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## Prime Series 1: Introduction to Prime Numbers

We have learned from elementary school mathematics that a prime number has only two factors, 1 and itself. For example, 2, 3, 5 and 7 are prime numbers, while 8 is not prime because it has four factors — 1, 2, 4, and 8. Numbers that are not prime are called composite numbers.

Geometric Interpretation of Prime and Composite Numbers

Mathematicians during the ancient times, particularly the Greeks, always make  use of geometric interpretations of numbers. Square numbers, for example, are represented with pebbles arranged with the same number of rows and columns.  The first five square numbers are 1, 4, 9, 16 and 25.

Rectangular numbers are also popular. For instance, the number 12 can be interpreted pebbles arranged in rectangles with the following dimensions: 3 by 4, 2 by 6 and 12 by 1. If we are going to use squares instead of pebbles, the geometric representations of these arrangements are shown in Figure 1.

Figure 1 – Different rectangular arrangements of 12 pebbles represented by squares.

Numbers that cannot be arranged as more than one rectangle are prime numbers. In our example above, 12 has three possible arrangements, while the numbers 3, 5 and 7 can only be arranged in a single row.

Figure 2 – Rectangular arrangements of 3, 5 and 7 pebbles represented by squares.

As we can see, this is the geometric interpretation of the definition that the factors of primes are only 1 and itself.

Sieve of Eratosthenes

The Greek philosopher and mathematician Eratosthenes was the first to be credited in identifying primes in a finite list by brute force.  The strategy is to list a finite set of counting numbers in increasing order, then starting with 2 eliminate all its multiples. This eliminates all even numbers except 2. Then we follow the same pattern: we eliminate multiples of 3 greater than 3, eliminate all multiples of 5 greater than 5 and so on, until all the numbers left are not multiples of any number smaller than it. The remaining numbers after all elimination are prime numbers.

The primes numbers less than 100 are shown in white cells in the table below. The numbers in the yellow cells are composite numbers.  Mathematicians agreed not to include 1 in the set of prime numbers or the set of composite numbers.

Figure 3 - The sieve of Eratosthenes shows primes less than 100 in white table cells..

If we look at the table above, we could not probably see a pattern about the number of primes in a given interval; however, if we investigate further, as the intervals increase, the number of primes is getting fewer and fewer. For example, there are 168 primes between 1 and 1000, 135 primes between 1000 and 2000, 127 primes between 2000 and 3000, and 120 primes between 3000 and 4000. With this observation, we want to ask the following question:

Is there a particular prime number, that after such number, we could no longer find primes?

or equivalently,

Are prime numbers finite?

We will answer this in the continuation of this article titled  “Infinitude of Primes.” The formal proof of this conjecture is also discussed in the third part.

## Is 0.999… really equal to 1?

Introduction

Yes it is. 0.999…  is equal to 1.

Before we begin our discussion, let me make a remark that the symbol “…” in the decimal 0.999… means that the there are infinitely many 9’s,  or putting it in plain language, the decimal number has no end.

For non-math persons, you will probably disagree with the equality, but there are many elementary proofs that could show it, some of which, I have shown below. A proof is a series of valid, logical and relevant arguments (see Introduction to Mathematical Proofs for details), that shows the truth or falsity of a statement.

Proof 1

$\frac{1}{3} = 0.333 \cdots$

$\frac{2}{3} = 0.666 \cdots$

$\frac{1}{3} + \frac{2}{3} = 0.333 \cdots + 0.666 \cdots$

$\frac{3}{3} =0.999 \cdots$

But $\frac{3}{3} = 1$, therefore $1 =0.999 \cdots$

Proof 2

$\frac{1}{9} = 0.111 \cdots$
Multiplying both sides by 9 we have

$1 = 0.999 \cdots$

Proof 3

Let $x = 0.999 \cdots$

$10x = 9.999 \cdots$

$10x - x = 9.999 \cdots - 0.9999 \cdots$

$9x = 9$

$x = 1$

Hence, $0.999 \cdots = 1$

Still in doubt?

Many will probably be reluctant in accepting the equality $1 = 0.999 \cdots$ because the representation is a bit counterintuitive.  The said equality requires the notion of the real number system, a good grasp of the concept of limits, and knowledge on infinitesimals or calculus in general.  If, for instance,you have already taken sequences (in calculus), you may think of the $0.999 \cdots$ as a sequence of real numbers $(0.9, 0.99, 0.999,\cdots)$. Note that the sequence gets closer and closer to 1, and therefore, its limit is 1.

Infinite Geometric Sequence

My final attempt to convince you that $0.999 \cdots$ is indeed equal to $1$ is by the infinite geometric sequence. For the sake of brevity, in the remaining part of this article, we will simply use the term “infinite sequence” to refer to an infinite geometric sequence.  We will use the concept of the sum of an infinite sequence, which is known as an infinite series, to show that $0.999 \cdots = 1$.

One example of an infinite series is $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$.  If you add its  infinite number of terms, the answer is equal to 1. Again, this is counterintuitive.

How can addition of numbers with infinite number of terms have an exact (or a finite) answer?

There is a formula to get the sum of an infinite geometric sequence, but before we discuss the formula, let me give the geometric interpretation of the sum above. The sum $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$ can be represented geometrically using a 1 unit by 1 unit square as shown below. If we divide the square into two, then we will have two rectangles, each of which has area $\frac{1}{2}$ square units. Dividing the other half into two, then we have three rectangles with areas $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{4}$ square units. Dividing the one of the smaller rectangle into two, then we have four rectangles with areas $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{8}$. Again, dividing one of the smallest rectangle into two, we have five rectangles with areas $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{16}$, and $\frac{1}{16}$ Since this process can go on forever, the sum of all the areas of all the rectangles will equal to 1, which is the area of the original square.

Now that we have seen that an infinite series can have a finite sum, we will now show that $0.999 \cdots$ can be expressed as a finite sum by expressing it as an infinite series. The number $0.999 \cdots$ can be expressed as an infinite series $0.9 + 0.09 + 0.009 + \cdots$. Converting it in fractional form, we have  $\frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots$.

We have learned that the sum of the infinite series with first term $\displaystyle a_1$ and ratio $r$ is described by $\displaystyle\frac{a_1}{1-r}$. Applying the formula to our series above, we have

$\displaystyle\frac{\frac{9}{10}}{1-\frac{1}{10}} = 1$

Therefore, the sum our infinite series is 1.

Implication

This implication of the equality $0.999 \cdots =1$ means that any rational number that is a non-repeating decimal can be expressed as a repeating decimal. Since $0.999 \cdots =1$, it follows that $0.0999 \cdots =0.1, 0.00999 \cdots=0.01$ and so on. Hence, any decimal number maybe expressed as number + 0.00…01. For example, the decimal $4.7$, can be expressed as $4.6 + 0.1 = 4.6 + 0.0999 \cdots = 4.6999 \cdots$. The number $0.874$ can also be expressed as $0.873 + 0.001 = 0.873 + 0.000999 \cdots = 0.873999 \cdots$

Conclusion

Any of the four proofs above is actually sufficient to show that $0.999 \cdots = 1$.  Although this concept is quite hard to accept, we should remember that in mathematics, as long as the steps of operations or reasoning performed are valid and logical, the conclusion will be unquestionably valid.

There are many counterintuitive concepts in mathematics and the equality $0.999 \cdots = 1$ is only one of the many.  In my post, Counting the Uncountable: A Glimpse at the Infinite, we have also encountered one:   that the number of integers (negative, 0, positive) is equal to the number of counting numbers (positive integers) and we have shown it by one-to-one pairing. We have also shown that the number of counting numbers is the same as the number of rational numbers. Thus, we have shown that a subset can have the same element as the “supposed” bigger set.  I guess that is what makes mathematics unique; intuitively, some concepts do not make sense, but by valid and logical reasoning, they perfectly do.

Notes:

1. You can find discussions about 0.999… = 1 here and here.
2. There is another good post about it here and here.
Related Articles
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# Alg1.4 Functions

### Lesson 1

• I can explain when a relationship between two quantities is a function.
• I can identify independent and dependent variables in a function, and use words and graphs to represent the function.
• I can make sense of descriptions and graphs of functions and explain what they tell us about situations.

### Lesson 2

• I can use function notation to express functions that have specific inputs and outputs.
• I understand what function notation is and why it exists.
• When given a statement written in function notation, I can explain what it means in terms of a situation.

### Lesson 3

• I can describe the connections between a statement in function notation and the graph of the function.
• I can use function notation to efficiently represent a relationship between two quantities in a situation.
• I can use statements in function notation to sketch a graph of a function.

### Lesson 4

• I can make sense of rules of functions when they are written in function notation, and create tables and graphs to represent the functions.
• I can write equations that represent the rules of functions.

### Lesson 5

• I can use technology to graph a function given in function notation, and use the graph to find the values of the function.
• I know different ways to find the value of a function and to solve equations written in function notation.
• I know what makes a function a linear function.

### Lesson 6

• I can identify important features of graphs of functions and explain what they mean in the situations represented.
• I understand and can use the terms “horizontal intercept,” “vertical intercept,” “maximum,” and “minimum” when talking about functions and their graphs.

### Lesson 7

• I understand the meaning of the term “average rate of change.”
• When given a graph of a function, I can estimate or calculate the average rate of change between two points.

### Lesson 8

• I can explain the average rate of change of a function in terms of a situation.
• I can make sense of important features of a graph and explain what they mean in a situation.
• When given a description or a visual representation of a situation, I can sketch a graph that shows important features of the situation.

### Lesson 9

• I can compare the features of graphs of functions and explain what they mean in the situations represented.
• I can make sense of an equation of the form $f(x)=g(x)$ in terms of a situation and a graph, and know how to find the solutions.
• I can make sense of statements about two or more functions when they are written in function notation.

### Lesson 10

• I know what is meant by the “domain” and “range” of a function.
• When given a description of a function in a situation, I can determine reasonable domain and range for the function.

### Lesson 11

• When given a description of a function in a situation, I can determine reasonable domain and range for the function.

### Lesson 12

• I can make sense of a graph of a piecewise function in terms of a situation, and sketch a graph of the function when the rules are given.
• I can make sense of the rules of a piecewise function when they are written in function notation and explain what they mean in the situation represented.
• I understand what makes a function a piecewise function.

### Lesson 13

• Given a set of numerical guesses and a target number, I can calculate absolute errors and create a scatter plot of the data.
• I can analyze and describe features of a scatter plot that shows absolute error data.
• I can describe the general relationship between guesses and absolute errors using words or equations.

### Lesson 14

• I can describe the effects of adding a number to the expression that defines an absolute value function.
• I can explain the meaning of absolute value function in terms of distance.
• When given an absolute value function in words or in function notation, I can make sense of it, and can create a table of values and a graph to represent it.

### Lesson 15

• I understand the meaning of “inverse function” and how it could be found.
• When given a linear function that represents a situation, I can use words and equations to describe the inverse function.

### Lesson 16

• I can explain the meaning of an inverse function in terms of a situation.
• When I have an equation that defines a linear function, I know how to find its inverse.

