman (1922) , p.2,

For example, the dimensional formula for acceleration $$\displaystyle g$$ is
 * {| style="width:100%" border="0"

$$  \displaystyle [g] = LT^{-2} = F_2 F_1^{-2} \ {\rm and} \ g  = \nu(g) [g] $$ Let the fundamental units for length and time be, respectively, $$\displaystyle [l] = F_2 = meter$$ and $$\displaystyle [t] = F_1 = second$$. Next consider a new set of fundamental units $$\displaystyle \widehat{[l]} = \hat F_2 = kilometer$$ and $$\displaystyle \widehat{[t]} = \hat F_1 = hour$$. We have $$\displaystyle \hat F_2 = 1000 F_2 $$ and $$\displaystyle \hat F_1 = 3600 F_1 $$, or $$\displaystyle F_2 = \hat F_2 / 1000$$ and $$\displaystyle F_1 = \hat F_1 / 3600$$, and
 * style="width:95%" |
 * style="width:95%" |
 * style= | (8)
 * }

\displaystyle g  = \nu(g) [g] =  \hat\nu(g) \widehat{[g]} $$

The old unit $$\displaystyle [g]$$ can be expressed in terms of the new unit $$\displaystyle \widehat{[g]}$$ as follows

\displaystyle [g] =  F_2 F_1^{-2} =  \frac{1}{1000} \hat F_2 (\frac{1}{3600} \hat F_1)^{-2} =  \frac{3600^2}{1000} \hat F_2 \hat F_1^{-2} =  (3.6)^2 \times 1000 \   \widehat{[g]} $$ Thus

\displaystyle g  = \nu(g) [g] =  (3.6)^2 \times 1000 \   \nu(g) \widehat{[g]} =  \hat\nu(g) \widehat{[g]} $$ and hence the new numerical value $$\displaystyle \hat\nu(g)$$ of $$\displaystyle g$$ in terms of the old numerical value $$\displaystyle \nu(g)$$ of $$\displaystyle g$$ is then given by

\displaystyle \hat\nu(g) =  (3.6)^2 \times 1000 \   \nu(g) $$

In other words,

\displaystyle g  = \nu(g) \   m/s^2 =  \hat\nu(g) \   km / hr^2 =  (3.6)^2 \times 1000 \   \nu(g) \   km / hr^2 $$

For example, the value of the acceleration of gravity is

\displaystyle g  = \underbrace{9.81}_{\nu(g)} \   \underbrace{m/s^2}_{[g]} =  \underbrace{ (3.6)^2 \times 1000 \times \underbrace{9.81}_{\nu(g)} }_{\hat \nu (g)} \   \underbrace{km / hr^2}_{\widehat{[g]}} $$

Alternative set of fundamental units
In the above, the traditional fundamental units that were used correspond to the physical quantities $$\displaystyle (R_1, R_2, R_3) = (t,l,m)$$. It is also possible to select from Table 1 a different set of (non-traditional) fundamental units, which are units of a different set of physical quantities. For example, we may want to select $$\displaystyle (R_2,R_3,R_4) = (l,m,g)$$, and take the corresponding units of these physical quantities to form a new set of fundamental units $$\displaystyle (F_2, F_3, F_4)$$. Let $$\displaystyle [R_4]=G$$ be the dimensional formula for acceleration of gravity $$\displaystyle g$$, then $$\displaystyle [R_1]= T$$ $$\displaystyle = \sqrt{L/G} = L^{1/2} G^{-1/2} = F_2^{1/2} F_4^{-1/2}$$ is the dimensional formula for time $$\displaystyle t$$ (which is now no longer a fundamental unit) in terms of the new fundamental units. In fact, the choice of the fundamental units is arbitrary; Hanche-Olsen (2004) , p.1.

Nondimensionalization by mathematical method
As a generalization of Eq.(8), the unit $$\displaystyle [R_j]$$ of the physical quantity $$\displaystyle R_j$$ can be written in terms of the fundamental units $$\displaystyle F_i$$, with $$\displaystyle i=1,\ldots,m$$ as follows
 * {| style="width:100%" border="0"

$$  \displaystyle [R_j] =  \prod_{i=1}^{i=m} (F_i)^{a_{ij}} \, \ {\rm with} \ j = 1, \ldots, n $$ where the powers $$\displaystyle a_{ij}$$ are rational numbers.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (15)
 * }

The fundamental units must satisfy the following independence condition:
 * {| style="width:100%" border="0"

$$  \displaystyle \prod_{j=1}^{j=m} (F_j)^{x_{i}} =  1   \Longrightarrow x_1 = \cdots = x_m = 0 $$ i.e., no single fundamental unit can be expressed as a product combination of the other fundamental units.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (16)
 * }

For the above two different sets of fundamental units, we have:
 * {| style="width:100%" border="0"

$$  \displaystyle T^{x_1} L^{x_2} M^{x_3} = 1 \Longrightarrow x_1 = x_2 = x_3 = 0 $$
 * style="width:95%" |
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 * style= | (17)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle L^{x_2} M^{x_3} G^{x_4}= 1 \Longrightarrow x_2 = x_3 = x_4 = 0 $$
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 * style="width:95%" |
 * style= | (18)
 * }

The dimension matrix is defined by
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A  := \left[ a_{ij} \right]_{m \times n} $$ where $$\displaystyle i = 1, \cdots, m$$ is the row index, and $$\displaystyle j = 1, \cdots, n$$ the column index.
 * style="width:95%" |
 * style="width:95%" |
 * style= | (19)
 * }

For the swinging pendulum with the $$\displaystyle m=3$$ fundamental units $$\displaystyle (T,L,M)$$, and $$\displaystyle n=5$$ physical quantities in Table 1, the dimension matrix is
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A  = \left[ \begin{array}{rrrrr} 1 & 0 & 0 & -2 & 0     \\      0 & 1 & 0 & 1 & 0      \\      0 & 0 & 1 & 0 & 0   \end{array} \right]_{3 \times 5} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (20)
 * }

A combination of the physical quantities $$\displaystyle R_j$$ is simply a product of these quantities raised to some powers, i.e., $$  \displaystyle R_1^{\lambda_1} \cdots R_n^{\lambda_n} $$, with the unit of such combination being
 * {| style="width:100%" border="0"

$$  \displaystyle \left[ R_1^{\lambda_1} \cdots R_n^{\lambda_n} \right] =  \prod_{i=1}^{i=m} (F_i)^{a_{i1} \lambda_1 + \cdots + a_{in} \lambda_n} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (21)
 * }

The combination $$  \displaystyle R_1^{\lambda_1} \cdots R_n^{\lambda_n} $$ is called dimensionless if its unit is equal to 1, i.e.,
 * {| style="width:100%" border="0"

$$  \displaystyle \left[ R_1^{\lambda_1} \cdots R_n^{\lambda_n} \right] = 1 $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (22)
 * }

which, as a result of Eq.(21), then implies that
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A  \boldsymbol \lambda =   \mathbf 0 \, \ {\rm with} \ \boldsymbol \lambda =  \left\lfloor \lambda_1, \cdots , \lambda_n \right\rfloor^T $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (23)
 * }

In other words, the column matrix $$\displaystyle \boldsymbol \lambda$$ is in the null space of the dimension matrix $$\displaystyle \mathbf A$$.

