Multidimensional Data Analysis of Deep–Language in J.R.R. Tolkien and C.S. Lewis Reveals Tight Mathematical Connections

1. Introduction

Unanimously, in a large number of papers – some of which are here recalled [1,2,3,4,5,6,7,8] from the vast literature on the topic – scholars of the English Literature state that J.R.R. Tolkien and C.S. Lewis influenced each other writings. The purpose of the present paper is not to review this large literature based on the typical approach used by scholars of literature – which is not our speciality – but to investigate this issue mathematically and statistically – study never done before – by using recent methods devised by researching the impact of the surface deep–language variables [9,10,11] and linguistic channels [12] in literary texts. Since scholars mention also the influence by George MacDonald on both, we consider also some novels written by this earlier author. To set all these novels in the framework of the English Literature, we consider also some novels written by other earlier authors, such as C. Dickens and others.

After this introduction, in Section 2 we introduce the literary texts (novels) studied. In Section 3 we report the series of words, sentences and interpunctions versus chapters for some novels, and define an index useful to synthetically describe regularity due to what we think is a consciouos design by authors. In Section 4 we start exploring the four deep−language variables; to avoid misunderstaning, these variables, and the linguistic channels derived from them, refer to the “surface” structure of texts, not to the “deep” structure mentioned in cognitive theory. In Section 5 we report results concerning the extended short–term memory and a universal readability index; both topics address human short–term memory buffers. In Section 6 we represent literary texts geometrically in the Cartesian plane by defining linear combinations of deep‒language variables and calculate the probability that a text can be confused with another. In Section 7 we show the linear relationships existing between linguistic variables in the novels considered. In Section 8 we report the theory of linguistic channels and in Section 9 we apply it to the novels presently studied. Finally, in Section 10 we summarise the main findings and conclude. Several Appendices report numerical data.

2. Database of Literary Texts (Novels)

Let us first introduce the database of literary texts used in the present paper. Table 1 lists some basic statistics of the novels by Tolkien, Lewis and MacDonald. To set these texts in the framework of earlier English Literature, we consider novels by Charles Dickens (Table 2) and other authors (Table 3). Notice that these statistics refer only to the main text because titles, footnotes and other extraneous material present in the digital texts have been deleted.

Some homogeneity can be noted in novels of the same author. The stories in The Space Trilogy and The Chronicles of Narnia, by Lewis, are told with about the same number of chapters, words and sentences, as is also for couples of MacDonald’s novels, such as At the Back of the North Wind and Lilith: A Romance. The stories in David Copperfield, Bleak House and Our Mutual Friend (by Dickens) are also told in this way as The Adventures of Oliver Twist and A Tale of Two Cities. These numerical values, we think, are not due to chance but consciously managed by the authors, a topic we purse more in the next Section.

3. Conscious Design of Texts: Words, Sentences and Interpunctions Versus Chapters

First, we study the linguistic variables which we think the authors deliberately designed. In the specific, we show the series of words, sentences and interpunctions versus chapter.

Let us consider a literary work (a novel) and its subdivision in disjoint blocks of text long enough to give reliable average values. Let $n_S$ be the number of sentences contained in a text block, $n_W$ the number of words contained in the $n_S$ sentences, $n_C$ the number of characters contained in the $n_{W }$words and $n_I$ the number of punctuation marks (interpunctions) contained in the $n_S$ sentences.

Figure 1 shows the series $n_W$versus the normalized chapter number for The Lord of the Rings, The Chronicles of Narnia, The Space Trilogy.

For example, the normalized value of chapter 10 in The Chronicles of Narnia, is $10/110=0.09$ in the $x-$scale of Figure 1. This normalization allows to show synoptically novels of different number of chapters.

In The Chronicles of Narnia (in the following Narnia, for brevity. We can notice a practically constant value $n_{W }$compared to The Lord of the Rings (Lord) and The Space Trilogy (Trilogy).

Let us define a synthetic index to describe the series drawn in Figure 1, namely the coefficient of dispersion $\delta$, given by the standard deviation ${\sigma }_{n_W}$divided by the mean value $<n_W>$

$$\delta =\frac{{\sigma }_{n_W}}{<n_W>}$$

(1)

Table 4 and Table 5 report $\delta$ for $n_W$, $n_S$ and $n_I$. Since $n_S$ and $n_I$(series not shown for brevity) are very well correlated with $n_S$, the three coefficients of dispersion are about the same.

In Narnia $\delta =0.16$, in Lord $\delta =0.34$ and in Trilogy$\delta =0.60$. Let us also notice the minimum value $\delta =0.07$ in The Screwtape Letters (Screwtape).