### Lesson 17

• I can write a linear function to model given data and find the inverse of the function.
• When given a linear function defined using function notation, I know how to find its inverse.

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# Model-Free Prediction

## Incremental Implementation

Let $$Q_n$$ denote the estimate of its action value after it has been selected n − 1 times,易得：
$$Q_{n+1}=\frac{1}{n}(R_n+\sum_{i=1}^{n-1}R_i)=Q_n+\frac{1}{n}[R_n-Q_n]$$。就是相当于把最后新来的一项平均分成n份和原来的相补。This implementation requires memory only for $$Q_n$$ and $$n$$, and only the small computation for each new reward.
The general form is

The expression [Target − OldEstimate] is an error in the estimate.The target is presumed to indicate a desirable direction in which to move, though it may be noisy. In the case above, for example, the target is the nth reward.

## Monte Carlo Methods

Monte Carlo methods require only experience——sample sequences of states, actions, and rewards from actual or simulated interaction with an environment.
Monte Carlo methods only for episodic tasks. we assume experience is divided into episodes, and that all episodes eventually terminate no matter what actions are selected. Only on the completion of an episode are value estimates and policies changed.

### Monte Carlo Prediction

An obvious way to estimate it from experience, then, is simply to average the returns observed after visits to that state. As more returns are observed, the average should converge to the expected value.
Each occurrence of state s in an episode is called a visit to s.

first-visit MC伪代码：

Both first-visit MC and every-visit MC converge to $$v_\pi(s)$$ as the number of visits (or first visits) to s goes to infinity. By the law of large numbers the sequence of averages of these estimates converges to their expected value.

Whereas the DP diagram includes only one-step transitions, the Monte Carlo diagram goes all the way to the end of the episode.
An important fact about Monte Carlo methods is that the estimates for each state are independent. The estimate for one state does not build upon the estimate of any other state, as is the case in DP. In other words, Monte Carlo methods do not bootstrap.
The computational expense of estimating the value of a single state is independent of the number of states. One can generate many sample episodes starting from the states of interest, averaging returns from only these states, ignoring all others.

## Temporal-Difference Learning

TD learning is a combination of Monte Carlo ideas and dynamic programming (DP) ideas.
TD methods update estimates based in part on other learned estimates, without waiting for a final outcome (they bootstrap)
constant-α MC:$$V(S_t)=V(S_t)+\alpha[G_t-V(S_t)]$$，$$G_t$$ is the actual return following time $$t$$, and $$alpha$$ is a constant step-size parameter.
TD methods need to wait only until the next time step. At time $$t + 1$$ they immediately form a target and make a useful update using the observed reward $$R_{t+1}$$ and the estimate $$V(S_t+1)$$. The simplest TD method makes the update：$$V(S_t)=V(S_t)+\alpha[R_{t+1}+\gamma V(S_{t+1})-V(S_t)]$$.

Sample updates differ from the expected updates of DP methods in that they are based on a single sample successor rather than on a complete distribution of all possible successors.
TD error：$$\delta_t=R_{t+1}+\gamma V(S_{t+s})-V(S_t)$$.
if the array V does not change during the episode (as it does not in Monte Carlo methods), then the Monte Carlo error can be written as a sum of TD errors:

### Advantages of TD Prediction Methods

TD methods have an advantage over DP methods in that they do not require a model of the environment, of its reward and next-state probability distributions.
The next most obvious advantage of TD methods over Monte Carlo methods is that they are naturally implemented in an on-line, fully incremental fashion.TD methods one need wait only one time step.
TD methods are much less susceptible to these problems because they learn from each transition regardless of what subsequent actions are taken.（这一部分的的对比没有看太懂，可能是和control之类的比）
For any fixed policy $$\pi$$, TD(0) has been proved to converge to vπ, in the mean for a constant step-size parameter if it is sufficiently small.
MC和TD这两种方法很难说谁先收敛到真正的解最快，但是一般：TD methods have usually been found to converge faster than constant-α MC methods on stochastic tasks.

## n-step Bootstrapping

The backup diagrams of n-step methods:

The methods that use n-step updates are still TD methods because they still change an earlier estimate based on how it differs from a later estimate.
The natural state-value learning algorithm for using n-step returns is thus,$$V_{t+n}(S_t)=V_{t+n-1}(S_t)+\alpha[G_{t:t+n}-V_{t+n-1}(S_t)]$$.Note that no changes at all are made during the first n−1 steps of each episode.

error reduction property：

Because of the error reduction property, one can show formally that all n-step TD methods converge to
the correct predictions under appropriate technical conditions.

## Eligibility Traces

The mechanism is a short-term memory vector, the eligibility trace $$Z_t \in R^d$$, that parallels the long-term weight vector $$W_t \in R^d$$. The rough idea is that when a component of $$W_t$$ participates in producing an estimated value, then the corresponding component of $$Z_t$$ is bumped up and then begins to fade away.
Forward views are always somewhat complex to implement because the update depends on later things that are not available at the time.MC以及n-step 的方法都属于这种。


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# Static gauge for strings in curved background

1. Aug 3, 2011

### haushofer

Hi,

I have a question about the static gauge in string theory, in which one sets

$$\tau = X^0$$

I understand that in the usual approach for strings in a flat target space, after the gauge fixing of the worldsheet metric gamma,

$$\gamma_{\alpha\beta} = \eta_{\alpha\beta}$$

one still has enough gauge symmetry left to choose the static gauge. See e.g. (2.3.7) of Green, Schwarz, Witten. My question is the following.

If one considers classical strings in curved target spaces, the wave equation for the embedding coordinates becomes

$$\gamma^{\alpha\beta}(\partial_{\alpha} \partial_{\beta} X^{\rho} + \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} \Gamma^{\rho}_{\mu\nu}) = 0$$

Here alpha,beta,... are 0,1, while mu,nu,rho run from 0,...,D-1. First, one can choose the conformal gauge,

$$\eta^{\alpha\beta}(\partial_{\alpha} \partial_{\beta} X^{\rho} + \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} \Gamma^{\rho}_{\mu\nu}) = 0$$

My question is: is one still able to choose after this conformal gauge for the worldsheet metric the static gauge for tau? The question arises, because the remaining gauge symmetries for tau and sigma show that the transformed tau obeys a free wave equation, see (2.3.7) of Green, Schwarz, Witten. However, X^0 obeys the equation above, which is not a free wave equation.

Any suggestions? :)

2. Aug 8, 2011

### haushofer

For general geometries, one apparently cannot choose this static gauge anymore (only in a point) because of the inhomogenous term in the geodesic equation. For the same reason the lightcone gauge cannot be chosen, so it seems to me that the usual argument that only the transverse coordinates $X^I$, where I=2,3,...,D-1 are "true degrees of freedom" might break down, although this seems a bit counterintuitive to me.

So my question is: does a classical string in a curved spacetime only have transverse degrees of freedom, as in the flat case?

Last edited: Aug 8, 2011
3. Aug 9, 2011

### haushofer

Nobody here knows of a general statement that one can always (or: for arbitrary backgrounds) gauge away the longitudinal coordinates such that the transverse coordinates $X^I$ stay as "true degrees of freedom"?

Is this just a silly question, or is it beyond textbook?

4. Aug 25, 2011


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# U=qv or u=0.5qv

1. Mar 8, 2010

### sonutulsiani

1. The problem statement, all variables and given/known data

Ok I am really confused!
Sometimes we use u=qv to find the potential energy
and sometimes u=0.5qv
Which one is correct?

2. Relevant equations

u= electro static potential energy
v= voltage
q is charge

3. The attempt at a solution

2. Mar 9, 2010

### nickjer

That question is very vague. You will have to be more specific of the problem.

U=qV is used for a charge (q) in an electrostatic potential (V). "U" is then the electric potential energy the charge has in that field.

I am not sure about U= 0.5qV, since that is very vague. If you can give an example where it is used, then you might be able to narrow it down.

3. Mar 9, 2010

### sonutulsiani

like for energy in a capacitor its used

4. Mar 9, 2010

### nickjer

The energy stored in a capacitor is very different than a single point charge in an electric field. The energy stored in a capacitor is actually an integral over many infinitesmal point charges, dq. Since it is a sum, you can't expect it to look like qV.

5. Mar 9, 2010

### sonutulsiani

Apparently my textbook writes potential energy of 2 point charges is u=qv
and electrostatic potential energy is u=1/2qv. What is the difference between them other than the word 'electrostatic'

6. Mar 10, 2010

### nickjer

Electrostatic just means the electric field isn't changing with time. So you can use it interchangeably with electric for most of your cases. You can say the electrostatic potential energy of 2 point charges is U = qV, since there is no time dependence that would cause magnetic fields.

For your U = 1/2 qV, you must be reading about capacitance and the stored energy in the electric field of the charges. They are both different, and shouldn't get them too confused from a question you might be asked.

7. May 14, 2011

### Alihammoud

u = q v is for electrical potential energy
for those who gave the example of capacitors:
u = 0.5 c v^2

8. May 14, 2011

### lepton5

I will try to sharp two eqn above based on common textbook,

1) $$W = q . [V(a) - V(b)]$$

this eqn tells us for the work done to move a charge from point (a) to point (b). V here means potential difference between points (a) and (b).

2) $$W = \frac{1}{2}\Sigma^{n}_{i=1} q_i V(r_i)$$

this eqn tells about work it takes to assemble a configuration of point charge. so W here represents energy stored in the configuration. factor 1/2 here is used to restore the eqn that initially count twice. (if you confused what I mean, you can check for yourself the derivation of this eqn). this eqn is more about the energy of a point charge distribution.

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Coronal hard X-ray sources revisited

This paper reports on the re-analysis of solar flares in which the hard X-rays (HXRs) come predominantly from the corona rather than from the more usual chromospheric footpoints. All of the 26 previously analyzed event time intervals, over 13 flares, are re-examined for consistency with a flare model in which electrons are accelerated near the top of a magnetic loop that has a sufficiently high density to stop most of the electrons by Coulomb collisions before they can reach the footpoints. Of particular importance in the previous analysis was the finding that the length of the coronal HXR source increased with energy in the 20 – 30 keV range. However, after allowing for the possibility that footpoint emission at the higher energies affects the inferred length of the coronal HXR source, and using analysis techniques that suppress the possible influence of such footpoint emission, we conclude that there is no longer evidence that the length of the HXR coronal sources increase with increasing energy. In fact, for the 6 flares and 12 time intervals that satisfied our selection criteria, the loop lengths decreased on average by 1.0 +/- 0.2 arcsec between 20 and 30 keV, with a standard deviation of 3.5 arcsec. We find strong evidence that the peak of the coronal HXR source increases in altitude with increasing energy. For the thermal component of the emission, this is consistent with the standard CHSKP flare model in which magnetic reconnection in a coronal current sheet results in new hot loops being formed at progressively higher altitudes. The explanation for the nonthermal emission is not so clear.