For the swing pendulum, using the matlab "null" command

> null(A, 'r')

where the argument 'r' was used to obtain the results in rational numbers instead of real numbers, one can find the null space of $$\displaystyle \mathbf A$$ to be
 * {| style="width:100%" border="0"

$$  \displaystyle Null(\mathbf A)  = \left[ \begin{array}{rr} 2 & 0	 \\	 -1 & 0	 \\	 0 & 0	 \\	 1 & 0	 \\	 0 & 1     \end{array} \right] =:  N(\mathbf A) $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (24)
 * }

Clearly,
 * {| style="width:100%" border="0"

$$  \displaystyle \mathbf A   \cdot N(\mathbf A)  = \left[ \begin{array}{cc} 0 & 0      \\      0 & 0       \\      0 & 0    \end{array} \right] $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (25)
 * }

Thus, the column matrix $$\displaystyle \boldsymbol \lambda$$ can be one of the two columns of the null-space matrix $$\displaystyle N(\mathbf A)$$, i.e., there are two possible dimensionless combinations of the physical quantities $$\displaystyle R_j$$:
 * {| style="width:100%" border="0"

$$  \displaystyle R_1^{2} R_2^{-1} R_4^{1} =  \frac{t^2 g}{l} \, \ {\rm with} \ \left[ \frac{t^2 g}{l} \right] =  T^2 L^{-1} [g] =   1 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (26)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle R_5^{1} =  \theta \, \ {\rm with} \ [\theta] =  1 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (27)
 * }

The dimensionless quantity in Eq.(26) is consistent with the result obtained in Eq.(7) by Bridgman (1922) , p.2,

As noted above, Eq.(7) is not yet in dimensionless form, as the quantities that appear in the list of arguments are not dimensionless. On the other hand, the mathematical approach in Hanche-Olsen (2004)

offers a systematic method to find the dimensionless quantities, which can be used in Eq.(7) to obtain a fully dimensionless relation. Eq.(7) can now be put in dimensionless form using Eqs.(26)-(27) as follows:
 * {| style="width:100%" border="0"

$$  \displaystyle \Phi(t \sqrt{g/l}, \theta) = 0 $$
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 * style="width:95%" |
 * style= | (28)
 * }

For example, the first part of Eq.(3) is already in dimensionless form, i.e.,
 * {| style="width:100%" border="0"

$$  \displaystyle t  \sqrt{\frac{g}{l}} =  \frac{\overline{T}}{4} \sqrt{\frac{g}{l}} =  \int\limits_{\alpha = 0}^{\alpha = \theta} \frac {     d \alpha }  {      \sqrt{\cos \alpha - \cos \theta} } $$ and Eq.(4) can be rewritten in dimensionless form as
 * style="width:95%" |
 * style="width:95%" |
 * style= | (29)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \overline{T} \sqrt{\frac{g}{l}} =  2 \pi $$
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 * style="width:95%" |
 * style= | (30)
 * }

Change of size in fundamental units
When the size of a fundamental unit changes, i.e., the relation between the old unit $$\displaystyle F_i$$ and the new unit $$\displaystyle \hat F_i$$ given by
 * {| style="width:100%" border="0"

$$  \displaystyle \hat F_i =   \frac {F_i} {x_i} \Longrightarrow F_i =  x_i \hat F_i \Longrightarrow [R_i] =  \nu(R_i) F_i =  \nu(R_i) \underbrace {     x_i \hat F_i }_{F_i} =  \underbrace {     \nu(R_i) x_i }_{ \hat \nu(R_i) } \hat F_i $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (35)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \hat \nu(R_i) =  \nu(R_i) x_i \, \    \hat \rho_i =  \rho_i x_i $$ where $$\displaystyle x_i$$ is a factor. For example, if the unit $$\displaystyle [l] = [R_2] = F_2$$ of length changes from kilometer to meter, then $$\displaystyle x_2 = 1000$$, or
 * style="width:95%" |
 * style="width:95%" |
 * style= | (36)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle F_2 =  x_2 \hat F_2 \Longrightarrow \underbrace{km}_{F_2} =  \underbrace{10^{3}}_{x_2} \    \underbrace{m}_{\hat F_2} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (36)
 * }

= The Pi theorem =

Developers: Rayleigh, Vaschy, Buckingham, et al.
Many scientists contributed to develop the Pi theorem, sometimes called "P theorem" ; a detailed historical account can be found in the chapter titled Dimensional exploration in Roche (1998) . The principal developers appear to be Rayleigh (1878) , Vaschy (1892) , and Buckingham (1914) .

Aime' Vaschy was probably a Frenchman, based on his first name and on the French journal in which his paper on dimensional analysis appeared, and based on the fact that a prominent French academician referred to the Pi theorem simply as the Vaschy theorem, ignoring the traditional attribution of the theorem to Buckingham. The Pi theorem was often referred to as the Buckingham Pi theorem, but it seems that since the 1960s , the contribution of Vaschy has been recognized, and the theorem referred to as the Vaschy-Buckingham theorem.

Vaschy is pronounced like "Vashy" in English; so many authors (including even French authors!) mispelled the original name of Vaschy as "Vashy" ; see, e.g., Greneche et al. (2005) , in which ironically these French authors only referred to the paper by Buckingham , but not the orginal paper by Vaschy (1892).

In The Buckingham Pi theorem in dimensional analysis, MIT OpenCourseWare, it was stated that F.M. White pointed out in his book Fluid Mechanics who were instrumental in developing the Pi theorem, in particular Lord Rayleigh (1877), A. "Vaschy" (1892), and D. Riabouchinsky (1911).

F.M. White, Fluid mechanics, McGraw-Hill, 2006. Google book, p.293; Chap 5, Dimensional analysis and similarity Amazon.com

F. Casarejos, J.F. Villas da Rocha, and R.M. Xavier, Fine Structure Constants in n-dimensional Physical Spaces through Dimensional Analysis, 2003.

A. Vaschy, Ann. Tel, Vol.12, p.25, 1892.

J. Fluid Mechanics Digital Archive (1968), 32 : 825-826. ... This last part covers dimensional analysis, internal .... The Buckingham theorem is now apparently the Vaschy-Buckingham theorem. ...

Presentation by Salencon A. Vaschy (1892). Annales télégraphiques, Vol.25-28, pp.189-211.

In T.G. Dobre and J.G. Sanchez Marcano, Chemical engineering: Modelling, simulation, and similitude, Wiley-VCH, 2007. Google book, p.465, stated that the first modern presentation of the Pi theorem was due to Vaschy.

M. Zlokarnik, Brief Historic Survey on Dimensional Analysis and Scale-up, Scale-Up in Chemical Engineering, 29 Jun 2006.

Finally: J.J. Roche, The mathematics of measurements, Springer, 1998, 422 pages. Google book, p.208, Chap 11. Whole history since Rayleigh, Vaschy, then Buckingham, with excerpts from original works.

Theorem statement
Consider the dimensional quantity $$\displaystyle a$$, ,p.27 defined as a function of the independent dimensional quantities $$\displaystyle {a_1},{a_2} \ldots ,{a_n}$$ by

 * $$\displaystyle a = f\left( {{a_1},{a_2} \ldots ,{a_k},{a_{k + 1}}, \ldots ,{a_n}} \right)$$

Let the first $$\displaystyle k$$ $$\displaystyle (k \le n)$$ of the dimensional quantities $$\displaystyle {a_1},{a_2} \ldots ,{a_n}$$ have independent dimensions $$\displaystyle($$the number of basic units should be larger than or equal to $$\displaystyle k)$$. Hence, the dimensions of quantities $$\displaystyle a,{a_{k + 1}}, \ldots ,{a_n}$$ can be expressed in terms of the dimensions of the paramaters $$\displaystyle {a_1},{a_2} \ldots ,{a_k}$$.

