Let $n_C$, $n_W$ and $n_I$ be respectively the number of characters, words and interpunctions (punctuation marks) calculated in disjoint blocks of texts, such as chapters or any other subdivisions, then four deep-language variables are defined (Appendix A lists the mathematical symbols used in the present paper).

The number of characters per word, $C_P$:

$C_P=\frac{n_C}{n_W}$ (1)

The number of words per sentence, $P_F$:

$P_F=\frac{n_W}{n_S}$ (2)

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

$I_P=\frac{n_I}{n_W}$ (3)

The number of word intervals, $n_{I_P}$, per sentence, $M_F$:

$M_F=\frac{n_{I_P}}{n_S}$ (4)

Equation (4) can be written also as $M_F=P_F/I_P$. Table 3 & Table 4 report the

mean values of these variables, the other variables in Table 3 and Table 4 will be discussed in the following sections.

Notice that all mean values have been calculated by weighing each text with its number of words to avoid that shorter texts weigh statistically as much as long ones. In other words, any text considered weighs the number of its words, compared to the total number of words. We have also used this method to calculate the mean values of the data bank of Greek-1 plus Greek-2 (last line in Table 3 ). In this case, for example, the statistical weight of Aeneas Tactitian is 13035/3,225,839 ≈ 0.004 (see Table 1 ) while the weight of Aristotle is 509646/3,225,839 ≈ 0.1580.

The mean values of these variables can be calculated from the sample totals listed in Table 1 and Table 2 . However, do not be misled; these values are not equal to the arithmetic or to the statistical means, as we prove now.

Let $M$ be the number of samples (i.e., number of disjoint blocks of text, such as chapters or books), then, for example, the statistical mean value $\langle P_F\rangle$, is given by

$\langle P_F\rangle ={\displaystyle {\sum }_{k=1}^M{P}_{F,k}\times \left(n_{W,k}/W\right)}$ (5)

where $W={\displaystyle {\sum }_{k=1}^M{n}_{W,k}}$ is the total number of words. Notice that $\langle P_F\rangle \ne \frac{1}{M}{\displaystyle {\sum }_{k=1}^M{P}_{F,k}}\ne {\displaystyle {\sum }_{k=1}^M{n}_{W,k}}/{\displaystyle {\sum }_{k=1}^M{n}_{S,k}}=W/S$, where $S$ is the total number of sentences.

For example, for Aristotle $W=509646$ and $S=17790$, Table 1 . These values would give the average $P_F=W/S=509646/17790=28.65$, while the statistical mean (calculated on the nine books listed in Table 1 ) $\langle P_F\rangle =29.29>28.65$.

In general, the average values calculated from sample totals are always smaller than their statistical means, therefore they give lower bounds, as we prove in the following.

Let us consider, for example, the parameter $P_F$. Because of Chebyshev’s inequality ( [58] , inequality 3.2.7), we can write Equation (5) as:

$\langle P_F\rangle ={\displaystyle {\sum }_{k=1}^M\frac{n_{W_k}}{n_{S_k}}\frac{n_{W_k}}{W}}\ge \frac{1}{M}{\displaystyle {\sum }_{k=1}^M\frac{n_{W_k}}{n_{S_k}}}{\displaystyle {\sum }_{k=1}^M\frac{n_{W_k}}{W}}=\frac{1}{M}{\displaystyle {\sum }_{k=1}^M\frac{n_{W_k}}{n_{S_k}}}$ (6)

Equation (6) states that the mean calculated with samples weighted $1/M$ (arithmetic mean) is smaller than (or equal to) the mean calculated with samples weighted $n_{W_k}/W$.

Now, again for Chebyshev’s inequality, we get:

${\displaystyle {\sum }_{k=1}^M\frac{n_{W_k}}{n_{S_k}}}\ge \frac{1}{M}{\displaystyle {\sum }_{k=1}^M{n}_{W_k}}{\displaystyle {\sum }_{k=1}^M\frac{1}{n_{S_k}}}=\frac{W}{M}{\displaystyle {\sum }_{k=1}^M\frac{1}{n_{S_k}}}$ (7)

Further, for Cauchy-Schawarz’s inequality (or by the fact that the harmonic mean is less than, or equal to, the arithmetic mean), we get:

${\displaystyle {\sum }_{k=1}^M\frac{1}{n_{S_k}}}\ge \frac{M^2}{{\displaystyle {\sum }_{k=1}^M{n}_{S_k}}}$(8)

Finally, by inserting these inequalities in (6), we get:

$\langle P_F\rangle \ge \frac{W}{M^2}\frac{M^2}{{\displaystyle {\sum }_{k=1}^M{n}_{S_k}}}=\frac{W}{S}$(9)

Equation (9) establishes that the statistical mean calculated with samples weighted $n_{W_k}/W$ is greater than (or equal to) the average calculated with sample totals. The values given by these three methods of calculation coincide only if all texts are perfectly identical, i.e., with the same number of characters, words, sentences and interpunctions, a case improbable.

The mean values of Table 3 and Table 4 (or their minimum values directly calculated from the totals, as discussed above) are useful for a first assessment of how “close”, or similar, mathematically, texts are by defining linear combinations of deep-language variables [1] . Texts are modeled as vectors, the representation of which is briefly recalled in the next section.