Original Article: http://arxiv.org/abs/1809.04631

The coronal volume of energetic particles in solar flares as revealed by microwave imaging

The spectrum of gyrosynchrotron emission from solar flares generally peaks in the microwave range. Its optically-thin, high-frequency component, above the spectral peak, is often used for diagnostics of the nonthermal electrons and the magnetic field in the radio source. Under favorable conditions, its low-frequency counterpart brings additional, complementary information about these parameters as well as thermal plasma diagnostics, either through gyrosynchrotron self-absorption, free-free absorption by the thermal plasma, or the suppression of emission through the so-called Razin effect. However, their effects on the low-frequency spectrum are often masked by spatial nonuniformity. To disentangle the various contributions to low-frequency gyrosynchrotron emission, a combination of spectral and imaging data is needed. To this end, we have investigated Owens Valley Solar Array (OVSA) multi-frequency images for 26 solar bursts observed jointly with Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) during the first half of 2002. For each, we examined dynamic spectra, time- and frequency-synthesis maps, RHESSI images with overlaid OVSA contours, and a few representative single-frequency snapshot OVSA images. We focus on the frequency dependence of microwave source sizes derived from the OVSA images and their effect on the low-frequency microwave spectral slope. We succeed in categorizing 18 analyzed events into several groups. Four events demonstrate clear evidence of being dominated by gyrosynchrotron self-absorption, with an inferred brightness temperature of $\geq10^8$~K. The low-frequency spectra in the remaining events are affected to varying degree by Razin suppression. We find that many radio sources are rather large at low frequencies, which can have important implications for solar energetic particle production and escape.

Original Article: http://arxiv.org/abs/1809.04753

Evolution of the solar Ly-$\alpha$ line profile during the solar cycle. II. How accurate is the present radiation pressure paradigm for interstellar neutral H in the heliosphere?

Following the derivation of a more accurate model of the evolution of the solar Lyman-$\alpha$ line with the changing solar activity by Kowalska-leszczynska et al. 2018 (IKL18) than the formerly used model by Tarnopolski et al. 2009 (ST09), we investigate potential consequences that adoption of the resulting refined model of radiation pressure has for the model distribution of interstellar neutral (ISN) H in the inner heliosphere and on the interpretation of selected observations. We simulated the ISN H densities using the two alternative radiation pressure models and identical models of all other factors affecting the ISN H distribution. We found that during most of the solar cycle, the IKL18 model predicts larger densities of ISN H and PUIs than ST09 in the inner heliosphere, especially in the downwind hemisphere. However, the density of ISN H at the termination shock estimated by Bzowski et al. 2008 obtained using ST09 does not need revision, and the detection of ISN D by IBEX is supported. However, we point out the existence of a considerable absorption of a portion of the solar Lyman-$\alpha$ spectral flux inside the heliosphere. Therefore, the model of radiation pressure for ISN H is still likely needs revision, and hence the available models of ISN H are not self-consistent.

Original Article: http://arxiv.org/abs/1809.04929

Casting the Coronal Magnetic Field Reconstruction Tools in 3D Using MHD Bifrost Model

Quantifying coronal magnetic field remains a central problem in solar physics. Nowadays the coronal magnetic field is often modelled using nonlinear force-free field (NLFFF) reconstructions, whose accuracy has not yet been comprehensively assessed. Here we perform a detailed casting of the NLFFF reconstruction tools, such as pi-disambiguation, photospheric field preprocessing, and volume reconstruction methods using a 3D snapshot of the publicly available full-fledged radiative MHD model. Specifically, from the MHD model we know the magnetic field vector in the entire 3D domain, which enables us to perform "voxel-by-voxel" comparison of the restored and the true magnetic field in the 3D model volume. Our tests show that the available pi-disambiguation methods often fail at the quiet sun areas dominated by small-scale magnetic elements, while they work well at the AR photosphere and (even better) chromosphere. The preprocessing of the photospheric magnetic field, although does produce a more force-free boundary condition, also results in some effective elevation’ of the magnetic field components. This elevation’ height is different for the longitudinal and transverse components, which results in a systematic error in absolute heights in the reconstructed magnetic data cube. The extrapolations performed starting from actual AR photospheric magnetogram are free from this systematic error, while have other metrics comparable with those for extrapolations from the preprocessed magnetograms. This finding favors the use of extrapolations from the original photospheric magnetogram without preprocessing. Our tests further suggest that extrapolations from a force-free chromospheric boundary produce measurably better results, than those from the photospheric boundary.

Original Article: http://arxiv.org/abs/1703.06360

The dynamic chromosphere: pushing the boundaries of observations and models

Tiago M. D. Pereira

The interface between the bright solar surface and the million-degree corona continues to hold the key to many unsolved problems in solar physics. Advances in instrumentation now allow us to observe the dynamic structures of the solar chromosphere down to less than 0.1" with cadences of just a few seconds and in multiple polarisation states. Such observational progress has been matched by the ever-increasing sophistication of numerical models, which have become necessary to interpret the complex observations. With an emphasis on the quiet Sun, I will review recent progress in the observation and modelling of the chromosphere. Models have come a long way from 1D static atmospheres, but their predictions still fail to reproduce several key observed features. Nevertheless, they have given us invaluable insight into the physical processes that energise the atmosphere. With more physics being added to models, the gap between predictions and observations is narrowing. With the next generation of solar observatories just around the corner, the big question is: will they close the gap?

Original Article: http://arxiv.org/abs/1809.04077

Disparity among low first ionization potential elements

The elemental composition of the solar wind differs from the solar photospheric composition. Elements with low first ionization potential (FIP) appear enhanced compared to O in the solar wind relative to the respective photospheric abundances. This so-called FIP effect is different in the slow solar wind and the coronal hole wind. However, under the same plasma conditions, for elements with similar FIPs such as Mg, Si, and Fe, comparable enhancements are expected. We scrutinize the assumption that the FIP effect is always similar for different low FIP elements, namely Mg, Si, and Fe. We investigate the dependency of the FIP effect of low FIP elements on the O7+/O6+ charge state ratio depending on time and solar wind type. We order the observed FIP ratios with respect to the O7+/O6+ charge state ratio into bins and analyze separately the respective distributions of the FIP ratio of Mg, Si, and Fe for each O7+/O6+ charge state ratio bin. We observe that the FIP effect shows the same qualitative yearly behavior for Mg and Si, while Fe shows significant differences during the solar activity maximum and its declining phase. In each year, the FIP effect for Mg and Si always increases with increasing O7+/O6+ charge state ratio, but for high O7+/O6+ charge state ratios the FIP effect for Fe shows a qualitatively different behavior. During the years 2001-2006, instead of increasing with the O7+/O6+ charge state ratio, the Fe FIP ratio exhibits a broad peak. Also, the FIP distribution per O7+/O6+ charge state bin is significantly broader for Fe than for Mg and Si. These observations support the conclusion that the elemental fractionation is only partly determined by FIP. In particular, the qualitative difference behavior with increasing O7+/O6+ charge state ratio between Fe on the one hand and Mg and Si on the other hand is not yet well explained by models of fractionation.

Original Article: http://arxiv.org/abs/1809.04345


text:
## PoofyPenguin Group Title Could someone please help me? Attachments to come... one year ago one year ago

1. PoofyPenguin Group Title

Here is the question:

2. PoofyPenguin Group Title

Here's what i got so far:

3. PoofyPenguin Group Title

I'm not sure if i did it right and i'm not sure where it's going?

4. PoofyPenguin Group Title

alright... i think i get it, but i don't get how or why the x becomes and e?

5. technopanda13 Group Title

fancy writing

6. PoofyPenguin Group Title

@technopanda13 thank you!

7. zepdrix Group Title

Haha I was thinking the same thing :D I love that multiply lil astrix XD

8. zepdrix Group Title

Hmm your steps look really good. I don't see any mistakes. For your final answer I would rewrite the ugly exponential term back as x^(4x^4 + 2), The one in front.

9. zepdrix Group Title

Oh need an explanation on the exponentiation + log thing?

10. baldymcgee6 Group Title

Yeah, it looks good to me too. But where did the e and the natural log all the sudden appear from?

11. PoofyPenguin Group Title

it was part of the hint...but i'm not sure why it works that way...

12. baldymcgee6 Group Title

13. PoofyPenguin Group Title

@baldymcgee6 have a look at the attachment! it's the one labeld problem 10

14. Fazeelayaz Group Title

X^(x2+2) is same as e^(4x2+2)lnx u can use calculator to verify

15. PoofyPenguin Group Title

@Fazeelayaz alright, but how would i go about simplifying it?

16. zepdrix Group Title

|dw:1351311147700:dw|

17. baldymcgee6 Group Title

ohh. sorry, I only saw you're answer :/

18. PoofyPenguin Group Title

@zepdrix i'm glad you like my astrix ;)

19. Fazeelayaz Group Title

simplyfying the answer or u want to know how they r equal

20. PoofyPenguin Group Title

@zepdrix great explanation! I understand it now! But how to simplify?

21. baldymcgee6 Group Title

@PoofyPenguin the reason they give you that hint is to avoid logarithmic differentiation. But for future derivatives, I would recommend learning log diffs. This is just a loophole around it! But good work, it looks good to me.

22. zepdrix Group Title

|dw:1351311341113:dw|

23. zepdrix Group Title

Ah yes logarithmic differentiation is a good skill to have c:

24. zepdrix Group Title

What you have written for f'(x) is perfect poof, you just want to rewrite the first part without the e and natural log, undo your FIRST step that you did to it.

25. Fazeelayaz Group Title

o i understand there is another way to solve take ln on both sides of ur question the equation becomes $\ln f(x)=(4x{4}+2)lnx$

26. Fazeelayaz Group Title

now differentiate both sides by x

27. technopanda13 Group Title

__━━____┓━╭━━━━━╮ ____━━____┗┓|::::::::::^━━^ ____━━____━┗|:::::::: |｡◕‿‿◕｡| ____━━____━━╰O--O----O-O-╯ here is a cat for for all you hard workers

28. zepdrix Group Title

hah :D

29. PoofyPenguin Group Title

alright! so would this be the final answer or can it be simplified more?

30. Fazeelayaz Group Title

$1/f(x)*(df(x)/dx)=...$use product rule on left side

31. PoofyPenguin Group Title

@technopanda13 cute kitty! ;P

32. zepdrix Group Title

You could get a common denominator on the inside if you wanted to, but it totally unnecessary c: Yah it looks good there.

33. Fazeelayaz Group Title

yes u have done it that's right

34. PoofyPenguin Group Title

ok! let me type it into the computer and get meh marks! :D

35. PoofyPenguin Group Title

Yayayayayayayay! I got it! Thanks so much everyone! All of you were so helpful! :D

36. zepdrix Group Title

yay team \:D/


text:
## JEE advanced important topics : derivative and growth of functions

JEE advanced important topics : The derivative of a function at a point indicates the rhythm of change of the function in said point in space. An immediate consequence is that the derivative will allow us to know if a function is increasing or decreasing in a determinate point, with only calculating the result of the derivative and looking if it’s greater or smaller than zero. If the derivative is positive (greater than zero), the variation of the function is positive, which means it’s growing in the point that we considered for the calculation. On the contrary, if the derivative is negative (smaller than zero), the variation in the function is negative, which means the function is decreasing in the considered point.