Let us introduce dimensions of the basic quantities:

 * $$\displaystyle \left[ \right] = {A_1},\,\,\,\left[  \right] = {A_2},\, \ldots ,\,\,\,\left[  \right] = {A_k}$$

The dimensions of the remaining quantities will be

 * $$\displaystyle \left[ a \right] = A_1^A_2^ \ldots A_k^,\,\,\,\left[ \right] = A_1^A_2^ \ldots A_k^, \ldots ,\,\,\,\left[  \right] = A_1^A_2^ \ldots A_k^$$

We now change the units of measurement of the quantities $$\displaystyle {a_1},{a_2} \ldots ,{a_k}$$ by factors of $$\displaystyle {\alpha _1},{\alpha _2} \ldots ,{\alpha _k}$$ respectively; the numerical values of these quantities and the quantities $$\displaystyle a,{a_{k + 1}}, \ldots ,{a_n}$$ in the new system of units, will be given by

 * $$\displaystyle a_1^' = {\alpha _1}{a_1},\,\,\,a_2^' = {\alpha _2}{a_2}, \ldots ,\,\,\,a_k^' = {\alpha _k}{a_k},$$

 * $$\displaystyle {a^'} = \alpha _1^\alpha _2^ \ldots \alpha _k^a,\,\,\,a_{k + 1}^' = \alpha _1^\alpha _2^ \ldots \alpha _k^{a_{k + 1}}, \ldots ,\,\,\,a_n^' = \alpha _1^\alpha _2^ \ldots \alpha _k^{a_n}$$

In the new system of units,

 * $$\displaystyle {a^'} = \alpha _1^\alpha _2^ \ldots \alpha _k^a = \alpha _1^\alpha _2^ \ldots \alpha _k^f\left( {{a_1},{a_2} \ldots ,{a_n}} \right) = f\left( {{\alpha _1}{a_1}, \ldots ,{\alpha _k}{a_k},\alpha _1^\alpha _2^ \ldots \alpha _k^{a_{k + 1}}, \ldots ,\alpha _1^\alpha _2^ \ldots \alpha _k^{a_n}} \right)$$

This relation shows that the function $$\displaystyle f$$ is homogeneous in the scales $$\displaystyle {\alpha _1},{\alpha _2} \ldots ,{\alpha _k}$$. Because the scales are arbitrary, we choose them in order to cut down the number of arguments in the function $$\displaystyle f$$. Let us assume:

 * $$\displaystyle {\alpha _1} = \frac{1},\,\,\,{\alpha _2} = \frac{1}, \ldots ,\,\,\,{\alpha _k} = \frac{1}$$

The numerical values of the parameters $$\displaystyle a,{a_{k + 1}}, \ldots ,{a_n}$$ are determined in this relative system of units by the formular:

 * $$\displaystyle \Pi = \frac{a},\,\,\,{\Pi _1} = \frac, \ldots ,\,\,\,{\Pi _{n - k}} = \frac$$

Where $$\displaystyle a,{a_1},{a_2}, \ldots ,{a_n}$$ are the numerical values of the quantities considered in the original system of units. It is obvious that the values $$\displaystyle \Pi ,{\Pi _1} \ldots ,{\Pi _{n - k}}$$ are independent of the choice of the original system of units of measurement since they have zero dimensions relative to the units $$\displaystyle {A_1},{A_2} \ldots ,{A_k}$$. Therefore, these quantities can be considered nondimensional.

Using the relative system of units, one can write:

 * $$\displaystyle \Pi = f\left( {1, \ldots ,1,{\Pi _1}, \ldots ,{\Pi _{n - k}}} \right)$$

Hence, the relation between the $$\displaystyle n+1$$ dimensional quantities $$\displaystyle a,{a_1},{a_2}, \ldots ,{a_n}$$ reduces to one between the $$\displaystyle n+1-k$$ quantities $$\displaystyle \Pi ,{\Pi _1} \ldots ,{\Pi _{n - k}}$$, which are a nondimensional combination of $$\displaystyle n+1$$ dimensional quantities.

Proof
J. B. Boyling, A short proof of the Pi theorem of dimensional analysis, ZAMP, Vol.30, No.3, May 1979.

= More examples =

Solid mechanics
A. Vaziri and L. Mahadevan, Localized and extended deformations of elastic shells, Proceedings of the National Academy of Sciences (USA), 105, 7913, 2008. Movie

$$\displaystyle \square \, w$$

Physics
On the use, by Einstein, of the Principle of Dimensional Homogeneity, in three problems of the Physics of Solids

Theorem of Buckingham Kepler's 3rd law.

= References =


text:
User:Egm6936.s09/Nondimensionalization (French thesis)

 The "loose" translation of Chapter 5 of the French thesis by E. Rieutord (1985) at the Institut National des Sciences Appliquees, Lyon (INSA de Lyon), France, forms only one of the many possible starting points for this wiki article on dimensional analysis.

By "loose" translation, I meant to take the freedom to improve the way certain ideas and concept were communicated in the thesis, which may or may not be well written (in the sense of being able to convey ideas in the most rigorous and crystal clear possible).

We will not hesitate to add material that would be helpful to help understand this topic. It is possible that the present article may evolve into a tutorial on dimensional analysis, at least for self study.

Tuan, you can contribute to this article by rewriting as many equations from the French thesis as possible in the wiki latex format. At the beginning, it is best to do your writing on wikiversity; in particular, see here. I'll show you how in an e-mail how to create a wikiversity account.

Contributing to writing this article is a good way to learn, not just dimensional analysis, but also latex. At the beginning, you may want to use the following software: Latex equation editor to generate the latex marked-up language. Once you are used to write equations in latex marked-up language directly, you won't need this web site.

This way, I can focus on translating French into English (but we also look at other references as well).

Another program to type wiki articles using firefox, or other browsers, is WikEd.

I identified some web material in English related to dimensional analysis for you to consult, as listed below.

Vql 20:39, 13 February 2009 (UTC)

= References =

 * E. Rieutord, Thesis title??, Institut National des Sciences Appliquees, Lyon (INSA de Lyon), France, 1985.

= Web references =

 * google search for "dimensional analysis"
 * Wikipedia: Dimensional analysis Buckingham's Pi theorem Similitude
 * E. Buckingham, On Physically Similar Systems; Illustrations of the Use of Dimensional Equations, Physical Review, Vol.4, No.4, pp.345-376, 1914.
 * P.W. Bridgman (Nobel laureate), Dimensional analysis (free-access internet book), by Yale University Press, 1922.
 * H. Hanche-Olsen, Buckingham's Pi theorem, 2004.
 * Measurement theory and dimensional analysis: Methodological impact on the comparison and evaluation process, E. Coatanea, J. Vareille, B. Yannou.
 * Conceptual modelling of life cycle design: A modelling and evaluation method based on analogies and dimensionless numbers, E. Coatanea, doctoral dissertation, Helsinki University of Technology, 2005.
 * Dynamos, Section 4, p.16, by T. Alboussiere, edited by P. Cardin and Cugliandolo, Elsevier, 2008.
 * Talk slides: 1. History, Equations 2. Energetics 3. Dimensionless parameters 4. MHD (Magneto-Hydro-Dynamics) effects

= Dimensional analysis -- similitude and model testing =

Mathematical analysis by itself is insufficient to solve all problems encountered in fluid mechanics and in other areas, despite enormous progress made in recent years in numerical analysis and in computational technology (large, powerful computers).

Experiments continue to be a most reliable, fastest, and less onerous method to solve a given problem. A recourse to experiments is also useful even when a theoretical solution had been found so to validate not just such solution, but also notably the hypotheses made in the problem statement.

On the other hand, when a problem possesses some degree of complexity, a judicious choice of the problem parameters and the analysis of the effects of such choice are essential not just in theoretical analysis, but also in the use of the experimental results.

To satisfy the above two requirements -- selection of a most rational formulation for theoretical analysis and experimental study or experimental verifications -- similitude theory and dimensional analysis are useful tools.

Let's begin with dimensional analysis, which is the study of the general relations that exist between different quantities that characterize a physical phenomenon.

Dimensional analysis
Dimensional analysis is based on a simple fundamental principle that a relation between different physical quantities should be dimensionally homogeneous, i.e., independent of the unit system being used.

Vashy-Buckingham Pi theorem
The above mentioned fundamental principle is given by the Vashy-Buckingham theorem, which can be stated as follows:

 The above statement of the Vashy-Buckingham theorem is rather obscure! It did not define quatities such as $$\displaystyle g_i$$ and $$\displaystyle \Pi_j$$.

It is likely that $$\displaystyle g_1, g_2, \cdots, g_q$$ in the above theorem are rational numbers, which form the powers for the independent variables $$\displaystyle x_1, x_2, \cdots, x_q$$. The quantities $$\displaystyle \Pi_{j}$$ are products (and thus the choice of the character $$\displaystyle \Pi$$, which is Greek for "P", and is mnemonic for "Product") of some independent variables, and are the dimensionless quantities.