The overal (words, sentences and interpunctions mixed together) mean value is $<\delta >=0.44$, the standard deviation ${\sigma }_{\delta }=0.18$. Therefore, Screwtape is practically more than ${2\times \sigma }_{\delta }$ from the mean – as also is Silmarillion on the other side – Narnia is at about $1.5\times {\sigma }_{\delta }$. In contrast, Trilogy, Lord and The Hobbit (Hobbit) are whithin $1\times {\sigma }_{\delta }$.

From these results, it seems that Lewis designed the chapters of Narnia and Screwtape with an almost uniform distribution of words, sentences and interpunctions, very likely because of the intended audience in Narnia (i.e., kids) and the “letters” fiction tool used in Screwtape. In Trilogy the design seems very different ($\delta =0.60$, well within $1\times {\sigma }_{\delta }$) likely due to the development of the science fiction story narrated.

Tolkien acted differently from Lewis, because he seems to have designed chapters more randomly and within $1\times {\sigma }_{\delta }$, as Hobbit and Lord show. An exception is The Silmarillion, published posthumously, a text far from being a “novel”.

Finally, notice that the novels by MacDonald show more homogeneous values, very similar to Hobbit and Trilogy and to the other novels listed in Table 5.

In conclusion, the analysis of series of words, sentences and interpunctions per chapter does not indicate likely connections between Tolkien, Lewis and MacDonald. Each author distributed words, sentences and interpunctions according to some plan different from author to author and also from novel to novel of the same author.

There are, however, linguistic variables that – as we have reported for modern and ancient literary texts [13,14,15] – are not consciously designed/managed by authors, therefore, these variables are the best candidates to reveal hidden mathematical/statistical connections between texts. In the next section we start dealing with these variables, with the specific purpose of comparing Tolkien and Lewis, although this comparison is set in the more general framework of the authors mentioned in Section 2.

4. Surface Deep–Language Variables

We start exploring the four stochastic variables we called deep−language variables, following our general statistical theory on alphabetical languages [9,10,11]. To avoid possible misunderstaning, these variables, and the linguistic channels derived from them, refer to the “surface” structure of texts, not to the “deep” structure mentioned in cognitive theory.

Contrarily to the variables studied in Section 3, the deep–language variables are likely due to unconscious design. As shown in [9,10,11], they reveal connections between texts far beyond writers’ awarness, therefore, the geometrical representation of texts [10] and the fine tuning of linguistic channels [12,13,14,15], are tools better suited to reveal connections. They can likely indicate also the influence of an author on another.

Defined the number of characters per chapter $n_C$ and the number of $I_P\text{'}\mathrm{s}$ per chapter $n_{I_P}$, the four deep−language variables are [9]:

The number of characters $C_P$:

$$C_P=\frac{n_C}{n_W}$$

(2)

The number of words per sentence $P_F$:

$$P_F=\frac{n_W}{n_S}$$

(3)

The number of interpunctions per word, referred to as the word interval, $I_P$:

$$I_P=\frac{n_I}{n_W}$$

(4)

The number of word intervals per sentence $M_F$:

$$M_F=\frac{n_{I_P}}{n_S}$$

(5)

Equation (5) can be written also as $M_F=P_F/I_P$.

Table 6, Table 7, Table 8 and Table 9 reports mean and standard deviation of these variables. Notice that these values have been calculated by weighing each chapter with its number of words to avoid that short chapters weigh as much as long ones. For example, chapter 1 of Lord has $10097$ words, therefore its statistical weight is $10097/472173\approx 0.021$ not $1/62\approx 0.016$. Notice, also, that the coefficient of dispersion used in Section 2 was calculated by weighting each chapter $1/62$, not $10097/472173,$to visually agree with the series drawn in Figure 1.

Specifically, let $M$ be the number of samples (i.e., chapters), then the mean value $<P_F>$ is given by:

$$<P_F>=\sum _{k=1}^M{P}_{F,k}\times \left(n_{W,k}/\sum _{k=1}^M{n}_{W,k}\right)$$

(6)

Therefore, notice, for not being misled, that $<P_F>\ne \frac{1}{M}\sum _{k=1}^M{P}_{F,k}\ne \sum _{k=1}^M{n}_{W,k}/\sum _{k=1}^M{n}_{S,k}$=$W/S$. In other words, $<P_F>$ is not given by the total number of words $W$divided by the total number of sentences $S$, or by assigning the weight $1/M$ to every chapter. The three values coincide only if all text blocks contain the same number of words and the same number of sentences, a case never found. The same observations apply to all other variables.

From Table 6, Table 7, Table 8 and Table 9 we can notice the following characteristcs. Lord and Narnia share the same $<P_F>$. Silmarillion is distinctly different from Lord and Hobbit, in agreement with the different coefficient of dispersion. Screwtape is distinctly different from Narnia and Trilogy. There is a great homogeneity in Dicken’s novels and a large homogeneity in ${<C}_P>$ in all novels.