If we know the derivative function, the problem in determining the intervals in which the function increases and decreases is reduced to the study of the sign of it’s derivative function and nothing more. The intervals in which the derivative function is positive correspond to the intervals in which the primitive function is increasing. The intervals in which the derivative function is negative correspond to the intervals in which the primitive function is decreasing.

When the derivative in a point is zero, the tangent in the function in said point is horizontal. But the tangent can be horizontal because of several reasons, that is why interpreting the null derivative tends to be more complex than when it is positive or negative. So, it may happen that you find a relative maximum or a relative minimum, or it can be an inflection point in the horizontal tangent or it simply is a point in which the function is constant.

The cutting points in the derivative function with the OX axis, meaning, the points where the derivative function is equal to zero tell us where the primitive function has points with horizontal tangent and also help to determine the increasing and decreasing intervals of the function.

If we derive the derivative function we obtain the second derivative function. This will give us information about the variation of the last function, meaning, the first derivative function. Following the same reasoning we have made so far, the second derivative will allow us to know whether or not the first derivative is decreasing or increasing, which in turn allows to know the type of curvature in the primitive function, like in which intervals it’s convex or concave. In consequence, the second derivative will allow us to determine the intervals in which the function is concave or convex, and also the inflection points, which are the points in which the type of curvature of the function changes.

## Application: Increase and decrease

If f is a derivable function in a, which is one point in the dominion such as $f'(a) \neq 0$.

• f is strictly increasing in a if and only if f’(a)>0.
• f is strictly decreasing in a if and only if f’(a)<0.

### Example 1.

Study the increase and decrease of the function $f(x)=\left | x^{2}-1 \right |$

##### Resolution:

The function f can be express as:

$f(x) = \left\{ \begin{array}{ll} x^{2}-1 & x\leq -1 \\ 1-x^{2} & -1\leq x\leq 1 \\ x^{2}-1 & 1\leq x \\ \end{array} \right.$

The function is continuous for every real number and is not derivable in -1 and 1.

$f'(x) = \left\{ \begin{array}{ll} 2x & x\leq-1 \\ -2x & -1\leq x\leq 1 \\ 2x & 1\leq x \\ \end{array} \right.$

By studying the derivative, we can see that the points in which the derivative function can change from positive and negative (and so, the primitive function can go from increasing to decreasing) are – 1 , 0 and 1.

Note: The point 0 is calculated from the fact that only when x=0 then f’(x)=0, which means f’(x) in never zero for the interval ( – ∞, – 1) nor in the interval (1, ∞).

• In the interval (-∞, -1), you obtain that f’(x)<0, which means that in that interval the function decreases.
• In the interval (1, ∞), you obtain that f’(x)>0, which means that in that interval the function increases.

f’(x) is equal to zero in the interval (-1, 1), in the point 0, that is why we need to study both intervals, (-1, 0) and (0,1), separately.

• In the interval (-1, 0), you obtain that f’(x) = -2x >0, which means that in that interval the function increases.
• In the interval (0, 1), you obtain that f’(x)=-2x<0, which means that in that interval the function decreases.

To put it short, f increases in the intervals (-1,0) and (1, ∞) but f decreases in the intervals (-∞,-1) and (0,1). In the points -1, 0 and 1 the function remains constant, so it doesn’t increase nor it decreases.

### Example 2.

Find out in which intervals the function $f(x)=\frac{x^{2}-4x+1}{2}$ increases or decreases.

Resolution.

First, we need to calculate f’(x), f’(x)=(2x-4)/2 = x-2

As f’(x) > 0 ↔ x – 2>0, then x>2, so f is increasing for x>2.

As f’(x) < 0 ↔ x – 2<0, then x<2, so f is decreasing for x<2.

We can easily observe this in the graph of the function f shown here:

### Example 3.

Find out in which intervals the function $f(x)= x^{2}+\frac{1}{x^{2}}$ increases or decreases with x is not equal to zero.

Resolution.

$f'(x)= 2x-2\frac{1}{x^{3}}$, which can be written as: $f'(x)= \frac{2(x-1)(x+1)(x^{2}+1)}{x^{3}}$

As $2(x^{2}+1)$ is positive for all real numbers,

then: $latex f'(x)=\frac{(x-1)(x+1)}{x^{^{3}}} > 0 \Rightarrow x\in (-1,0)\cup (1,\infty)$
and
$f'(x)=\frac{(x-1)(x+1)}{x^{^{3}}} < 0 \Rightarrow x\in (-\infty,-1)\cup (0,1)$

Then: f’(x) > 0 if x ϵ [-1,0] U [1,+ ∞), which means that the function f increases in the interval [-1,0] U [1,+ ∞).

Also, f’(x)<0 if (-∞,-1] U [0,1] which means the function f decreases in the interval (-∞,-1] U [0,1].

The graphical representation of this function is shown in this diagram:

### Example 4.

Find out in which intervals the function f(x)=(x + 1)/(x – 1) increases or decreases with $x\neq 0$.

Resolution.

The derivative of f is $f'(x)=\frac{-2}{(x-1^{2})}$

As $(x-1)^{2}$ is bigger than zero for all the real numbers when $x\neq 1$, and also -2<0 then f’(x) < 0 for all the real numbers when $x\neq 1$. Here we have the graphical representation of this function as a visual aid:


text:
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Regular version of the site

## Baxter-Belavin R-matrices as non-abelian generalization of elliptic functions

A. Levin, Olshanetsky M., Zotov A.
It was shown in our previous paper that quantum ${\rm gl}_N$ $R$-matrices satisfy noncommutative analogues of the Fay identities in ${\rm gl}_N^{\otimes 3}$. In this paper we extend the list of $R$-matrix valued elliptic function identities. We propose counterparts of the Fay identities in ${\rm gl}_N^{\otimes 2}$, the symmetry between the Planck constant and the spectral parameter, quasi-periodicities with respect to these variables, the Kronecker double series representation of the R-matrix. As an application we construct $R$-matrix valued $2N^2\times 2N^2$ Lax pairs for the Painlev\'e VI equation (in the elliptic form) with four free constants using ${\mathbb Z}_N\times {\mathbb Z}_N$ elliptic $R$-matrix. More precisely, the four free constants case appears for an odd $N$ while even $N$'s correspond to a single constant.


text:
+0

# Bisectors in a Triangle

-1
58
1
+204

The triangle $\triangle ABC$ is an isosceles triangle where $AB = 4\sqrt{2}$ and $\angle B$ is a right angle. If $I$ is the incenter of $\triangle ABC,$ then what is $BI$?

Express your answer in the form $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by any perfect square other than $1.$

Mar 3, 2021

#1
+56
+2

https://web2.0calc.com/questions/i-need-some-help-thank-you

Same question

Search up your question online before asking for help on here, it probably has already been solved before

Mar 3, 2021


text:
# Limit of a sequence (TIFR 2013 problem 32)

Question:

True/False?

$$\lim_{n\to \infty } (n+1)^{1/3} -n^{1/3} = \infty$$

Hint:

Simplify the given expression.

Discussion:

We feel that $$(n+1)^{1/3}$$ goes to infinity at the same speed as $$n^{1/3}$$. So in fact, the above limit should be zero.

We make this little bit more rigorous.

$$(n+1)^{1/3} -n^{1/3} = \frac{n+1-n}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} }$$

$$=\frac{1}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3} } \to 0$$ as $$n\to \infty$$.


text:
# Boiling Point Elevation Discussion

Navigation: User Guide -> Boiling Point Elevation -> Boiling Point Elevation Discussion

If we plot the curves for both pure water (dashed) and a liquid containing impurities (such as salt water or Bayer Liquor), we find the latter has moved up at any point. The difference between these two curves is the Boiling Point Elevation (BPE) at that particular pressure.

Note that according to this definition, β is intrinsically a function of pressure. Formally

$\beta = T_L^{s}(p) - T_w^{s}(p)$

where:

$T_w^{s}(p)$ is the boiling point of water expressed as a function of pressure,
$T_{L}^{s}(p)$ is the boiling point for any liquor.

We have a number of possible ways of interpreting BPE:

## BPE at Stream Pressure

The BPE at stream pressure is difference between the temperature at which the liquor would boil and that which water would boil at, at the stream pressure.

## BPE at Stream Temperature

Care must be taken if the boiling point elevation is specified as a function of temperature - Does the correlation refer to the liquor boiling temperature or water boiling temperature?

See the graph below:

The BPE at stream temperature can be interpreted in two ways:

• We can choose the liquor vapour pressure at that temperature as the basis for saturation (point 1 in the above diagram); or
• We can choose the water vapour pressure at that temperature as the basis for saturation (point 2 in the above diagram).

Please see Boiling Point Elevation for a description of the Boiling Point Elevation equation used in the Standard Species model in SysCAD.


text:
# Problem solving in linear equation

By | September 9, 2020

To solve (4), apply euclid’s principle to obtain. solve the equation solve the linear equation $$\frac{3}{8}a=3$$. they are good for combining them with linear equations, since they introduce the concept of valid and invalid answers for an equation (what problem solving in linear equation will be later called the domain of problem solving in linear equation a function) then we overprescribed mental illness essay solve for n. use the addition, subtraction, multiplication, and division properties of equalities to solve linear equations when solving linear systems, you have two methods at your disposal, planning a business event and which one you woodlands junior homework help choose depends on online casino business plan the problem: for example, senior essay let us choose (0, 82) and (1, 80). the same rules apply. linear systems – word problems when what should a thesis avoid it comes to free research papers on information security using linear systems to solve word problems, the biggest problem is recognizing the important elements research paper writer and setting up the equations. ——————————–. rewrite the equations, if necessary, to make like variables line up underneath problem solving in linear equation each other. c(x) = 125x 1500 and r(x) = 1500x – 1000, canoes has a fixed cost of \$20,000. read the problem carefully and determine argumentative essay examples for high school pdf what is being asked. step 2 : a linear equation is simply an algebraic expression that represents a …. linear equations in word problems problem solving in linear equation bob has three sacks of apples and three more apples in his pocket.

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text:
# Find the limit algebraically lim ((x-1)/(x^2-1)) x ->

1. Aug 19, 2004

### Haftred

Find the limit algebraically lim ((x-1)/(x^2-1)) x -->

Find the limit algebreically:

lim ((x-1)/(x^2-1))
x --> 1

Thanks

2. Aug 19, 2004

Hello.