Vql 20:39, 13 February 2009 (UTC)

 Who was Vashy? google search for "Vashy Buckingham". Many authors referred to the Vashy-Buckingham Pi theorem, but then only cited the paper by Buckingham; there was no citation of any paper by Vashy, e.g., R. Greneche, Y. Ravalard, and D. Coutellier, A Method for Crash Tests on Laminated Composite Scaled-Down Models, Applied Composite Materials, Vol.12, pp.355-368, 2005. What was the contribution of Vashy?

In the above google search, I noticed that many articles written by French researchers referred to the Vashy-Buckingham theorem. In the English literature, the theorem was often referred to as the Buckingham Pi theorem; it is possible that Vashy was a Frenchman who contributed to develop this theorem, but did not publish his (her?) result in an archival journal like Buckingham did. Vql 22:53, 13 February 2009 (UTC)

In practice, we select $$\displaystyle x_1, x_2, \cdots, x_q$$ as the parameters that we consider essential for the problem at hand, and which we want to see appear explicitly in the expression for $$\displaystyle G$$.

At this point, I became a bit disillusioned with the French thesis, and suggest to look at the book by a master:

 * P.W. Bridgman (Nobel laureate), Dimensional analysis (free-access internet book), by Yale University Press, 1922.

and a more recent reference:

 * H. Hanche-Olsen, Buckingham's Pi theorem, 2004.

and perhaps eventually the classic paper by Buckingham himself:

 * E. Buckingham, On Physically Similar Systems; Illustrations of the Use of Dimensional Equations, Physical Review, Vol.4, No.4, pp.345-376, 1914.

I like the general and rigorous mathematical exposition of the VB (short for Vashy-Buckingham) Pi theorem in Hanche-Olsen (2004) and the simple examples that Bridgman (1922) used to begin his exposition of the VB Pi theorem before stating the theorem on p.41.

The above expression in Rieutord (1985), i.e.,

\displaystyle G   = x_1^{g_1} x_2^{g_2} \cdots x_q^{g_q} F ( \Pi_{q+1} \Pi_{q+2} \cdots \Pi_{p}) $$ actually corresponds to Eq.(6) in Hanche-Olsen (2004) and the subsequent arguments:

\displaystyle \begin{align} \Phi [	 (	   x_1^{a_{11}}	    \cdots	    x_m^{a_{m1}}	    \rho_1	 ) ,	 \ldots ,	 (	   x_1^{a_{1n}}	    \cdots	    x_m^{a_{mn}}	    \rho_n	 ) ]     &      =      x_1^{b_{1}} \cdots x_m^{b_{m}} \Phi (	 \rho_1	 ,	 \ldots	 ,	 \rho_n     ) \\     &      =      R_1^{c_1} \cdots R_n^{c_n} \Phi (	 R_1	 ,	 \ldots	 ,	 R_n     ) \\     &      =      R_1^{c_1} \cdots R_n^{c_n} \psi (	 R_1	 ,	 \ldots	 ,	 R_k	 ,	 \Pi_{1}	 ,	 \ldots	 ,	 \Pi_{n-r}     ) \\     &      =      R_1^{c_1} \cdots R_n^{c_n} \Psi (	 \Pi_{1}	 ,	 \ldots	 ,	 \Pi_{n-r}     ) \end{align} $$

and the equation at the top of p.41 in Bridgman (1922):

\displaystyle \alpha =  \beta^{x_1} \gamma^{x_2} \cdots \Phi (     \Pi_2     ,      \Pi_3      ,      \cdots   ) $$

Loosely speaking, the theorem stated in Rieutord (1985) can be restated as follows:

See also Buckingham's Pi theorem (Wikipedia):

Plan: Explain dimensional analysis by starting with the example in Bridgman (1922) but couch it in the language of Hanche-Olsen (2004) thus understand both documents, "one stone for two birds". We will follow the notation in Hanche-Olsen (2004).

See Topic:Dimensional analysis tutorial

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Collaborative play writing/Cardenio

The following text is based on Double Falsehood (1727) by Lewis Theobald, an adaptation of the lost "The History of Cardenio" by John Fletcher and William Shakespeare (1613). Only a small part of the original version of the play, perhaps 1%, sounded like either of the 17th century authors. The purpose of the project is to make it more Shakespeare-like, or at least more Fletcher-like. At the present moment, the entire play has been redone and sounds much more like an 17th century work. The names of most of the characters have been changed back to the original version of the story, namely the Cardenio episodes (chapters 24, 27, 28, 29, and 36) in Miguel de Cervantes' "Don Quixote" (1605 and 1615).

Cardenio

Dramatic characters (14)

Cardenio

Luscinda

The duke of Osuna

Fernando, his elder son

Rodrigo, his younger son

Violante

Bernardo, father to Luscinda

Camillo, father to Cardenio

Fabian, Camillo's brother

Giraldo, servant to Fernando

Ancianida, duenna to Violante and then Luscinda

Master of the goats

A goatherd

Laurencia, an abbess

Servants

SCENE. The province of Andalusia in Spain

TIME: Early 17th century

/Act_1/

/Act_2/

/Act_3/

/Act_4/

/Act_5/

title: Collaborative play writing/Cardenio


text:
Collaborative play writing/Cardenio/Cardenio/Act 1

Act 1. Scene I. The ducal palace

Enter the duke of Osuna and Rodrigo

Rodrigo. My gracious father, this unwonted strain

Visits my heart with sadness.

Osuna. Why, my son?

Making my death familiar to my tongue

Digs not my grave one jot before the date.

I've worn the garland of my honours long,

And would not leave it withered to thy brow,

But flourishing and green; worthy the man,

Who, with my dukedoms, heirs my better glories.

Rodrigo. This praise, which is my pride, spreads me with blushes.

Osuna. Think not, that I can flatter thee, my Rodrigo;

Or let the scale of love o'er-poize my judgment.

Like a fair glass of retrospection, thou

Reflect'st the virtues of my early youth,

Making my old blood mend its pace with transport:

While fond Fernando, thy irregular brother,

Sets the large credit of his name at stake,

A truant to my wishes, and his birth.

His taints of wildness hurt our nicer honour,

And call for swift reclaim.

Rodrigo. I trust, my brother

Will, by the vantage of his cooler wisdom,

E'er-while redeem the hot escapes of youth,

And court opinion with a golden conduct.

Osuna. Be thou a prophet in that kind suggestion!

But I, by fears weighing his unweighed course,

Interpret for the future from the past.

And strange misgivings, why he hath of late

By importunity, and strained petition,

Wrested our leave of absence from the court,

Awake suspicion. Thou art inward with him;

And, haply, from the bosomed trust can'st shape

Some formal cause to qualify my doubts.

Rodrigo. Why he hath pressed this absence, sir, I know not;

But have his letters of a modern date,

Wherein by Cardenio, good Camillo's son

(Who, as he says, shall follow hard upon;

And whom I with the growing hour expect)

He doth sollicit the return of gold

To purchase certain horse, that like him well.

This Cardenio he encountered first in France,

And lovingly commends him to my favour;

Wishing, I would detain him some few days,

To know the value of his well-placed trust.

Osuna. O, do it, Rodrigo; and assay to mould him

An honest spy upon thy brother's riots.

Make us acquainted when the youth arrives;

We'll see this Cardenio, and he shall from us

Receive the secret loan his friend requires.

Bring him to court.

Exeunt Osuna and Rodrigo

Act 1. Scene 2. Prospect of a village at a distance

Enter Camillo with a letter

Camillo. How comes the duke to take such notice of my son, that he must needs have him in court, and I must send him upon the view of his letter. Horsemanship! What horsemanship has Cardenio? I think, he can no more but gallop a hackney, unless he practised riding in France. It may be, he did so; for he was there a good continuance. But I have not heard him speak much of his horsemanship. That's no matter: if he be not a good horseman, all's one in such a case, he must bear. Princes are absolute; they may do what they will in anything, save what they cannot do.

Enter Cardenio

O, come on, sir; read this paper: no more ado, but read it:  it must not be answered by my hand, nor yours, but, in gross, by your person; your sole person. Read aloud.

Cardenio. 'Please you, to let me first o'erlook it, sir.

Camillo. I was this other day in a spleen against your new suits: I do now think, some fate was the taylour that hath fitted them: for, this hour, they are for the palace of the duke. - Your father's house is too dusty.