In the next sections we use $<P_F>$, ${<I}_P>$ and $<M_F>$ to calculate interesting indices connected to the short–term memory of readers–

5. Extended Short–Term Memory of Writers/Readers and Universal Readability Index

In this section we deal with the linguistic variables that, very likely, are not consciously managed by writers who, of course, act also as readers of their own text. We first report findings concerning the extended short–term memory and then those concerning a universal readability index. Both topics address human short–term memory buffers.

5.1. Extended Short–Term Memory and Multiplicity Factor

In [16,17], we have conjectured that the human short–term memory is sensitive to two independent variables, which apparently engage two short–term memory buffers in series, constituents of what we have called the extended short–term memory (E–STM). The first buffer is modelled according to the number of words between two consecutive interpunctions—i.e., the variable $I_P$, the word interval—which follows Miller’s $7\pm 2$ law [18]; the second buffer is modelled according to the number of word intervals, $I_P\text{'}s$, contained in a sentence —i.e., the variable $M_F$ – ranging approximately for 1 to 7.

In [17] we studied the patterns (which depend on the size of the two buffers) that determine the number of sentences that theoretically can be recorded in the E–STM of a given capacity. These patterns were then compared with the number of sentences actually found in novels of Italian and English literature. We have found that most authors write for readers with short memory buffers and, consequently, are forced to reuse sentence patterns to convey multiple meanings. This behavior is quantified by the multiplicity factor $\alpha$, defined as the ratio between the number of sentences in a novel and the number of sentences theoretically allowed by the two buffers, a function of $I_P$ and $M_F$.

We found that $\alpha >1$ is more likely than $\alpha <1$ and often $\alpha \gg 1$. In the latter case, writers reuse many times the same pattern of number of words. Few novels show $\alpha <1$, in this case writers do not use some or most of them. The values of $\alpha$ found in the novels presently studied are reported in Table 10 and Table 11.

5.2. Universal Readability Index

In Reference [19], we have proposed a universal readability index given by:

$$G_U=89-10k{C}_P+300/P_F-6\left(I_P-6\right)$$

(7)

$$k=<C_{P,ITA}>/<C_{P,Eng}>$$

(8)

In Eq. (8) ${<C}_{p,ITA}>=4.48$, ${<C}_{p,ENG}>=4.24$. By using Eqs. (7)(8), the average value ${<kC}_P>$ of any language is forced to be equal to that found in Italian, namely $4.48$. The rationale for this choice is that $C_P$ is a parameter typical of a language which, if not scaled, would bias $G_U$ without really quantifying the reading difficulty of readers, who in their language are used, on the average, to read shorter or longer words than in Italian. This scaling, therefore, avoids changing $G_U$ for the only reason that a language has, on the average, words shorter (as English) or longer than Italian. In any case, $C_p$ affects Eq.(7) much less than $P_F$ or $I_P$.

The values of $<G_U>$ – calculated as the other linguistic variables, i.e., by weighting chapters (samples) according to the number of words – are reported in Table 10 and Table 11. The reader may be tempted to calculate Eq.(7) by introducing the mean values reported in Table 6, Table 7, Table 8 and Table 9. This, of course, can be done but it should be noted that the values so obtained are always less or equal (hence they are lower bounds) to the means calculated from the samples (see Appendix A). For example, for Lord, instead of 64.9 we would obtain 61.9.

It is interesting to “decode” these mean values into the minimum number of school years, $Y$ necessary to make a novel “easy” to read, according to the Italian school system, assumed as the reference, see Figure 1 of [19]. The results are also listed in Table 10 and Table 11.

5.3. Discussion

There are several interesting facts to be noticed from the results reported in the previous sub–sections.

Silmarillion with $\alpha =0.2$ is quite diverse from other Tolkien’s writings. Mathematically, this is due to its large $<M_F>=3.62$ and $<I_P>=8.58$. In practice, the number of theoretical sentences allowed by the E–STM to read this text is only $1/\alpha =5$ times the number of sentence patterns actully used in the text. The reader needs a powerful E–STM and reading ability, since $G_U=38.7$ and $Y>13$. This does not occur for Hobbit ($\alpha =39.4,$ $G_U=52.4$, $Y=$9.9) and Lord ($\alpha =368.1,G_U=64.2,$ $Y=7.4$) in which Tolkien reuses patterns many times, especially in Lord.