To find your limit we cannot just set x equal to 1. This would make the expression indeterminate, i.e. 0/0. Hence we have to use L ' Hopital's Rule.

lim f ' (x) / g ' (x) = 1 / 2x. Setting x = 1 we get 1/2 for the limit.
x-->1

If you want to solve it algebraically:

lim (x-1)/((x^2 - 1) = lim (x-1)/(x-1)(x+1) = lim 1 / (x+1) = 1/2
x-->1 x-->1 x-->1

Last edited: Aug 19, 2004
3. Aug 19, 2004

### Hurkyl

Staff Emeritus
That's not doing it algebraically. The questions wants you to do something algebraic with (x-1)/(x^2-1)

4. Aug 19, 2004

### Corneo

Algebra is the real key here. Factor the denominator

$$x^2-1=(x-1)(x+1)$$

Then we have

$$\lim\limits_{x \to 1} \frac {x-1}{x^2-1} = \lim\limits_{x \to 1} \frac {x-1}{(x-1)(x+1)} = \lim\limits_{x \to 1} \frac {1}{x+1} = \frac {1}{2}$$


text:
# Search for new phenomena with multiple charged leptons in proton–proton collisions at $\sqrt{s} = 13$ TeV

CMS Collaboration; Canelli, Florencia; Kilminster, Benjamin; Aarestad, Thea; Caminada, Lea; De Cosa, Annapaoloa; Del Burgo, Riccardo; Donato, Silvio; Galloni, Camilla; Hinzmann, Andreas; Hreus, Tomas; Ngadiuba, Jennifer; Pinna, Deborah; Rauco, Giorgia; Robmann, Peter; Salerno, Daniel; Schweiger, Korbinian; Seitz, Claudia; Takahashi, Yuta; Zucchetta, Alberto; et al (2017). Search for new phenomena with multiple charged leptons in proton–proton collisions at $\sqrt{s} = 13$ TeV. European Physical Journal C - Particles and Fields, C77(9):635.

## Abstract

Results are reported from a search for physics beyond the standard model in final states with at least three charged leptons, in any combination of electrons or muons. The data sample corresponds to an integrated luminosity of 2.3 $fb^{−1}$ of proton–proton collisions at $\sqrt{s} = 13$ TeV, recorded by the CMS experiment at the LHC in 2015. Two jets are required in each event, providing good sensitivity to strong production of gluinos and squarks. The search regions, sensitive to a range of different new physics scenarios, are defined using the number of jets tagged as originating from bottom quarks, the sum of the magnitudes of the transverse momenta of the jets, the imbalance in the overall transverse momentum in the event, and the invariant mass of opposite-sign, same-flavor lepton pairs. The event yields observed in data are consistent with the expected background contributions from standard model processes. These results are used to derive limits in terms of R-parity conserving simplified models of supersymmetry that describe strong production of gluinos and squarks. Model-independent limits are presented to facilitate the reinterpretation of the results in a broad range of scenarios for physics beyond the standard model.

## Abstract

Results are reported from a search for physics beyond the standard model in final states with at least three charged leptons, in any combination of electrons or muons. The data sample corresponds to an integrated luminosity of 2.3 $fb^{−1}$ of proton–proton collisions at $\sqrt{s} = 13$ TeV, recorded by the CMS experiment at the LHC in 2015. Two jets are required in each event, providing good sensitivity to strong production of gluinos and squarks. The search regions, sensitive to a range of different new physics scenarios, are defined using the number of jets tagged as originating from bottom quarks, the sum of the magnitudes of the transverse momenta of the jets, the imbalance in the overall transverse momentum in the event, and the invariant mass of opposite-sign, same-flavor lepton pairs. The event yields observed in data are consistent with the expected background contributions from standard model processes. These results are used to derive limits in terms of R-parity conserving simplified models of supersymmetry that describe strong production of gluinos and squarks. Model-independent limits are presented to facilitate the reinterpretation of the results in a broad range of scenarios for physics beyond the standard model.

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text:
## What problem was Ladd trying to solve?

Last weekend I was at an amazing conference on Feminist Philosophy and Formal Logic (which I’ve written about elsewhere), during which there were a number of papers on the history of women and logic. Frederique Janssen-Lauret gave a paper on women in early analytic philosophy, and spoke about Christine Ladd-Franklin’s contribution to logic.

Ladd (as she was at the time) studied at Johns Hopkins under Peirce but was not allowed to formally receive her doctorate. Her dissertation, “On the Algebra of Logic” was published 1883 (and is available from archive.org), and in it, she solved a problem in syllogistic which had purportedly baffled logicians since Aristotle:

The argument of inconsistency
$(a\bar{\vee}b)(b\bar{\vee}c)(c\vee a)\bar{\vee}$
is therefore the single form to which all the ninety-six valid syllogisms (both universal and particular can be reduced) (p.~40).

I knew a bit about Ladd and her work before Janssen-Lauret’s talk, but this mention of a solution to a long-standing syllogism puzzle piqued my interest immediately — not the least because I wasn’t at all sure what the problem was. As presented, it was a problem about reducing all forms of syllogism to one form, but the idea of “form” here was confusing: Syllogisms are typically spoken of as having mood and figure. Now, “form” here clearly can’t be “figure”, since it is already well-known, and due to Aristotle himself, that every non-first figure syllogism can be reduced to a first-figure syllogism, so the long-standing problem cannot about this sort of reduction, interpreting “form” as “figure”.

If what is meant is, however, “mood”, then the question whether it is possible to reduce all the valid moods to a single valid mood is certainly an interesting one, but I’m not sure that it’s one that has exercised logicians for two centuries — certainly I’d be hard pressed to find a medieval logician who was particularly worried about such a fine-grained reduction, most being content to reduce just to the four perfect syllogisms, Barbara, Celarent, Darii*, and Ferio. So even interpreted in this way, I’m stumped. (And interpreting “form” as something other than figure or mood makes it even more unlikely that the question is one that had bothered logicians for millenia.)

It’s not that I doubt that Ladd provided a solution to some technical problem in the syllogistic — not doubt at all about that — and I’m sure that a precise statement of her problem can be given. What intrigues me here is the social history of the problem: Not what it was but how/when/why did it get recognised as a problem, and how/when/why did people come to think of it as a problem that had plagued logicians since the time of Aristotle?

Looks like I’ve got my next paper topic lined up…

Footnote
* I very nearly wrote “Dario” here, showing that the distinction between names of syllogisms and names of Game of Thrones characters is not as great as you might think. (Come to think of it, “Bocardo” would be a good name for a roughly-and-tumble pirate warrior, donchya think?)

Posted in Uncategorized | | 2 Comments

## Not quite medieval, not quite logic: Proclus’s commentary on Euclid

Earlier this week I was following up on some notes I’d scribbled down over a month ago in a meeting with a colleague, one of which was “Proclus: Image in the water”.

A bit of googling later, and I found myself with a translation into English of Proclus’s commentary on the first book of Euclid’s Elements, along with, per the title page, “A History of the Restoration of Platonic Theology” by the later Platonists, and a translation of Proclus’s “Theological Elements”, courtesy of googlebooks. (Thanks, googlebooks!) Cool! On the face of it, it’s neither medieval nor logic, but it’s definitely something related to both and thus of interest to me.

Someone asked me on twitter who the translator was, and the answer turns out to be…I have no idea. The translator does not name himself on the title page, despite the fact that the publication info is given as “London, printed for the author” and I’m sure that the ‘author’ here isn’t Proclus!, and that there is an extensive preface, where the translator discusses the “great difficulty and labour” that attended his work, arising from the problematic state of his sources. (He laments the “great incorrectness” of the Greek edition and notes that he was substantially assisted by the translation into Latin by Francis Barocius the Venetian, done in 1560. There is otherwise no evidence as to the translator in the frontmatter, and there isn’t much out there elsewhere on google about 18th C translators of Proclus, so I thought I’d take a skim through the book to see if I could find any other clues.

Following the preface there is a “Dissertation on the Platonic Doctrine of Ideas” and after that a “Dissertation on the Demonstrative Syllogism” (!) which includes as part of it a substantial epistemological discussion. This is followed by a third discussion, “On the Nature of the Soul”, and a fourth on “The True End of Geometry”.

These four dissertations concluding, we next have a translation of Marinus’s “Life of Proclus, Or, Concerning Felicity”, and at this point I’m half-way through the PDF and am beginning to wonder just exactly when we’re going to get to the actual texts of Proclus. After Marinus’s biography, the translator gives us a bibliography of Proclus’s works, both available (whether fragmentary or not) and lost. (One of the lost books is a commentary on On Interpretation, which would’ve been really interesting!). Finally, nearly 2/3 of the way through the PDF, we get to Proclus’s commentaries, which comprise the rest of the volume. One can only hope that the next volume has the answer — I shall have to see if I can find it. Meanwhile, I might devote a future post to a discussion of the Dissertation on the Demonstrative Syllogism because it appears to be quite interesting.

Posted in Uncategorized | | 3 Comments

## Moody 2018: William of Ockham and his Milieu

From the 2nd to the 4th of March, the University of California Los Angeles will be hosting the yearly Moody Workshop in Medieval Philosophy. This meeting is named after Earnest Addison Moody, a former member of the Philosophy Department and a founding member of the UCLA Centre for Medieval and Renaissance Studies – which are jointly sponsoring the event.

Professor Moody was one of the the pioneers of Medieval Philosophy in North America and one of the first scholars to approach the study of medieval philosophical theories in their own right, without subordinating them to theological views. But for those who have even a passing interest in the history of logic, E. A. Moody is first and foremost the author of  Truth and Consequence in Medieval Logic (19531), one of the most influential volumes in the historiography of medieval logic and of medieval philosophy as a whole.

A Williams College graduate (class 1924), E. A. Moody obtained his PhD from Columbia in 1936, with a thesis on William of Auvergne’s Treatise De anima. While teaching at Columbia, he developed an interest in the history of medieval logic and science. Having retired for a few years to a ranch in Texas, in 1958 he joined the Department of Philosophy at UCLA; there he found a department not only already traditionally strong in logic and language but that was undergoing further changes toward that direction as well – in other words, the perfect fit for Moody’s own research interests.

Above all, those research interest left a significant imprint on the way we look at medieval philosophy in general and at the history of medieval logic in particular, especially in North America. Since the 19th century, the history of the study of medieval philosophy had been almost entirely shaped by quasi-nationalistic and quasi-theological concerns. Leo XIII’s Aeterni Patris encyclical (1879), by restoring “Christian Philosophy in Catholic Schools in the spirit of the Angelic Doctor, St. Thomas Aquinas”, was pivotal in determining a dominant picture of medieval philosophy that had both the kind of uniformity that scholars like Maurice De Wulf ascribed to terms like “scholasticism” and Aquinas as the central figure. True enough, this picture of medieval philosophy as fundamentally homogenous began to break down when Étienne Gilson and others started to realise that it couldn’t manage to account for roughly a thousand years of philosophical speculations. But even so, after Aeterni Patris typically there was still an idea that medieval philosophy had seemingly emerged out of the wreckage of the Roman Empire, grown slowly to a peak with Aquinas, and then began to decay – Gilson, for example, thought that the decay culminated in Descartes. Various people had slightly different pictures, but it was always as a curve and Aquinas was typically at its peak. Now, Moody had a different agenda and a different set of thoughts: he pursued his interests in medieval logic, language and science in connection to the contemporary discussions in logic and language – an whatever Aquinas was up to, an emphasis on logic and language wasn’t really that. This pushed Moody’s interests later, well into the 14th century, and had the consequence that the people who where interested in what Moody was interested in started to look more closely and seriously at the later middle ages, finding in 14th century philosophy a veritable goldmine, while it simultaneously became a common practice to develop those interests in the history of science, logic and language in connection with analogous contemporary endeavours. That ultimately set the tone for Medieval Philosophy on this side of the Pond. E. A. Moody remained at UCLA until his retirement in 1969 and in 1972 was succeeded by Marilyn McCord Adams. Even if McCord Adams later on became much more interested in philosophical theology, when she was at UCLA her major work was on Ockham – and it wasn’t really a theological work at all, rather it focused on Ockham’s logic. Overall, what Moody’s influence in North America amounted to was creating a climate in which Medieval Philosophy could be moved out of theology and toward philosophical issues relevant for contemporary discussions; from that kind of point of view, the 14th century was one of the most interesting things one could study, and in many ways, still is. Nobody else among the major figures who were doing medieval philosophy in Moody’s generation had anything like that on their agenda or achieved similar results.