Cardenio. (aside

Hem!- to court? Which is the better, to serve a mistress, or a duke? I am sued to be his slave, and I sue to be Luscinda's.

Camillo. You shall find your horsemanship much praised there. Are you so good a horseman?

Cardenio. I have been,

E'er now, commended for my seat, or mocked.

Camillo. Take one commendation with another, every third's a mock.--Affect not therefore to be praised. Here's a deal of command and entreaty mixt; there's no denying; you must go, peremptorily he inforces that.

Cardenio. (aside

What fortune soever my going shall encounter, cannot be good fortune; What I part withal unseasons any other goodness.

Camillo. You must needs go; he rather conjures, than importunes.

Cardenio. (aside

No moving of my love-suit to him now?

Camillo. Great fortunes have grown out of less grounds.

Cardenio. (aside

What may her father think of me, who expects to be sollicited this very night?

Camillo. Those scattered pieces of virtue, which are in him, the court will solder together, varnish, and rectify.

Cardenio. (aside

He will surely think I deal too slightly, or unmannerly, or foolishly, indeed; nay, dishonestly; to bear him in hand with my father's consent, who yet hath not been touched with so much as a request to it.

Camillo. Well, sir, have you read it over?

Cardenio. Yes, sir.

Camillo. And considered it?

Cardenio. As I can.

Camillo. If you are courted by good fortune, you must go.

Cardenio. So it please you, sir.

Camillo. By any means, and tomorrow: Is it not there the limit of his request?

Cardenio. It is, sir.

Camillo. I must bethink me of some necessaries, without which you might be unfurnished: And my supplies shall at all convenience follow You. Come to my closet by and by. I would there speak with you.

Exit Camillo

Cardenio. I do not see that fervour in the maid,

Which youth and love should kindle. She consents,

As 'twere to feed without an appetite;

Tells me, she is content; and plays the coy one,

Like those that subtly make their words their ward,

Keeping address at distance. This affection

Is such a feigned one, as will break untouched;

Die frosty, e'er it can be thawed; while mine,

Like to a clime beneath Hyperion's eye,

Burns with one constant heat. I'll strait go to her;

Pray her to regard my honour: but she greets me.-

Enter Luscinda

See, how her beauty doth inrich the place!

O, add the music of thy charming tongue,

Sweet as the lark that wakens up the morn,

And make me think it paradise indeed.

I was about to seek thee, Luscinda,

And chide thy coldness, love.

Luscinda. What says your father?

Cardenio. I have not moved him yet.

Luscinda. Then do not, Cardenio.

Cardenio. Not move him? Was it not your own command,

That his consent should ratify our loves?

Luscinda. Perhaps it was: but now I've changed my mind.

You purchase at too dear a rate, that puts you

To woo me and your father, too. Besides,

As he, perchance, may say, you shall not have me;

You, who are so obedient, must discharge me

Out of your fancy. Then, you know, 'twill prove

My shame and sorrow, meeting such repulse,

To wear the willow in my prime of youth.

Cardenio. Oh! do not rack me with these ill-placed doubts;

Nor think, though age has in my father's breast

Put out love's flame, he therefore has not eyes,

Or is in judgment blind. You wrong your beauties,

Venus will frown if you disprize her gifts,

That have a face would make a frozen hermit

Leap from his cell, and burn his beads to kiss it;

Eyes, that are nothing but continual births

Of new desires in those that view their beams.

You cannot have a cause to doubt.

Luscinda. Why, Cardenio?

When you that dare not chuse without your father,

And, where you love, you dare not vouch it; must not,

Though you have eyes, see with 'em- can I, think you,

Somewhat, perhaps, infected with your suit,

Sit down content to say, you would, but dare not?

Cardenio. Urge not suspicions of what cannot be;

You deal unkindly; misbecomingly,

I'm loth to say: For all that waits on you,

Is graced, and graces. - No impediment

Shall bar my wishes, but such grave delays

As reason presses patience with; which blunt not,

But rather whet our loves. Be patient, sweet.

Luscinda. Patient! What else? My flames are in the flint.

Haply, to lose a husband I may weep;

Never, to get one: when I cry for bondage,

Let freedom quit me.

Cardenio. From what a spirit comes this?

I now perceive too plain, you care not for me.

Duke, I obey thy summons, be its tenour

Whate'er it will: If war, I come thy souldier:

Or if to waste my silken hours at court,

The slave of fashion, I with willing soul

Embrace the lazy banishment for life;

Since Luscinda has pronounced my doom.

Luscinda. What do you mean? Why talk you of the duke?

Wherefore of war, or court, or banishment?

Cardenio. How this new note is grown of me, I know not;

But the duke writes for me. Coming to move

My father in our business, I did find him

Reading this letter; whose contents require

My instant service, and repair to court.

Luscinda. Now I perceive the birth of these delays;

Why Luscinda was not worth your suit.

Repair to court? Ay, there you shall, perhaps,

(Rather, past doubt) behold some choicer beauty,

Rich in her charms, trained to the arts of soothing,

Shall prompt you to a spirit of hardiness,

To say, So please you, father, I have chosen

This mistress for my own.-

Cardenio. Still you mistake me:

Ever your servant I profess myself;

And will not blot me with a change, for all

That sea and land inherit.

Luscinda. But when go you?

Cardenio. Tomorrow, love; so runs the duke's command;

Stinting our farewell-kisses, cutting off

The forms of parting, and the interchange

Of thousand precious vows, with haste too rude.

Lovers have things of moment to debate,

More than a prince, or dreaming statesman, know:

Such ceremonies wait on Cupid's throne.

Why heaved that sigh?

Luscinda. O Cardenio, let me whisper

What, but for parting, I should blush to tell thee:

My heart beats thick with fears, lest the gay scene,

The splendors of a court, should from thy breast

Banish my image, kill my interest in thee,

And I be left, the scoff of maids, to drop

A widow's tear for thy departed faith.

Cardenio. O let assurance, strong as words can bind,

Tell thy pleased soul, I will be wond'rous faithful;

True, as the sun is to his race of light,

As shade to darkness, as desire to beauty:

And when I swerve, let wretchedness o'ertake me,

Great as e'er falshood met, or change can merit.

Luscinda. Enough. I'm satisfied: and will remain

Yours, with a firm and untired constancy.

Make not your absence long: old men are wavering;

And swayed by int'rest more than promise giv'n.

Should some fresh offer start, when you're away,

I may be prest to something, which must put

My faith, or my obedience, to the rack.

Cardenio. Fear not, but I with swiftest wing of time

Will labour my return. And in my absence,

My noble friend, and now our honoured guest,

The Lord Fernando, will in my behalf

Hang at your father's ear, and with kind hints,

Poured from a friendly tongue, secure my claim;

And play the lover for thy absent Cardenio.

Luscinda. Is there no instance of a friend turned false?

Take heed of that: no love by proxy, Cardenio.

My father.

Enter Bernardo

Bernardo. What, Cardenio, in public? This wooing is too urgent. Is your father yet moved in the suit, who must be the prime unfolder of this business?

Cardenio. I have not yet, indeed, at full possessed

My father, whom it is my service follows;

But only that I have a wife in chase.

Bernardo. Chase! Let chase alone: No matter for that.- You may halt after her, whom you profess to pursue, and catch her too; marry, not unless your father let you slip.- Briefly, I desire you (for she tells me, my instructions shall be both eyes and feet to her) no farther to insist in your requiring, till, as I have formerly said, Camillo make known to me, that his good liking goes along with us; which but once breathed, all is done; till when, the business has no life, and cannot find a beginning.

Cardenio. Sir, I will know his mind, e'er I taste sleep:

At morn, you shall be learned in his desire.

I take my leave.- O virtuous Luscinda,

Repose, sweet as thy beauties, seal thy eyes;

Once more, adieu. I have thy promise, love;

Remember, and be faithful.