Lord and Narnia show very large values, $\alpha =368.1$ and $\alpha =297.7$, and very similar $G_U\text{'}\mathrm{s}$ and school years: $G_U=64.2$, $Y=7.4$ and $G_U=61.1$, $Y=7.9$, respectively. Sentence patterns are reused many times by Lewis in this novel, but not in Screwtape ($\alpha =1.4)$, which is more difficult to read ($G_U=33.5$) and requires more years of schooling, $Y>13$. Moreover, Lord and Narnia have practically the same $<P_F>\approx 14$.

In general, Narnia is closer to Lord than to Trilogy, although the number of words and sentences in Trilogy and Narnia are quite similar (Table 1). This difference between Trilogy ($G_U=56.2$, $Y=9$) and Narnia ($G_U=61.1$, $Y=7.9$) might depend on the different readers addressed, kids for Narnia, adults for Trilogy, with different reading ability, as $G_U$ indicates.

The novels by MacDonald show values of $\alpha$ and $G_U$ very similar to those of the other English novels.

Notice the homogeneity in Dicken’s novels, which require about $Y=7~8$ years of school and readability index $<G_U>=59~65$.

In conclusion, Lord and Narnia are the novels that address readers with very similar E–STM buffers, re–use sentence patterns in similar way, contain the same number of words per sentence, require the same reading ability and school years compared to other Tolkien’s and Lewis’ novels. The mathematical connections between Lord and Narnia will be further pursued in the next Section, where the four deep–language parameters are used to represent texts geometrically.

6. Geometrical Representation of Texts

The mean values of Table 6, Table 7, Table 8 and Table 9 can be used to assess how texts are “close”, or mathematically similar, in the Cartesian coordinate plane, by defining linear combinations of deep‒language variables. Texts are then modelled as vectors, representation discussed in detail in [9,10] and here briefly recalled. An extension of this geometrical representation of texts allows to calculate the probability that a text may be confused with another one, an extension in two dimensions of the problem discussed in [20].

6.1. Vector Representation of Texts

Let us consider the following six vectors of the indicated components of deep‒language variables$:$ $\overrightarrow{R_1}=({<C}_P>,{<P}_F>$), $\overrightarrow{R_2}=({<M}_F>,{<P}_F>$), $\overrightarrow{R_3}=(<I_P>,<P_F>$), $\overrightarrow{R_4}=({<C}_P>,{<M}_F>$), $\overrightarrow{R_5}=({<I}_P>,{<M}_F>$), $\overrightarrow{R_6}=({<I}_P>,<C_P>$) and their resulting vector sum:

$$\overrightarrow{R}=\sum _{k=1}^6\overrightarrow{R_k}=x\overrightarrow{i}+y\overrightarrow{j}$$

(9)

The choice of which parameter represents the component in abscissa and ordinate axes is not important because, once the choice is made, the numerical results will depend on it, but not the relative comparisons and general conclusions.

In the 1st quadrant of the Cartesian coordinate plane two texts are likely mathematically connected ‒ they show close ending points of vector (9) ‒ if their relative Pythagorean distance is small. A small distance means that texts share a similar mathematical structure, according to the four deep–language variables.

By considering the vector components $x$ and $y$ of Eq. (9), we obtain the scatterplot shown in Figure 2 where $X$ and $Y$ are normalized coordinates calculated by setting Lord at the origin $(X=0, Y=0)$ and Silmarillion at $(X=1, Y=1)$, according to the linear tranformations:

$$X=\frac{x-x_{Lord}}{x_{Silma}-x_{Lord}}$$

(10)

$$X=\frac{y-y_{Lord}}{y_{Silma}-y_{Lord}}$$

(11)

From Figure 2 we can notice that Silmarillion and Screwtape are distinctly very far from all other texts examined, marking their striking diversity, as already remarked, therefore in the following we neglect them. Moreover, Pride, Vanity, Moby and Floss are grouped together and far from Trilogy, Narnia and Lord, therefore in the following we will not consider them further.

The complete set of the Pythagorean distance $d$ between couples of texts is reported in Appendix B. These data synthetically describes proximity of texts and may indicate to scholars of literature connections between texts not considered before.

Figure 3 shows example of these distances concerning Lord (text 1), Narnia (text 3) and Trilogy (text 4). By referring to the cases in which $d<0.2$, we can observe:

The closest texts to Lord are Narnia (3), Back (5), Lilith (6), Mutual (11) and Peter (14).

The closest texts to Narnia (1) are Lord (1), Lilith (6), Bleak (9), Martin, Peter (14)

The closest texts to Trilogy (3) are Hobbit (2), Martin (12), Peter (14).

Besides the proximity with earlier novels, Lord and Narnia show close proximity with each other and with two novels by MacDonald.

These remarks, however, refer to the “average” display of vectors whose ending point depends only on mean values. The standard deviation of the four deep–language variables, reported in Table 6, Table 7, Table 8 and Table 9, do introduce data scattering, therefore in the next sub–section we study and discuss this issue by calculating the probability (called “error” probability) that a text may be mathematically confused with another one.