The Moody Workshop was instituted by Calvin Normore, when he took over the mantle of Medieval Philosophy at UCLA in 1998. The first speaker in the first Moody Workshop was Martin Tweedale, Moody’s last student at UCLA. Since then a small group of scholars meets every year to discuss a particular topic in medieval philosophy, usually among those that would have met Moody’s interest. This year’s meeting, on William of Ockham and his Milieu, is dedicated to the memory of Marilyn McCord Adams.

Programme:

FRIDAY,MARCH 2ND
DODD HALL -ROOM 399

3:30 PM Peter King (Toronto)
Mental Without the Mind: Ockham’s Radical Revolution

SATURDAY,MARCH 3RD
ROYCE HALL -ROOM 306

10:00 AM Jenny Pelletier (Leuven)
What is Dominium? Ockham and the Ontology of Lordship

11:30 AM Mikko Yrjönsuuri (Jyväskylä)
Valid on Formal Ground: Burley, Ockham and Buridan

1:00 PM Lunch

2:00 PM Magali Roques (Helskinki)
Ockham’s Theory of Real Definitions

3:30 PM Graziana Ciola (UCLA)
Relativa grammaticalia and the Regimentation of Latin in 14th Century Logic

SUNDAY,MARCH 4TH
ROYCE HALL -ROOM 306

10:00 AM André Martin (McGill)
What Can the Ockham/Chatton Debate on Self‐Awareness Tell us About Medieval Accounts of Consciousness?

11:30 AM Josh Blander (The King’s College)
Does Ockham Have Priorities?

1:00 PM Lunch

2:00 PM Tom Ward (Baylor)
Ockham on Omnipotence, ‘LogicalImpossibility’, and Hating God

3:30 PM Christopher Martin (Auckland)
Suppose God did not Exist, or that the Eye Were an Animal: Impossible Positio and its use by Scotus, Ockham and Chatton

DISCUSSANTS WILL INCLUDE MILO CRIMI (UCLA) AND BOAZ SCHUMAN (UNIVERSITY OF TORONTO).

## Three-year lectureship in medieval philosophy, Charles University, Prague

Boosting the signal:

Job offer: Lecturer in philosophy
Department of Philosophy and Religious Studies
Faculty of Arts, Charles University

The Department of Philosophy and Religious Studies, Faculty of Arts, Charles University invites applications for one full-time three-year position at the lecturer level (starting at AP2 in the Czech scale), beginning October 1st 2018.

The area of specialization for the position will be medieval philosophy possibly with another area of specialisation.

#### Responsibilities for the position include:

• teaching philosophy courses at the undergraduate as well as graduate level: 7 courses (course = 90 min per week) in one academic year (academic year = two terms), i.e. 3 courses in one term, four courses in the other term
• supervising students
• student’s examination
• contributing to the curriculum development as appropriate
• establishing and maintaining independent research
• participating in the academic life of the department

#### Qualifications:

• PhD in philosophy or a related field
• teaching experience
• publication record
• working knowledge of Latin (the curriculum includes Latin reading classes)
• excellent command of English (or Czech) both in writing and speaking; knowledge of Czech is not required but working knowledge of it is expected after two years (in the case the contract should be prolonged); good knowledge of one other language (preferably German or French)

#### Conditions of employment

The Faculty of Arts offers a salary up to 30 000 CZK with possibility of further increase depending on the research outcomes of the employee. The Faculty covers social and medical insurance according to EU regulations. According to the university rules the contract can be prolonged after three years; the prolongation is either for another period of three years or tenure. The probationary period is three months.

Starting date: 1st October 2018

#### Application

You may apply for this position until 15th March 23:59h / before 16th March 2018 Prague local time. Submission by mail or in person: the application with all its attachments can be sent by mail or delivered in person to the following address: Hana Vamberská, Human Resources Dept., FF UK, nám. Jana Palacha 1/2, 116 38 Praha, Czech Republic

Submission by e-mail: the application with all its attachments can be sent by e-mail to the address hana.vamberska@ff.cuni.cz.

The application must include the following:

1. a cover letter containing your motivation for the position
2. cv and list of publications
3. the names and contact details of two referees and an indication whether we can contact them at this stage
4. a sample of a written work that is relevant to this position, not more than 10 000 words; it may be a published or unpublished sample and may be an extract from a longer piece
5. teaching evaluation (if available)
6. suggestion of a syllabus for an undergraduate course on medieval philosophy (two terms, 20-24 lectures (one lecture = 90 min); the aim of the course should be introduction to medieval philosophy (possibly its sources and impacts on later tradition) for philosophy students on BA level

The application process will be in two rounds. The committee appointed by the dean of the Faculty of Arts decides in the first round on the basis of submitted materials. The candidates invited for the second round will be asked to (a) deliver a public lecture in the Department of Philosophy and Religious Studies, Faculty of Arts, Charles University and (d) be available for an interview with the committee.

The shortlist of candidates for the second round will be available by early April 2018; lectures and interviews are scheduled for late April and early May 2018.

#### Information

For additional information you can contact:
Dr. Jakub Jirsa, Head of department, jakub.jirsa@ff.cuni.cz (job profile and academic matters)

Hana Vamberská, hana.vamberska@ff.cuni.cz (application procedures)

## Brill’s “Investigating Medieval Philosophy” series

In case any of our readers don’t already know about it, Brill has a book series “Investigating Medieval Philosophy”, edited by John Marenbon and with an editor board: consisting of Margaret Cameron, Simo Knuuttila, Christopher J. Martin, and our very own Martin Lenz. The series aims to publish two volumes a year, and:

The series aims to provide a peer-reviewed forum for high-quality monographs and coherent collective volumes on medieval philosophy, written in such a way as to make them comprehensible and interesting to mainstream philosophers and historians of philosophy in Anglophone philosophy departments. Volumes in the series are not required to use medieval philosophy to make a direct contribution to debates in contemporary analytical philosophy (although this is one possibility), but the manner in which the medieval texts are treated should reflect, in an historically sensitive way, the methods and the language of contemporary analytical philosophy – in especial, its ideals of clarity and unpretentiousness. There are many different varieties of this general ‘analytical’ approach, and the series is open to any of them. The scope of medieval philosophy is taken widely, to include the Arabic, Greek and Jewish traditions, as well as the Latin one, and to run from c.500 to c.1500; works which go on even so far as 1700 may be considered, if they are at least equally concerned with the period before 1500.

## John Trevisa on the rational soul

Earlier this week I picked up John Trevisa’s Middle English translation of Bartholomeus Anglicus’s De Proprietatibus Rerum ‘On the Properties of Things’, for reasons entirely unrelated to logic and philosophy (mostly because I wanted to learn things like “why do you never seen angels’ thighs when they are depicted by painters in corporeal form?” and “what does the shape of your soul say about you?” and “why is the stomach ridged and ribbed instead of smooth?” and other such important questions). But what Anglicus/Trevisa has to say about ‘rationality’ is interesting.

Bartholomeus Anglicus was a 13th-century English Franciscan who studied at the university of Paris. His compendium De Proprietatibus Rerum, an early forerunner of the modern encyclopedia, was written around 1240 at Magdeburg, and draws upon a tremendous wealth of resources, classic and medieval, western and eastern, dipping into philosophy, theology, medicine, natural sciences, biology, astronomy, geography, etymology, etc. What I’ve found fascinating about it so far is how many of his sources he cited by name, far, far more than the usual “some say” that you often get in medieval texts.

John Trevisa, a Cornishman who was educated at Exeter College, Oxford, and worked in the circle of Wyclif at Queen’s College, translated De Proprietatibus Rerum into Middle English while he lived and worked in Berkeley, Gloucestershire, around 1397-1399. The first printed edition of the Middle English text was produced by Wynkyn De Worde, around 1495, attesting to its importance.

I’m finding Trevisa’s Middle English somewhat more accessible than Chaucer’s; most of the words are easily identifiable as their modern counterparts, though archaic forms of the being verb like ‘buþ’ and ‘beþ’ threw me at first until I figured them out. One verb which turns up quite a bit that I was not familiar with previously is fongen, which has proven to be quite a useful and versatile word and I’m sorry it has fallen out of common currency in English. (I also lament the loss of the variant ‘noseþirl’ for ‘nostril’.)

All that by means of introduction. I thought it would be interesting and semi-relevant for the topic of this blog to say something about Anglicus/Trevisa’s view of rationality and the rational soul, which is not the typical view I’m used to.

First, in Book III, Chapter 6 Trevisa (or rather Trevisa-translating-Anglicus, but I’m just going to say “Trevisa” from here on out) notes that there “fyue maner myȝtes and vertues” that the soul has: (1) “felinge”, (2) “bodiliche wit”, (3) “ymaginacioun”, (4) “racio ‘reason'”, and (5) “intellectus ‘vndirstondinge and inwit'”. Reason is glossed as the power to “demeþ betwene gode and euel and soþ and fals”. The first three powers are common to both men and other beasts, and depend upon a composite of body and soul. But the latter two powers are found in men alone, and do not require a body; for they “beþ in þe soule, in þat he may be departid from þe body and abide departid as an aungel.” The power in such a disembodied soul which “biholdeþ þe ouer þinges” is intellectus, and the power which “beholdiþ þe neþer þinges” is racio.

When we consider the ends of the soul, three properties or virtues of the soul can be distinguished, namely (1) “racionalis, þerby he takeþ hede to þing þat is soþ and trewe”, (2) “concupissibilis and þerby he takeþ hede to þing þat is good”, and (3) “irascibilis and þerby he takeþ hede to þing þat is grete and huge and to þing þat is euerlastinge”. We’ll ignore (2) and (3) for now; but of (1), Trevisa says that “in þe racional is knowinge [soþ].”