Exit Cardenio

Bernardo. His father is as unsettled, as he is wayward, in his disposition. If I thought young Cardenio's temper were not mended by the metal of his mother, I should be something crazy in giving my consent to this match. And, to tell you true, if my eyes might be the directors to your mind, I could in this town look upon twenty men of more delicate choice. I speak not This altogether to unbend your affections to him: But the meaning of what I say is, that you set such price upon yourself to him, as many, and much his betters, would buy you at; (and reckon those virtues in you at the rate of their scarcity) to which if he come not up, you remain for a better mart.

Luscinda. My obedience, sir, is chained to your advice.

Bernardo. 'Tis well said, and wisely. I fear, your lover is a little folly-tainted; which, shortly after it proves so, you will repent.

Luscinda. Sir, I confess, I approve him of all the men I know; but that approbation is nothing, till seasoned by your consent.

Bernardo. We shall hear soon what his father will do, and so proceed accordingly. I have no great heart to the business, neither will I with any violence oppose it. But leave it to that power which rules in these conjunctions, and there's an end. Come, haste we homeward, girl.

Exeunt Bernardo and Luscinda

Act 1. Scene 3. Before Violante's house

Enter Fernando and Giraldo

Fernando. Bear the lights close: where is the music, sirs?

Giraldo. Coming, my lord.

Fernando. Let 'em not come too near. This maid,

For whom my sighs ride on the night's chill vapour,

Is born most humbly, tho' she be as fair

As nature's richest mould and skill can make her,

Mended with strong imagination.

But what of that? Th'obscureness of her birth

Cannot eclipse the lustre of her eyes,

Which make her all one light.- Strike up,

But touch the strings with a religious softness;

Teach sound to languish through the night's dull ear,

Till melancholy start from her lazy couch,

And carelessness grow convert to attention.

(Giraldo plays music

She drives me into wonder, when I sometimes

Hear her discourse; the court, whereof report,

And guess alone inform her, she will rave at,

As if she there sev'n reigns had slandered time.

Then, when she reasons on her country state,

Health, virtue, plainness, and simplicity,

On beauties true in ttitle, scorning art,

Freedom as well to do, as think, what's good;

My heart grows sick of birth and empty rank,

And I become a villager in wish.

Play on; she sleeps too sound.- Be still, and vanish.

Exit Giraldo

A gleam of day breaks sudden from her window:

O taper, graced by that midnight hand!

Enter Violante above at her window

Violante. Who is't, that wooes at this late hour? What are you?

Fernando. One, who for your dear sake-

Violante. Watches the starless night!

My Lord Fernando, or my ear deceives me.

You've had my answer, and 'tis more than strange

You'll combat these repulses. Good my lord,

Be friend to your own health; and give me leave,

Securing my poor fame, nothing to pity

What pangs you swear you suffer. 'Tis impossible

To plant your choice affections in my shade,

At least, for them to grow there.

Fernando. Why, Violante?

Violante. Alas! Sir, there are reasons numberless

To bar your aims. Be warned to hours more wholesome;

For, these you watch in vain. I have read stories,

(I fear, too true ones) how young lords, like you,

Have thus besung mean windows, rhymed their sufferings

Ev'n to th'abuse of things divine, set up

Plain girls, like me, the idols of their worship,

Then left them to bewail their easy faith,

And stand the world's contempt.

Fernando. Your memory,

Too faithful to the wrongs of few lost maids,

Makes fear too general.

Violante. Let us be homely,

And let us too be chaste, doing you lords no wrong;

But crediting your oaths with such a spirit,

As you profess them: so no party trusted

Shall make a losing bargain. Home, my lord,

What you can say, is most unseasonable; what sing,

Most absonant and harsh. Nay, your perfume,

Which I smell hither, cheers not my sense

Like our field-violet's breath.

Fernando. Why this dismission

Does more invite my staying.

Violante. Men of your temper

Make ev'ry thing their bramble. But I wrong

That which I am preserving, my maid's name,

To hold so long discourse. Your virtues guide you

T'effect some nobler purpose!

Exit Violante

Fernando. Stay, bright maid!

Come back, and leave me with a fairer hope.

She's gone.- Who am I, that am thus contemned?

The second son to a prince?- Yes, well, what then?

Why, your great birth forbids you to descend

To a low alliance: here's is the self-same stuff,

Whereof we dukes are made; but clay more pure!

And take away my title, which is acquired

Not by my self, but thrown by fortune on me,

Or by the merit of some ancestour

Of singular quality, she doth inherit

Deserts t'outweigh me.- I must stoop to gain her,

Throw all my gay comparisons aside,

And turn my proud additions out of service,

Rather than keep them to become my masters.

The dignities we wear, are gifts of pride,

And laughed at by the wise, as mere outside.

Exit Fernando


text:
Collaborative play writing/Cardenio/Act 1

Act 1. Scene I. The ducal palace

Enter the duke of Osuna and Rodrigo

Rodrigo. My gracious father, these unwonted strains

Of death can visit saddest hearts with tears.

Osuna. To make my death familiar to my tongue

Perhaps will make it pleasanter to all the rest

Of my shrunk body. Garlands in my life

I have worn long, unwithered on my brow,

Though never green most worthy of the man.

Who better than yourself, a son of pride,

Can better glories with my dukedom's charge?

Not to be known, unless your brother dies.

Rodrigo. Such praise, my pride and sadness, covers me

With tears that seem like blushes.

Osuna. To flatter young ones in these gaudy times,

When painted tinsel is accounted gold

And old men to be dropped off like their hair,

Much savors of designed senility.

Let leaden weights of old love counterpoise

My noble judgment. Like that Holland glass

Which turns milk-drops into a thousand stars,

Your love resolves the virtues of my youth,

Makes sluggish-lazy blood increase its pace,

Like wearied soldiers seeing from afar

Their welcome in the smoking chimney, while

Your blood-clot of a brother, stirring but

In frolics, drinkings, escapades of lusts,

A truant to my wishes and his birth,

Makes hearts like mine murmur erratically,

Sends credits of our fame to bankruptcy,

His arms of wildness thrashing all about

To hurt our glassy honor silken-wrapped.

Rodrigo. Fernando, I trust, will by ventages

Of wisdom cool the hot escapes of youth.

Osuna. Like two demented prophets backward-wise,

Both you and I interpret but the past.

Fernando leaves our court to fornicate,

In used holes spurting Guadalhorce streams

More plenteously, as I must always hear,

Than tears of my physicians when I die.

How is this seemly as my son and heir?

Rodrigo. I have his letters of a modern date,

In which Cardenio, old Camillo's son,

His true bordello -friend in Paris met,

Is hotly sent here for obtaining gold

To buy six jennets pleasing him too well.

Osuna. Pay him, Rodrigo. In return, attempt

To use Cardenio as our honest spy

On loose Fernando's riots. To our court

Bring friend-Cardenio, let him stay as long

As we might wish.

Rodrigo. I'll write to his much sighing father now.

Exeunt Osuna and Rodrigo

Act 1. Scene 2. Camillo's house

Enter Camillo and Fabian

Camillo. My son, noticed by the duke! He'll have Cardenio in his palace, and I to send him on view of this letter.

Fabian. By which capacity?

Camillo. Horsemanship! What horsemanship has Cardenio? To my certain knowledge, he gallops in a coach when his coachmen are commanded to hurry, unless he practiced riding with you in France.

Fabian. No.

Camillo. No matter in such a case. The duke has spoken and we must hear.

Fabian. A visitation likely to bring much honor to your house!

Camillo. I believe so, should Cardenio think so.

Fabian. Have you reason to fear your son incapable of gilding our name in great men's houses?

Camillo. No, unless desire of advancement lags behind love-pursuits.

Fabian. He has noticed, I hear, Luscinda, neighbor Bernardo's daughter.

Camillo. He has more than noticed her.

Fabian. So do we.

Camillo. Not with the eye of youth that will have more of her. I violently suspect my son will request me to use violence on Bernardo till obtaining Luscinda as his wife.

Fabian. Is the father so averse to this marriage?

Camillo. Just so-so, enough to desesperate Cardenio.

Fabian. That mellow evening proposition must fade before the bright new morning of the duke's commands.

Camillo. Great men are absolute, doing as they wish in anything, even in what they cannot do.