6.2. Error Probability

Besides the vector $\overrightarrow{R}$ of Eq. (9) – due to mean values – we can consider another vector $\overrightarrow{\rho }$, due to the standard deviation of the four deep–language variables, that adds to $\overrightarrow{R}$. In this case, the final random vector describing a text is given by:

$$\overrightarrow{T}=\overrightarrow{R}+\overrightarrow{\rho }$$

(12)

Now, to get some insight into this new description, we consider the area of a circle centered at the ending point of $\overrightarrow{R}$.

We fix the magnitude (radius)$\rho$ as follows. First, we add the variances of the deep–language variables that determine the components $x$ and $y$ of $\overrightarrow{R}$ – let them be ${\sigma }_x^2$, ${\sigma }_y^2$ – then we calculate the average value ${\sigma }_{\rho }^2=0.5\times ({\sigma }_x^2+{\sigma }_y^2)$ and finally we set:

$$\rho ={\sigma }_{\rho }$$

(13)

Now, since in calculating the coordinates $x$ and $y$ of $\overrightarrow{R}$ a deep–language variable can be summed twice or more, we add its standard deviation (referred to as sigma) twice or more times before squaring. For example, in the $x-$ component $I_P$ appears three times, therefore its contribution to the total variance in the $x-\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}$ is 9 times the variance calculated from the standard deviation reported in Table 6, Table 7, Table 8 and Table 9. For Lord, for example, it is $9\times {0.51}^2$. After these calculations, the values of the 1–sigma circle are transformed into the normalized coordinates $X, Y$ according to Eqs (10)(11).

Figure 4 shows a significant example involving Lord, Narnia, Trilogy, Back and Peter. We see that Lord can be almost fully confused with Narnia, and partially with Trilogy, but not vice versa. Lord can also be confused with Peter and Back, therefore indicacting strong connections with these earlier novels.

Now, we can estimate the (conditional) probability that a text is confused with another by calculating ratios of areas. This procedure is correct if we assume that the bivariate density of the normalized coordinates ${\rho }_X, {\rho }_Y$, centred at $\overrightarrow{R}$, is uniform. By assuming this hypothesis we can calculate probabilities as ratios of areas [21,22].

The hypothesis of substantially uniformity around $\overrightarrow{R}$ should be justified by noting that the coordinates $X, Y$ are likely distributed according to a log–normal bivariete density because the logarithm of the four deep–language variables, which combine in Eq.(9) linearly, can be modelled as Gaussian. For the central limit theorem we should expect approximately a Gaussian model on the linear values, but with a signficantly larger standard deviation that that of the single variables. Therefore, in the area close to $\overrightarrow{R}$ the bivariate density function should not be very peaked, hence the uniform density modelling.

Now we can calculate the following probabilities. Let $A$ be the common area of two 1–sigma circles (i.e., the area proportional to the joint probability of two texts), let $A_1$ be the area of 1–sigma circle of text 1 and $A_2$ the area of 1–sigma circle of text 2. Now, since probabilities are proportional to areas, we get the following relationships:

$$\frac{A}{A_1}=\frac{P(A_1,A_2)}{P\left(A_1\right)}=\frac{P(A_2/A_1)P\left(A_1\right)}{P\left(A_1\right)}=P(A_2/A_1)$$

(14)

$$\frac{A}{A_2}=\frac{P(A_1,A_2)}{P\left(A_2\right)}=\frac{P(A_1/A_2)P\left(A_2\right)}{P\left(A_2\right)}=P(A_1/A_2)$$

(15)

In other words, $A/A_1$gives the conditional probability $P(A_2/A_1)$ that part of text 2 can be confused (or “contained”) in text 1; $A/A_2$ gives the conditional probability $P(A_1/A_2)$that part of text 1 can be confused with text 2. Notice that these conditional probabilities depend on the distance between two texts and on the 1–sigma radii (Appendix C)

Of course, these joint probabilities can be extended to three or more texts, e.g., in Figure 4 we could calculate the area shared by Lord, Narnia and Trilogy and the corresponding joint probability, not done in the present paper.

We think that the conditional probabilities and the visual display of 1–sigma circles give useful clues to establish possible hidden connections between texts and, maybe, even between authors, because the variables involved are not consciously managed by them.

In Table 12 the conditional probability $P(A_2/A_1)$ is reported in the columns, therefore $A_1$ refers to the text indicated in the upper row; $P(A_1/A_2)$ is reported in the rows, therefore $A_2$ refers to the text indicated the left column.