However, if we consider the soul with respect to its workings (and this is the topic of Chapter 7), we can distinguish three virtues that are found in the soul:

1. vegetabilis þat ȝeueþ lif”
2. sensibilis þat ȝeueþ felinge”
3. racionalis þat ȝeueþ resoun”

Whatever has a sensible soul also has a vegetative soul, and whatever has a rational soul also has a sensible one.

For Trevisa and Anglicus, the only reason to exercise reason is to become closer to God. I’m kind of disappointed that there is no more explicit discussion of reasoning, of logic, of dialectics, or argument, or anything of how one can use reason to know what is “soþ and trewe” — because surely isn’t that the interesting bit? Maybe this topic will come up again — the work does have, after all, nineteen books each of which is divided up into many chapters — so there’s still hope that I can revisit the topic again!

In the meantime, I’m live-tweeting my way through the book under the hash-tag #PropThings, if you want to follow along!

## Syllogism Mnemonics

The other day a colleague of mine asked if I had anything I could send him regarding the medieval syllogism mnemonics. I told him there was some info in the textbook I’m writing, but it’s rather idiosyncratic to the way I present the syllogistic, and that instead I’d write up something specific for him. When he protested, I told him it’d be a useful thing to have on the Medieval Logic blog, which we hope to update every Thursday. It’s this sort of thing precisely that blogging is good for! I may not need this collected information now, but I’m sure I’ll use it sometime in the future.

Syllogisms are characterised by their figure — the relative arrangement of the terms — and their mood — the triple of copulae that connect the terms.

Figure 1 Figure 2 Figure 3
P__M M__P P__M
M__S M__S S__M
P__S P__S P__S

(where P is the predicate of the conclusion (the major term), S is the subject of the conclusion (the minor term), and M is the middle term).

William of Sherwood in his Introduction to Logic (c1240, [1]) provides his students with a mnemonic verse to remember which figure is which:

The difference of the figures is retained in this verse: Sub pre prima bis pre secunda tertia bis sub (66).

That is, in the fist figure, the middle term is first the subject and then the predicate; in the second, it is the predicate twice; in the third, it is the subject twice. (This same verse occurs in Roger Bacon [5, ∥ 290], and Lambert of Auxerre, their contemporary, has the same content in a much more verbose form [5, 139].)

Following this, he then says:

The moods and their reductions, on the other hand, are retained in these verses:

Barbara celarent darii ferio baralipton
Celantes dabitis fapesmo frisesomorum
Cesare camestres festino baroco
Darapti felapton disamis datisi bocardo ferison
(66)

Kretzmann notes that this is “the oldest known surviving version of these famous mnemonic verses, and Sherwood may have been the inventor of them…However, there were earlier attempts at the same sort of device in the thirteenth century, and the word ‘Festino’ appears ‘in a MS dating at the latest from 1200′” [citing Bochenski] (66-67), though later in the same footnote he compares Sherwood’s verses to those in Peter of Spain and concludes that the differences between the two “strongly suggest at least one earlier version on which both men were drawing” (67).

Sherwood explains the mnemonic names as follows:

In these lines ‘a’ signifies a universal affirmative proposition, ‘e’ a universal negative, ‘i’ a particular affirmative, ‘o’ a particular negative, ‘s’ simple conversion [conversio simplex], ‘p’ conversion by limitation [conversio per accidens], ‘m’ transposition of the premisses, and ‘b’ and ‘r’ when they are in the same word signify reduction per impossibile. The first two lines are devoted to the first figure, the four words of the third line to the second figure, and all the other words to the third figure (67).

(Note that, conveniently, ‘a’ and ‘i’ are the first two vowels in Latin affirmo ‘I affirm’, and ‘e’ and ‘o’ are the two vowels in Latin nego ‘I deny’.) While the names that Sherwood introduces are typical/traditional, the explanation that he gives of the names is not. First, he doesn’t note that the first letter of the name indicates to which of the four basic syllogisms, Barbara, Celarent, Darii, and Ferio, the others can be reduced to. Second, his identification of ‘b’ and ‘r’ as the cues for reductio per impossibile are atypical (it’s not much of a mnemonic; why not, e.g., ‘r’ alone?)

Other roughly contemporary texts offer a slightly different version of the verses and explanation of the names. Roger Bacon in his Art and Science of Logic [5] has a verse for remembering the types of conversions (Maloney notes that similar verses also occur in the 12th-century Logica Ut Dicit and the Logic ‘cum sit nostra’la):

Simpliciter feci convertitur, eva per acci,
Acto per contra. Sic fit conversio tota (∥ 279)

and he notes that “‘E’ signifies a universal negative, ‘I’ a particular affirmative, ‘A’ a universal affirmative, and ‘O’ a particular negative” (∥ 279). The list of mnemonic names, identical to Sherwood’s, appears in ∥ 295, and is explained in ∥ 296; he corrects Sherwood’s apparent mistake and identifies ‘c’ as the code for reductio per impossibile. Lambert also has the same list of names (143), except that he names the second mood of the second figure “Campestres”; the addition of the ‘p’ is not warranted.

Peter of Spain in his Summaries of Logic, written shortly after Sherwood’s, [2] says:

Barbara Celarent Darii Ferio Baralipton
Celantes Dabitis Fapesmo Frisesomorum
Cesare Cambestres Festino Barocho Darapti
Felapto Disamis Datisi Bocardo Ferison

In these four verses are nineteen words representing the nineteen moods of the three figures, so that by the first word we understand the first mood of the first figure, by the second word the second mood, and so on for the others…

…By the vowel A we understand the universal affirmative, by E the universal negative, by I the particular affirmative, and by O the particular negative…

…It must be understood that all the moods indicated by a word beginning with ‘B’ are to be reduced to the first mood of the first figure, and ll the moods signified by a word beginning with C to the second mood, those beginning with D to the third, and those with F to the fourth. Also, wherever an S is put in these words, it signifies that the proposition understood by the immediately preceding vowel is to be converted simply. And by P it signifies that the proposition is to be converted accidentally [converti per accidens]. Wherever M is put, it signifies that a tranposition in premisses is to be done…Where C is put, however, it signifies that the mood understood by that word is to be confirmed by impossibility (191, 193).

Peter of Spain’s identification of “C” (by which we should understand NOT the “C” which begins the word) with reductio per impossibile is orthodox.

Copenhaver et al. also have a footnote, noting that “since the verses scan as Latin hexameters, Latin was almost certainly the language in which they were first written” (contra Prantl who “contends that the first mnemonic lines were Greek and formulated by Psellus (b. 1020)” [3, 519]), and that Sherwood’s verses “appear in the Dialectica monacensis and the Logica ‘cum sit nostra’, treatises dated before or around 1200 by de Rijk…note that de Rijk thinks that BARBARA CELARENT is a later interpolation in the Dialectica monacensis” (191). They also mention related material in the Ars burana and Ars emmerana, also edited by de Rijk, and some mnemonic verses found in a 9th C text [3]. These verses, appearing in Codex Sti. Galli 64, “though they do not contain the technical words, Barbara, etc., or their equivalent, yet served the purpose for which those words were afterwards invented” (519). (Because the verses are long and can be read in full, for free, on JSTOR, we do not quote them here.) Turner notes that

it would, of course, be idle to look for evidences of an original contribution to the science of logic. It belongs to an age in which originality was not a dominant characteristic of teachers of logic. It simply sums up what was to be found in the treatises of Apuleius, Martianus Capella, Cassiodorus, and Isidore. Its terminology does not vary essentially from that which was current in the schools of the ninth and tenth centuries (525).

In the few cases where the terminology used is not standard, this is due to the need to pick terms that fit the desired meter.

Let us return to the matter of the verses and material in the Dialectica monacensis, the Logica ‘cum sit nostra’, and the Artes burana et emmerana. De Rijk notes that “I have not been able to find these verses in twelfth century treatises. It should be noted, however, that the famous verses BARBARA, CELARENT had a few forerunners in two twelfth century tracts on syllogism” (401), namely the Ars emmerana and the Ars burana, which we will discuss below.

The Logica ‘cum sit nostra’ contains chapter on the syllogisms which in de Rijk’s edition [4] contains:

Barbara Celarent Darii Ferio
Baralipton Celantes Dabitis Fapesmo Frisesomorum
Cesare Campestres Festimo Baroco Drapti
Felaptdon Disamis Datisi Bocardo Ferison.

A notat universalem affirmativam, E notat universalem negativam, I particularem affirmativam, O particularem negativam…

…Consequenter dicendum est de reductione. Et sciendum quod omnes modi debent reduci in quatuor primos modos prime figure, quia omnes modi incipientes per B debent reduci in Barbara, per C in Celarent, per D in Darii, per F in Ferio.

Et reducuntur per tria, scilicet per conversionem, per transpositionem, et per impossibile. Unde sciendum quod S denotat conversionem simplicem, P conversionem per accidens, M transpositionem propositionum, C reductionem per impossibile. Unde versus:

S simplx, P per acc.
M transpos., C notat impossibile (436).

In the Dialectia Monacensis, we have the following:

Hec omnia facilius possunt haberi per hos versus:

Barbara * Celarent * Darii * Ferio * Baralipton *
Celantes * Dabitis * Fapesmo * Frisesmomorum

In hiis versibus sunt novem dictiones novem modis prime figure deservientes, prima primo et secunda secundo, et sic deinceps. Horum autuem versuum triplex est utilitas. Prima est quia scitur quales et quante debeant esse propositiones in qualibet materia; et hoc per vocales istarum dictionem: per A intelligitur universalis affirmativa, per E universalis negativa, per I particularis affirmativa, per O particularis negativa…Secunda utilitas est quia scitur qui sillogismi in quos habeant reduci; et hoc per initiales litteras istarum dictionum. Tertia utilitas est quia scitur per quid unusquisque sillogismus reducator: hec enim littera S ubicumque inventiur est signum simplicis conversionis; B vero significant conversionem per accidens; M vero significat transpositionem premissarum, idest quod de maiore fiat minor et econverso (494).

Note the use of “B” to stand for accidental conversion; this cannot be correct since (a) it leaves Fapesmo and Baralipton wholly unexplained and (b) Dabitis cannot be proven if the major premise is accidentally converted.

A separate verse is given for the second and third figure syllogisms:

Et sciendum quod omnia que dicta sunt de secunda et tertia figura, faciliter possunt haberi per hos versus:

Cesare * Camestres * Festino * Baroco *
Darapti * Felapton * Disamis * Datisi * Bocardo * Feriso *

In hiis versibus sunt decem dictiones. Inter quas prime quatuor deserviunt secunde figure et alie que secuntur deserviunt tertie figure, ita quod prima primo, et ita deinceps (497-498).

Of these verses, de Rijk notes “something peculiar must be noticed” (413) about their placement; the verse concerning the first figure appears after the discussion of the first figure, but the verses discussing the second and third figure occur together after the discussion of the third figure. He says that “this peculiar arrangement of the mnemonic verses strongly suggests that they were interpolated in the original treatise” (413) and that there was insufficient margin space to add the verses at the end of the discussion of the second figure, which is why they were saved for a later space.