Enter Cardenio

O, come, Cardenio, read this letter, no more ado, but read at once. It must not be answered by my hand or yours but by your complete person. Read aloud for your uncle's sake.

Cardenio. Should it please you, let me first overlook the paper alone.

Camillo. Here with a darting eye, Cardenio. I was this other day in hot anger against precocious love-suits, which, I now think, have found the tailor fitting them to the honor of our house, too dusty next to a duke's palace.

Cardenio. Hum! To court? Which is better, to serve a mistress or great ones? I must beg be the duke's slave, or Luscinda's.

Fabian. Friendship with Lord Fernando serves you entirely with the father. I find your horsemanship much praised in his house. How is this?

Cardenio. I have ridden well with Fernando above various mistresses, that's true. Commended for a seat because of those, or mocked!

Camillo. If you compare promotions in the world, every third's a mockery. Do not therefore wait in affection till you are better praised next time but go. Here is an ounce of entreaty mixed with a pound of command. No denying puissance in a hurry! Go, peremptorily at your slowest pace, when a duke's suggestion enforces.

Cardenio. What fortune howsoever my going encounters, it cannot be good, for what I part with unseasons any other dish.

Camillo. He rather orders than asks, I think.

Cardenio. Love-suits lie cold this summer.

Camillo. Why do you speak of love now?

Fabian. Sun-flowers grow on poorer grounds than ours:

There may be honor in your going now.

Cardenio. What should I do when a woman expects to be solicited this very day?

Camillo. Who thinks of women now?- I hope, brother, that those scattered pieces of mettle in Cardenio can be soldered together and varnished at court.

Fabian. No doubt.

Cardenio. Too slightly, unmannerly, foolishly, or dishonestly carried out on the part of any type of so-called lover! A father's consent can be requested with no loss of precious honor.

Camillo. A father's consent you already have, unless I fail to understand myself. Have you read the letter over?

Cardenio. I have.

Fabian. And considered it with your brain?

Cardenio. As I can.

Camillo. So courted by good fortune, speedily

Away without another word of text!

Cardenio. Should it please you, already far away.

Camillo. By any means tomorrow at the latest, the limit of his request, no?

Cardenio. It is.

Camillo. I must think of superfluities, necessary no doubt at court, without which a young man seems unfurnished. Further supplies will at my convenience follow. Come to my room later in the afternoon, for more in the way of a father's tearful recommendations to his departing son.

Exeunt Camillo and Fabian, enter Luscinda

Cardenio. See how bright beauties evermore enrich

Our foil! Add but the soundings of your tongue,

The music-box of love, to make me think

I live in artificial paradise.

Luscinda. What does your father say to marriage, sir?

Cardenio. Hum, hah! I have not pressed that question yet.

Luscinda. Why then, do not, Cardenio.

Cardenio. I was about to seek love as you came,

To chide her coldness.

Luscinda. Mine?

Cardenio. I do not see that virgin-seeming heat

Which youth and love should kindle. You consent

To feed without the edge of appetite,

Revealing your content like coyer ones,

Who subtly make love-words their only wards,

Thus keeping open passion farther off.

Your affectation plays, like coward swords

Too loudly martial, to break off untouched.

Your love lies frosty in the bud all night,

While mine, a clime beneath Hyperion's eye

Burns in one constant place. Your own command

Desired my father's will should ratify

With many mounds of earth our garden loves.

Luscinda. Perhaps it did, but now my mind seems changed.

You seek to purchase at too dear a rate

When wooing maidens and your father, too.

Besides, some say he does not like my face.

If so, a son's obedience must discharge

A girl from fancy. That will prove to be

My shame and sorrow, knowing what I lose,

To wear the willow in my prime of youth.

Cardenio. Do not rack love with heretic misdoubts,

Or think, because age freezes ancient breasts,

He can put out love's flame. He has no eyes,

Or counts gold in the dark. You always wrong

Your beauties. Venus-favored fame must frown

If you disprize her gifts, enough to make

A frozen curate leap out from his cell

And burn his beads to kiss them oftener:

Eyes, nothing less than more continual births

Of new desires than we can fondle, ears,

Much like the shell of Venus when she first

Saw her light brightening the seas of love.

Luscinda. Why should I think as you do, stupidly,

When you without a father dare not choose,

Or, if so, dare not show me as your own?

If you dare not, though you have eyes and mouth,

Should I sit satisfied, daydreaming that

My lover likes but dares not say he likes?

Cardenio. Urge no suspicion of what cannot be.

You deal unkindly or misbecomingly,

Because the man I wish to be depends

On you, both graced and gracing evermore.

Impediments can never hold my wish,

But our delays press patience to the ground

Almost to death, so that sex-passion's edge,

Too blunt as yet, must rather whet his tongue

To murder them for us.

Luscinda. Cold patience is asleep and takes our place

In bed. You are in love with her, not me.

Thus, my flames waver in the flint, choked off.

I'll lose a husband if I weep too loud,

Never to get one. When I cry for bonds,

Let freedom quit me, though I weep much more.

Cardenio. From which tomb does this inexistent ghost

Arise? I now perceive you have no care

For me. Duke, I obey your summons here,

Whether of war or peace, tomorrow march

As soldiers do. If to waste silken hours

At court, as fashion's slave with willing soul

I will embrace my lazy banishment,

Since my Luscinda's spirit dooms our love.

Luscinda. What do you mean? Why do you speak of dukes,

Of war, or court, or brainless banishment?

Cardenio. How new notes from forgotten instruments

Strike at our ears I do not care to know,

But yet the duke commands me to his court.

Luscinda. I now perceive the spring-time of your stop

And go, your hesitations and delays,

Why pale Luscinda is invisible.

To court? I understand. There you will seek

Past any doubt some choicer beauty, rich

In being new, trained in the arts of love,

What is considered so at palaces,

To prompt you into bolder hardiness,

Enough to say: "Should it please you, dear dad,

I choose at last a mistress of my own."

Cardenio. Mistaken still! As a slave I protest

I will arrive and leave. No mistress ink

Can blot me from your page, for all I know

The sea and land inherits in our world.

Luscinda. When do you go?

Cardenio. Tomorrow, sweet: so resonates the duke,

Our farewell kisses almost choking off

Before we think of parting. Interchange

Of far more than a thousand vows must hold,

By courrier haste cut short, though lovers' speech

Contains far heavier subjects of debate

Than dreaming statesmen, knowing little that

They dream, for ceremonies always wait

On Venus' throne.- Was that a sigh I heard

Or winds on grasses of forgotten tombs?

Luscinda. Cardenio, let me ponder lucidly

What, but for parting, I should blush to tell:

My heart beats thick with fears, lest richer scenes,

The splendors of a court, should from your breast

And mine my image banish, murdering

Your interest in me, or yours in mine,

And I be left the scoff of maidens, with

A widow's tear for our departed faith.

Cardenio. No, let assurance, as strong as words bind,

Tell your pleased soul I will be faithful still,

As true as sunlight in its lines of beams,

As shade to darkness, as desire to love.

Thus, if I swerve, let wretchedness take me,

As deep as dungeons falsehood ever found.

Luscinda. Enough. I'm satisfied, remaining yours,

Untired in constancy. But, truest love,

Do not delay: old men say yes and no,

Swayed more by interest than promises.

Should fresher offers like battalions come,

I may be pressed to something I dislike,

A father's faith in my obedience racked

Because of you.

Cardenio. With swiftest bulls of time I'll labor till

I turn again this way. Meantime, missed one,

My noble friend, our very honored guest,

Fernando, on whom I build trust on top

Of trust, will, for our sake, if you agree,

Hang heavily against your father's ear

With many hints of love, securing me

Above all marriage-vows you may obtain.

Enter Fernando

Here is Fernando, lending us to love

And happiness. Say, best of friends, can you

Replace Cardenio in a father's ear,

Fulfilling my hopes in her as you would

Your very own?

Fernando. Say that I am remiss if I fail to

Advance love's progress in her moistest cell,

Especially for your Luscinda, prize

Unseen since Paris's choice of goddesses

Among all women I have ever known.