Notice that $P(A_2/A_1)=1$ means $A=A_1$, therefore text 1 can be fully confused with text 2 and $P(A_1/A_2)=1$ means ${A=A}_2$, therefore text 2 can be fully confused with text 1.

For example, assuming Lord as text 1 (column 1 of Table 12) and Narnia as text 2 (row 3), we find $P(A_2/A_1)=0.974$. Viceversa, if we assume Narnia as text 1 (column 3) and Lord as text 2 (row 1), we find $P(A_2/A_1)=0.356$. These data indicate that Lord can be confused with Narnia with probability close to 1, but not vice versa. In other words, in the data bank considered in this paper, if a machine randomly extracts a chapter from Lord, another machine, unaware of this choice, could attribute it to Lord, of course, but also, with decreasing probability, to Back, Peter, Narnia and Lilith.

On the contrary, if the text is extracted from Narnia, then it is more likely attributed to Peter or Trilogy than to Lord or other texts.

Now, we can define a synthetic parameter which highlights how much, on the average, two texts can be erroneously confused with each other. The parameter is the average conditional probability (see [20] for a similar problem):

$$p_e=P(A_2/A_1)P\left(A_1\right)+P(A_1/A_2)P\left(A_2\right)$$

(16)

Now, since in comparing two texts we can assume $P\left(A_1\right)=P\left(A_2\right)=0.5$, we get:

$$p_e=0.5\times \left[P\right(A_2/A_1)+P(A_1/A_2\left)\right]$$

(17)

If $p_e=0$, there is no intersection between the two 1–sigma circles, the two texts cannot be each other confused, therefore there is no mathematical connection involving the deep–language parameters (this happens for Screwtape and Silmarillion, which can be each other confused, but not with the other texts). If $p_e=1$ the two texts can be totally confused, the two 1–sigma circles coincide. Appendix D reports the values of $p_e$ for all couples of novels.

Now, just to allow some rough analysis, it is reasonable to assume $p_e=0.5$ as a reference threshold, i.e., the probability of getting heads or tails in flipping a fair coin: if $p_e>0.5$, then two texts can be confused not by chance; if $p_e\le 0.5$, then two texts cannot be likely confused.

To visualize $p_e$, Figure 5 draws $p_e$when text 1 is Lord (column 1 of Table 12), Narnia (column 3) or Trilogy (column 4). We notice that$p_e>0.5$ in the following cases:

Lord as text 1: Narnia, Back, Lilith, Mutual, Peter.

Narnia as text 1: Lord, Trilogy, Back, Lilith, Bleak, Mutual, Martin, Peter.

Trilogy as text 1: Hobbit, Narnia, Bleak, Martin, Bask.

We can reiterate that Tolkien (Lord) appears significantly connected to Lewis (Narnia), to MacDonald (Back, Lilith) and Barrie (Peter), but not to Dicken’s novels as, on the contrary, does Lewis.

In the next Section the four deep–language variables are singled out to consider linguistic channels existing in texts. This is the analysis we have called the “fine tuning” of texts [12].

7. Linear Relationships in Literary Texts

The theory of linguistic channels, which we will recall in the next Section, is based on the regression line between linguistic variables:

$$y=mx$$

(18)

Therefore, we show examples of these linear relationships found in Lord and Narnia.

Figure 6a shows the scatterplot of $n_S$ versus $n_W$ of Lord and Narnia. In Narnia the slope of the regression line is $m=0.0729$, the correlation coefficient $r=0.7610.$In Lord, $m=0.0731$and $r=0.9199$, therefore the average relationships – i.e., Eq.(18) – are practically identical – see also the values of $<P_F>$ in Table 6 and Table 7 – while the correlation coefficients – i.e., the scattering of the data – are not and this fact will impact on the sentence channel discussed in Section 9.

Similar observations can be done for Figure 6b, which shows $n_I$ versus $n_S$ in Lord and Narnia. We find $m=2.0372,$ $r=0.9609$ in Lord, and $m=1.9520$and $r=$0.9384 in Narnia. Appendix E reports the complete set of these parameters.

Figure 7 shows the scatterplots of Lord and Trilogy. In Trilogy, for $n_S$ versus $n_W$ $m=0.0672$, $r=0.9325$; for $n_I$ versus $n_S$ $m=1.9664,r=0.9830$.

Figure 8 shows the scatterplots for Lord and Back or Lilith. We see similar regression lines and data scattering. In Back (left panel), the regression line between $n_S$ and $n_W$ gives $m=0.0681$, $r=0.9416$; in Lilith (right panel), $m=0.0676$, $r=$0.8890. These results likely indicate the influence of MacDonald on Tolkien’s writing because they are different from most other novels.