What about the early hints towards the later mnemonics that de Rijk found in the Ars Emmerana and the Ars Burana? We have no verses in AE, but there is a clear attempt to encode the relevant information of a mood in terms of letters:

Notandum quod universalis affirmative designantur his quatuor literis E I O U; universales negative his: L M N R; particularis affirmative tribus: A S T; particulares negative tribus: B C D.

Secundum hoc he novem voces designent novem modos prime figure:

VIO * NON * EST * LAC * VIA * MEL * VAS * ERB * ARC

he quatuor voces designant quatuor modos secunde figure:

REN * ERM * RAC * OBD

sex modi tertie figure his sex designantur:

EVA * NEC * AUT * ESA * DUC * MAC (173).

Here, only the quality and the quantity of the propositions involved in the syllogisms are mentioned.

The mnemonic material in the Ars Burana is similar:

He littere E I O V significant universales affirmativas; et he littere L M N R significant universales negativas; et he A S T significant particulares affirmativas; et he B C D significant particulares negativas (199).

With this convention established, the author goes on to say that:

De modis igitur prime figure talis assignatur versus:

VIO NON EST LAC VIA MEL VAS ERB ARC (200).

Each of these syllogisms is then exemplified; VIO is Barbara, NON is Celarent, etc. A similar verse is given for the second figure: “REN ERM RAC OBD” (203), and the third: “EVA NEC AVT ESA DVC NAC” (205).

So much for the period up through the 13th century. What about afterwards? I have fewer 14th-century treatises on my shelves, but Buridan has a chapter on syllogisms in his Summulae de Dialectica [7]. In that chapter, the mnemonic names are taken completely for granted; Buridan refers to Disamis and Bocardo before he even introduces the concepts of figure and mood, thus assuming that these are already familiar to the reader (310). The mnemonic verse itself doesn’t appear for another 10 pages or so, where he introduces the verse (320), explains how to form a syllogism from the names given (321), and then explains “the verse insofar as it indicates the reduction of imperfect to perfect syllogisms” (321). His explanation follows that of Peter, Bacon, Lambert, etc.

The final person I’d like to mention in this post is Paul of Venice, if for no other reason than that I’m in the process of editing and translating his chapter on syllogisms from the Logica Magna, written at the very end of the 14th century. Paul also assumes familiarity with the names in that he uses them before he explains them:

Having seen what is required for a syllogism in the first figure, it remains to show the moods which are syllogized in the same [figure]. And they are six, namely: Barbara, Celarent, Darii, Ferio, Fapesmo, Frisesomorum.

(Note that he omits Dabitis and Celantes). He says:

Whence it must be noted that in whatever mode, some of these four vowels, namely a, e, i, o, are put in. ‘a’ denotes the universal affirmative, implicitly or explicitly. ‘e’ [denotes] the universal negative, implicitly or explicitly, ‘i’ the particular, indefinite, or singular affirmative, implicitly or explicitly. ‘o’ denotes the particular, indefinite, or singular negative, implicitly or explicitly.

(Paul differs from earlier writers in explicitly including indefinites and singulars in his syllogistic, and he assimilates them to the particulars, rather than to the universals, as some other commentators did.)

There is one way in which Paul’s discussion of the mnemonics is utterly unlike anything that I have seen preceding him. He says:

It is considered whether in a mood there are four such letters by the multiplication of one, or not…However, if this is so [that you have four letters rather than three, e.g., Barbarai], it will happen that you know the conclusion is able to follow according to the third or fourth denotation. Hence in Barbara is ‘a’ triplicated, and a simple solitary ‘i’ for that reason may be denoted by such mood that the premisses are implicitly or explicitly universals according to a two-fold precession of that term ‘a’, that is, because of their position. The conclusion which is able to be inferred either implicitly or explicitly is denoted by the third vowel, but because there is a final ‘i’, for this reason it may be denoted by the particular, indefinite, or affirmative itself, it is possible to infer either explicitly or implicitly, just as is stated in the example.

What’s going on in this section is not entirely clear, though it became clearer when we worked through my draft translation in the St Andrews medieval logic/Latin reading group a year or two ago. What we think is going on is that Paul is explicitly noting the validity of “Barbari”, a syllogism which has otherwise not been mentioned at all in any of the mnemonics above.

It is not until Paul introduces Fapesmo, the first non-perfect syllogism in hist list above, that he explains the other letters in the names:

Whereby any mood beginning with ‘B’ is able to be reduced to Barbara, everyone beginning with ‘C’ to Celarent, and everyone beginning with ‘D’ to Darii. And everyone beginning with ‘F’ to Ferio: But, as this is able to done, you ought to note that whenever ‘S’ is put down in these words, it signifies that the proposition indicated by the vowel immediately preceding ought to be converted simply. Wherever a ‘P’ is put down it signifies that the proposition indicated before the ‘P’ ought to be converted per accidens, and wherever an ‘M’, it is denoted that a transposition of the premises ought to be made, that is, to make the minor of the major and conversely. However, where ‘C’ is put down, it is denoted that this mood or a syllogism formed in the same should be reduced per impossibile.

He then gives this verse:

Simpliciter verti universaliter, ‘S’, ‘P’ vero per accidens, ‘M’ vult transponi, ‘C’ per impossibile duci.

I had intended in this post to also write about the verses relating to the square of opposition, especially the modal one; but I think that this point, that’s best saved for another post! One other aspect that I haven’t mentioned at all here is the mnemonic tradition in commentaries on the Prior Analytics, and that is because I have no idea what such a tradition is, or if it even existed. Again, a topic for another post! One final topic for another post, if only because I am running out of time, is how long these syllogism mnemonics persisted as pedagogical tools; in particular, I’d like to look at these and other 16th-century logic textbooks to see if the syllogism mnemonics show up in them.

#### Notes

[1] William of Sherwood, Introduction to Logic, translated with an introduction and notes by Norman Kretzmann (University of MN Press: 1966).

[2] Peter of Spain, Summaries of Logic, text, translation, introduction, and notes by Brian P. Copenhaver with Calvin Normore and Terence Parsons (OUP: 2014).

[3] William Turner, “Mnemonic Verses in a Ninth Century MS.: A Contribution to the History of Logic”, The Philosophical Review Vol. 16, No. 5 (Sep., 1907), pp. 519-526, JSTOR link

[4] L. M. de Rijk, ed., Logica Modernorum: A Contribution to the History of Early Terminist Logic vol. II, parts one & two (Van Gorcum, 1967).

[5] Roger Bacon, Art and Science of Logic, trans. Thomas S. Maloney, (PIMS 2009).

[6] Lambert of Auxerre, Logica or Summa Lamberti, T. S. Maloney, trans., (University of Notre Dame, 2015).

[7] John Buridan, Summulae de Dialectica, trans. with an introduction by Gyula Klima (Yale University Press, 2001).

[8] Paul of Venice, Logica Magna (Venice, 1499).


text:
# Two images to fill one beamer frame: most appropriate method?

Suppose I have two images: one a solid blue square and another a yellow. What is the most appropriate way to divide and fill an entire beamer frame with these images? Horizontally (i.e., left half blue right half yellow)? Vertically? (i.e., top half blue bottom half yellow)

This post provides a method, but it involves low-level operations. Another post (can't find the link now) suggests using TikZ.

Is there a third that doesn't involve low-level operations or complicated external packages? I'm confused. I'd think this task wouldn't be all that difficult.

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What do you mean by entire? Do you still wish to have the theme bars on the top/bottom? Margins? Place for the navigation buttons? Or do you want it to look just like someone took a page out of another pdf file and pasted it to your carefully-themed document? –  Federico Poloni Nov 7 '11 at 22:34
\frame{\includegraphics[width=.5\textwidth}{image1}\includegraphics[width=.5\textwidth}{image2}}? Might not fill the full height, though. –  Martin Scharrer Nov 7 '11 at 22:35
You can use \usebackgroundtemplate before the frame –  Marco Nov 7 '11 at 22:39
For other examples see this or this or this –  Marco Nov 7 '11 at 22:46
@FedericoPoloni I mean the latter, like someone took a page out of another pdf file. –  lowndrul Nov 8 '11 at 0:26

You can use the standard \includegraphics command setting the proper values for the width and height in terms of \paperwidth and \paperheight; a box (in this case \makebox) will prevent an overfull box; using plain for the frames will remove the headline, footline:

\documentclass{beamer}
\usepackage{graphicx}

\begin{document}

\begin{frame}[plain]
\makebox[\linewidth]{%
\includegraphics[width=0.5\paperwidth,height=\paperheight]{cat1}%
\includegraphics[width=0.5\paperwidth,height=\paperheight]{cat2}}
\end{frame}

\begin{frame}[plain]
\makebox[\linewidth]{%
\includegraphics[width=\paperwidth,height=0.5\paperheight]{cat1}}\\\nointerlineskip
\makebox[\linewidth]{%
\includegraphics[width=\paperwidth,height=0.5\paperheight]{cat2}
}
\end{frame}

\end{document}

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\paperwidth is hardly correct here because \paperwidth is larger than \linewidth. –  Igor Kotelnikov Nov 8 '11 at 9:10
@Igor: you're wrong: since I used \makebox[\linewidth]{...}, LaTeX doesn't care if the contents of the box overflows the dimension. –  Gonzalo Medina Nov 8 '11 at 12:44
@GonzaloMedina: First, there's still a thin white horizontal line in the second. I tried \setlength{\baselineskip}{0pt} and \setlength{\parskip}{0pt} locally and had no success. Secondly, how did that work? Could you explain? Is this a hack or an intended use of \makebox? What do you mean "will prevent an overfull box"? It looks like the graphics overfilled the box to fill the frame. Thx for the help! –  lowndrul Nov 8 '11 at 15:12
@brianjd: you said "there's still a thin white horizontal line in the second". In the second what? Please explain in more detail what are you referring to with "in the second". On the other hand, the use of \makebox it's not a hack; it's an intended use. As I said in my previous reply, using \makebox[\linewidth]{...} makes LaTeX let the contents of the box overflow its dimension without producing an overfull box. –  Gonzalo Medina Nov 8 '11 at 15:25
@GonzaloMedina: The top (almost) half is blue. The bottom (almost) half is yellow. And there's a thin white horizontal line separating the two. I'm guessing this is just the standard white-space that goes between two lines. –  lowndrul Nov 8 '11 at 16:17

Here is a very simple solution via columns.

\documentclass{beamer}
\usepackage{lmodern}
\begin{document}
\begin{frame}[plain]
\begin{columns}
\column{.5\paperwidth}
\textcolor{blue}{\rule{.5\paperwidth}{\paperheight}}
\column{.5\paperwidth}
\textcolor{yellow}{\rule{.5\paperwidth}{\paperheight}}
\end{columns}
\end{frame}
\begin{frame}[plain]
\begin{columns}
\column{\paperwidth}
\textcolor{blue}{\rule{\paperwidth}{.5\paperheight}}\\\nointerlineskip
\textcolor{yellow}{\rule{\paperwidth}{.5\paperheight}}
\end{columns}
\end{frame}
\end{document}

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