Cardenio. And thereby breathes my terrors in the night,

Reflecting others may look as you do.

Fernando. No doubt some will. I'll wait for you outside,

To lend you for a while to your best self,

Till riding post-haste to my father's court.

Exit Fernando

Luscinda. Is there no instance of a friend turned false?

No love by proxy, my Cardenio.

Cardenio. I kiss such fears away.

Luscinda. My father!

Enter Bernardo

Bernardo. What, Cardenio, in public?

Cardenio. But not yet in pubis, Don Bernardo.

Bernardo. A wooing much too urgent, nevertheless! Is your father yet apprised of your suit, the prime unfolder of love's contract?

Cardenio. I have not yet in full informed that man

I call my father, whom my services

Should follow all my days but not the nights,

Except to promulgate I chase a wife.

Bernardo. Let chase alone. You may stumble after the girl whom you profess to pursue, and yet catch her, but not unless a father lets you slip.- To be briefer than I wish, because my opinion is in Luscinda's view the eyes and feet of her obedience, I desire you to proceed no farther, till, as formerly said, Camillo makes known to me whether his liking marches along with ours, which, but once breathed, all is done, till which time, our business has no life, or the end cannot find its beginning.

Cardenio. I will once know his mind before I dream

Of sleep, and thus I take my leave.- My love,

Repose in all your beauties, sealed in hope.

Once more, adieu. I have your promises:

Remember, and be faithful.

Exit Cardenio

Bernardo. The father is as unsettled as the son is wayward. If I thought Cardenio's temper unmended by his mother's sense, I would suffer somewhat under the effects of an old man's folly in giving my consent to this match. To yield you tardily some snatches of truth, if eyes direct the mind, I could look in this city on twenty men of a more refulgent aspect. I do not say this to unbend your affections altogether away from his desire, my meaning being that you should set such a price on yourself as many more men, perhaps choicer, may be inclined to buy, reckoning your virtues at the rate of its rareness in society, to which if father and son do not come up, you remain available for a more favorable mart.

Luscinda. Am I your merchandise?- How, startled, sir?

Recall what I once said. I do not dream

To be reported as so many girls

We grievingly hear of in Spanish streets:

Bold mouths in looser petticoats, but yet

Consider I have always loved your mind

Because you have respected mine. Do I

Bear judgment in this matter as you have

Allowed in others? Show it now, but know,

In any case, my dear obedience's sway

Is chained against the post of your advice.

Bernardo. Well said and wisely, female Machiavel. Your lover may be a little folly-tainted, I fear, which shortly after it proves so, you will repent.

Luscinda. I confess I approve of him more hotly than all the men I know, but that liking tastes tartly, till seasoned by your consent.

Bernardo. We'll soon hear what his father does, and so proceed accordingly. I have no great heart in this business, but neither do I with violence oppose it, leaving it to those powers ruling women's conjunctions, which philosophers since Socrates must despair of understanding. In regard to a more important matter: food, let us haste homeward, girl.

Exeunt Bernardo and Luscinda

Act 1. Scene 3. Before Violante's house at night

Enter Fernando and Giraldo with a torch and a lute

Fernando. Bear your light low. Where is your music, fool?

Giraldo. Here, at your elbow, never in your voice.

Fernando. After your tune, let no one near her house.

Giraldo. No, not her father.

Fernando. This Violante, my own Violante-

Can man love names before once meeting them?-

For whom my sighs ride hot on nightime's breath,

Is born too lowly, though she is as fair

As nature's richest mold which skill creates,

Improved with my imagination's force.

But what of that? Obscurenesses of birth

Cannot eclipse the heaven in her eyes,

Which make her all one light.- Strike up, fond slave.

In touching strings with a religious hand,

Teach sound to languish through a virgin ear,

Till melancholy startles from her bed,

And carelessness converts to love's repose.

(Giraldo plays

She drives me into wonder. I sometimes

Hear glad replies from Violante where

She never can be found, of whose report

I guess how she may lie, still raving on,

As if with seven reigns she slanders time.

When she discourses on her country state,

Health, virtue, plainness, and simplicity,

On beauties true in title, false in art,

Her freedom to do and to think assured,

My head grows sick of birth and rank, and I

Become in mind a rutting villager.

Play on; she sleeps too soundly.- Vanish, slave.

A gleam like hope most sudden on her door,

Her taper graced by heaven's midnight hand!

Exit Giraldo, enter Violante and Ancianada above

Violante. What man woos at this late hour? Who are you?

Fernando. One who composes one part of your dreams.

Violante. Who let you in?- Not Ancianada, ha?

Ancianada. Somewhat, girl.

Violante. Once more, who are you, sir? Fernando, or

The ear deceives as men most often do.

You have your answer, sir, before I speak.

Acteon boldy entering at night

And I without a hound to punish him!

Ancianada. Unless duennas may aptly termed

Dogs of your honor.

Violante. I dare not, Ancianada.- To you, sir.

Befriend your virtues better, give me leave,

Securing reputation, not to know

What pangs a lover suffers. Labor lost

On dirt and stones it is when lovers seek

To plant their rose-affections in my shade,

Not least for them to grow there.

Fernando. Why, Violante?

Violante. Alas! There are such reasons, numberless,

To bar your aims. Be warned to love or hope

More wholesomely at virgin-clearer hours

Than these watched-for in vain. I have read tales-

I fear, too true- how many rakish lords,

Besing their way in houses, rhyme their hearts

In gross abuse of things divine, set down

Plain girls as idols of their worshipped fane,

Then leave them to bewail their easy faith,

And stand alone against the world's contempt.

Fernando. Your memory, too faithful to the wrongs

Of willing women, makes fear general.

Violante. Let women's faces rest more homely chaste,

Attracting lords demurely, venting speech

Like breathing, not with open laughing mouths,

But crediting their oaths with such a tune

As you profess them: thus, no party's trust

Bemoans a losing bargain. Home, my lord.

What you should say is too unseasonable

And absonant. Moreover, your perfume,

Too near my nose, does not rejoice the sense

Like freshest violets in a loved one's grave.

Fernando. A harsh rebuke invites.

Violante. Men of your temper, I regret to see,

Make everything their brambles. But I wrong

The place I am preserving, virgin's cell,

To hold so long a speech. May virtues guide

You to some nobler purposes tonight.

Exit Violante

Fernando. Stay, stay. By leaving, you attract me more.

Abandon lovers later with some hope.-

She's gone.- Who am I, frothing, too contemned?

The first son of a duke? Hum, what of that?

Our greater birth forbids us to descend

To low alliances: the self-same stuff

Knits up our shirts and coats, but clay like hers

Is pure, and takes away my title, got

Not by myself, but heaped by fortune's sway,

Or by the merit of some ancestor

Of unknown quality. Her face and mind

Inherit virtues to outweigh my own,

So that I need to stoop to win her here,

Throw all my gay comparisons aside,

And turn my proud additions out of pay,

Rather than keep them to become their slave.

The dignities we wear seem gifts of pride,

Much laughed at by the wise as mere outside.

I itch with lust.- No, keep away, far, far.

I tingle to the very tip of it.

No word, or else I use two swords tonight.

Exit Fernando inside the house and re-enter Giraldo

Giraldo. So, is she won at last?

Ancianada. Not in the way I hoped. O Virgin, help!

Giraldo. A maquarella prays, when she laid out

The sheets her startled mistress must bleed on.

Ancianada. Life's first syllable is woe.

Giraldo. Hot deeds are stirring. I hear their sounds, but this can in no fashion be called love. I barely contain myself to play with myself.

Ancianada. Salacious-lolling cur, wriggling weasel, will you remove your ear from the door?

Giraldo. I do, involuntary bawd. What he is doing now no one should attempt to know about.

Ancianada. What have I done? Sacrificed my mistress to ribaldry and loathsomeness! For what? Mere coins, vanished tomorrow for an ear-ring!

Giraldo. Console your mountain breasts by letting me share a little in the pile, best reward of filthy stratagems.

Ancianada. Your gold for sinning well.

Giraldo. The world's most common way, old remonstrance! Thank your hypocrisy for our riches.

Exeunt Ancianada and Giraldo