In conclusion, the regression lines of Lord, Narnia and Trilogy are very similar, but they can differ in the scattering of the data. Regression lines, however, describe only one aspect of the relationship, namely the relationship between conditional average values Eq. (18), they do not consider the other aspect of the relationship, namely the scattering of data, which may not be the same even when two regression lines almost coincide, as shown above. The theory of linguistic channels, discussed in the next Section, on the contrary, considers both slopes and correlation coefficients and provides a “fine tuning” tool to compare two sets of data by singling out each of the four deep–language parameters.

8. Theory of Linguistic Channels

In this section we recall the general theory of linguistic channels [12]. In a literary work, an independent (reference) variable $x$ (e.g., $n_W)$ and a dependent variable $y$ (e.g., $n_S)$ can be related by the regression line given by Eq. (18).

Let us consider two different text blocks $Y_k$ and $Y_j$, e.g., the chapters of work $k$and work $j$. Eq. (18) does not give the full relationship between two variables because it links only conditional average values. We can write more general linear relationships, which take care of the scattering of the data, – measured by the correlation coefficients $r_k$ and $r_j$, respectively – around the average values (measured by the slopes $m_k$ and $m_j$):

$$y_k=m_{k}x+n_k$$

(19)

$$y_j=m_{j}x+n_j$$

(20)

The linear models Eqs. (19)(20) introduce additive “noise” through the stochastic variables $n_k$ and $n_{j }$, with zero mean value [9,12,20]. The noise is due to the correlation coefficient $\left|r\right|\ne 1$.

We can compare two literary works by eliminating $x$, therefore we compare the output variable $y$ for the same number of the input variable $x$. For example, we can compare the number of sentences in two novels ‒ for an equal number of words ‒ by considering not only the average relationship, Eq. (18), but also the scattering of the data, measured by the correlation coefficient, Eqs. (19)(20). We refer to this communication channel as the “sentences channel”, S–channel and to this processing as “fine tuning” because it deepens the analysis of the data and can provide more insight into the relationship between two literary works, or any other texts.

By eliminating $x$, from Eqs. (19)(20) we get the linear relationship between, now, the input number of sentences in work $Y_k$ (now the reference, input text) and the number of sentences in text $Y_j$ (now the output text):

$$y_j=\frac{m_j}{m_k}{y}_k-\frac{m_j}{m_k}{n}_k+n_j$$

(21)

Compared to the new reference work $Y_k$, the slope $m_{jk}$is given by:

$$m_{jk}=m_j/m_k$$

(22)

The noise source that produces the correlation coefficient between $Y_k$ and $Y_j$ is given by:

$$n_{jk}=-\frac{m_j}{m_k}{n}_k+n_j={-m}_{jk}{n}_k+n_j$$

(23)

The “regression noise−to−signal ratio”, $R_{\mathrm{m}},$ due to $\left|m_{jk}\right|\ne 1$, of the new channel is given by:

$$R_m={(m_{jk}-1)}^2$$

(24)

The unknown correlation coefficient $r_{jk}$ between $y_j$ and $y_k$ is given by:

$$r_{jk}=\mathrm{c}\mathrm{o}\mathrm{s}\left|\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\left(r_j\right)-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\left(r_k\right)\right|$$

(25)

The “correlation noise−to−signal ratio”, $R_{\mathrm{r}}$, due to $\left|r_{jk}\right|<1$, of the new channel from text $Y_k$ to text $Y_j$ is given by:

$$R_r=\frac{1-r_{jk}^2}{r_{jk}^2}{m}_{jk}^2$$

(26)

Because the two noise sources are disjoint and additive, the total noise−to−signal ratio of the channel connecting text $Y_k$ to text $Y_j$ is given by:

$$R=R_m+R_r$$

(27)

Notice that Eq. (27) can be represented graphically [10]. Finally, the total and the partial signal−to−noise ratios are given by:

$${\mathsf{\Gamma }}_{dB}=-10\times {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\mathrm{R}$$

(28)

$${\mathsf{\Gamma }}_{m,dB}=-10\times {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{R}_m$$

(29)

$${\mathsf{\Gamma }}_{r,dB}=-10\times {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{R}_r$$

(30)

Of course, we expect that no channel can yield $\left|r_{jk}\right|=1$and $\left|m_{jk}\right|=1$, therefore ${\mathsf{\Gamma }}_{dB}=\infty$, a case referred to as the ideal channel, unless a text is compared with itself. In practice, we always find $\left|r_{jk}\right|<1$and $\left|m_{jk}\right|\ne 1$. The slope $m_{jk}$ measures the multiplicative “bias” of the dependent variable compared to the independent variable; the correlation coefficient $r_{jk}$measures how “precise” the linear best fit is.

In conclusion, the slope $m_{jk}$ is the source of the regression noise $R_m$, the correlation coefficient $r_{jk}$is mostly the source of the correlation noise of the channel $R_r$.

9. Linguistic Channels

In long texts (such as novels, essays etc.) we can define at least four linguistic linear channels [12], namely:

Sentence channel (S–channel)

Interpunctions channel (I–channel)

Word interval channel, WI–channel

Characters channel (C–channel).

In S‒channels, the number of sentences of two texts is compared for the same number of words. These channels describe how many sentences the author of text $j$ writes, compared to the writer of text $k$ (reference text), by using the same number of words. Therefore these channels are more linked to $P_F$ than to other parameters. Very likely they reflect the style of the writer.

In I‒channels, the number of word intervals of two texts is compared for the same number of sentences. These channels describe how many short texts between two contiguous punctuation marks (of length $I_P$) two authors use, therefore these channels are more linked to $M_F$ than to other parameters. Since $M_F$ is very likely connected with the E–STM, I‒channels are more related to the second buffer of readers‘ E–STM than to the style of the writer.

In WI‒channels, the number of words contained in a word interval (i.e., $I_P$) is compared for the same number of interpunctions. These channels are more linked to $I_P$ than to other parameters. Since $I_P$ is very likely connected with the E–STM, WI‒channels are more related to the first buffer of readers‘ E–STM than to the style of the writer.

In C‒channels, the number of characters of two texts is compared for the same number of words. They are more related to the language used, e.g., English, than to the other parameters, unless essays or scientific/academic texts are considered because these latter texts use, on the average, longer words [9].

As an example, Table 13 reports the total and the partial signal–to–noise ratios ${\mathsf{\Gamma }}_{dB}$, ${\mathsf{\Gamma }}_{m,dB}$, ${\mathsf{\Gamma }}_{r,dB}$ in the four channels by considering Lord as reference (input) text. In other words text $j$ is compared to text $k$ (reference text, i.e., Lord).

Appendix F reports ${\mathsf{\Gamma }}_{dB}$ for all novels considered in the paper.

Let us make some fundamental remarks on Table 13, applicable whichever is the reference text. The signal–to–noise ratios of C‒channels are practically the largest ones, ranging from 19.17 dB (Lilith) to 31.19 dB (Back). These results are simply saying that all authors use the same language and write texts of the same kind, novels not essays or scientific/academic papers. These channels are not apt to distinguish or assess large differences between texts or authors.

In the three other channels we can notice that Trilogy, Back and Lilith have the largest signal–to–noise ratios, about $~19 \mathrm{t}\mathrm{o} ~22$ dB, therefore these novels are very similar to Lord. In other words, these channels seem to confirm the likely influence by MacDonald on both Lord and Trilogy, and the connection between Lord and Trilogy.

On the contrary, Narnia shows poor values in the S–Channel (10.12 dB) and WI–Channel (7.94 dB). These low values are determined by the correlation noise because $R=R_m+R_r\approx R_r$. If we consider only ${\mathsf{\Gamma }}_{m,dB}$ – i.e., only the regression line – then we notice a strong connection with Lord since ${\mathsf{\Gamma }}_{m,dB}=51.26$ dB. As we have already observed regarding Figure 6, the regression lines are practically identical but the spreading of the data is not. Lewis in Narnia is less “regular” than in Trilogy or Tolkien in Lord in shaping (unconsciously) these two linguistic channels.

10. Summary and Conclusions

Scholars of the English Literature unanimously say that J.R.R. Tolkien and C.S. Lewis influenced each other writings. For the first time we have investigated this issue mathematically by using an original multi–dimensional analysis of linguistic parameters, based on the surface deep–language variables and linguistic channels.

To set our investigation in the framework of the English Literature, we have also considered some novels written by earlier authors, such as Charles Dickens and others, including George MacDonald, because scholars mention his likely influence on Tolkien and Lewis.

In our multidensional analysis, only the series of words, sentences and interpunctions per chapter, in our opinion, were consciously planned by the authors and, specifically, they do not indicate strong connections between Tolkien, Lewis and MacDonald. Each author distributed words, sentences and interpunctions differently from author to author and, sometimes, even from novel to novel of the same author.

On the contrary, the deep–language variables and the linguistic channels, discussed in the paper, are likely due to unconscious design and can reveal connections between texts far beyond writers’ awarness.

In summary, the buffers of the extended short–term memory required to readers, the universal readability index of texts, the geometrical representation of texts and the fine tuning of linguistic channels – all tools largely discussed in the paper – have revealed strong connections between The Lord of the Rings (Tolkien), The Cronicles of Narnia and The Space Triology (Lewis) on one side, and the strong connection also with some novels by MacDonald, on the other side, therefore substantially in agreement with what scholars of the English Literature say.