In texts, we can always define at least four linguistic linear channels [3] [11] , namely:

1) Sentence channel (S-channel)

2) Interpunctions channel (I-channel)

3) Word interval channel (WI-channel)

4) Characters channel (C-channel).

In S‒channels, the number of sentences of two texts is compared for the same number of words. Notice that, as far as we know, only the theory of lingusitic channels allows this comparison. 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 the other variables. Very likely they reflect the style of the writer.

In I‒channels, the number of word intervals $I_P$'s 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 $I_P$ words) two authors use; therefore, these channels are more linked to $M_F$ than to the other variables. Since $M_F$ is connected to 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 (i.e., $I_P$) contained in a word interval is compared for the same number of interpunctions. These channels are more linked to $I_P$ than to other variables, therefore 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. These channels are more related to the language used, e.g. Greek in this case, than to the other variables.

An independent (reference) variable $x$ (e.g., $n_W$ in S-channels) and a dependent variable $y$ (e.g., $n_S$) can be related by a regression line (slope $m$) passing through the origin:

$y=mx$(12)

Let us consider two texts $Y_k$ and $Y_j$. For we can write Equation (12) for the same couple of parameters. In both cases, Equation (12) does not give their full relationship because it only connects the mean conditional values. More general linear relationships must also consider the scattering of the data—measured by the correlation coefficients $r_k$ and $r_j$—around the regression lines (slopes $m_k$ and $m_j$):

$y_k=m_{k}x+n_k$(13)

$y_j=m_{j}x+n_j$

Equation (12) connects $y$ to $x$only on the average. Equation (13) introduces additive “noise” $n_k$ and $n_j$, with zero mean value. The noise is due to the correlation coefficient $\left|r\right|\ne 1$.

In these channels, which consider two scatterplots, we compare two texts by eliminating $x$, e.g., the number of sentences in two texts—for an equal number of words—by considering not only the mean relationship but also the scattering of the data.

As recalled before, we refer to this communication channel as the “sentences channel” and to this processing as “fine tuning” because it deepens the analysis of the data and provides more insight into the relationship between two texts. The mathematical theory follows.

By eliminating $x$, from Equation (13) we obtain the linear relationship between—now—the sentences in text $Y_k$ (reference, input text) and the sentences in text $Y_j$ (output text):

$y_j=\frac{m_j}{m_k}{y}_k-\frac{m_j}{m_k}{n}_k+n_j$(14)

Compared with the independent (input) text $Y_k$, the slope $m_{jk}$ is given by

$m_{jk}=\frac{m_j}{m_k}$(15)

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

$n_{jk}=-\frac{m_j}{m_k}{n}_k+n_j=-m_{jk}{n}_k+n_j$ (16)

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

$R_m={\left(m_{jk}-1\right)}^2$(17)

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

$r_{jk}=\text{cos}\left|\text{arcos}\left(r_j\right)-\text{arcos}\left(r_k\right)\right|$(18)

The “correlation noise-to-signal ratio”, $R_r$, due to $\left|r_{jk}\right|<1$, of the channel that connects the input text $Y_k$ to the output text $Y_j$ is given by:

$R_r=\frac{1-r_{jk}^2}{r_{jk}^2}{m}_{jk}^2$(19)

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

$R={\left(m_{jk}-1\right)}^2+\frac{1-r_{jk}^2}{r_{jk}^2}{m}_{jk}^2$(20)

Finally, the total signal-to-noise ratio is given by

$\gamma =1/R$(21)

$\text{Γ}=10\times {\log }_{10}\gamma$

$\text{Γ}$ is in dB.

Notice that no channel can yield $\left|r_{jk}\right|=1$ and $\left|m_{jk}\right|=1$ (i.e., $\text{Γ}=\infty$), a case referred to as the ideal channel, unless a text is compared with itself (self-comparison, self-channel). 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. The slope $m_{jk}$ is the source of the regression noise of the channel, the correlation coefficient $r_{jk}$ is the source of the correlation noise. Finally, notice that since the probability of finding $\left|r\right|=1$ and $m=1$ is practically zero, all channels are always noisy.

To clarify what we mean by a single scatterplot and one-to-one correspondence, let us consider the translation of a text in which we can draw a scatterplot for each linguistic variable. For example, we can display the number of words per chapter in the translated text versus that of the original text. Now, if there were a perfect ideal translation, the translated text would have the same number of words per chapter of the original text, therefore in this case $m=1$ and $r=1$ because the scatterplot would collapse to the deterministic 45˚C linear relationship between $x$ and $y$:

$y_k=x$ (22)

Now, if in the expressions of the previous sub-section we set $m_k=1$ and $r_k=1$ (hence, $n_k=0$), we study a special case of the general theory for which we get the following expressions:

$m_{jk}=m_j$ (23)

$R_m={\left(m_j-1\right)}^2$ (24)

$r_{jk}=\text{cos}\left|\text{arcos}\left(r_j\right)-0\right|=r_j$ (25)

$R_r=\frac{1-r_j^2}{r_j^2}{m}_j^2$ (26)

$R={\left(m_j-1\right)}^2+\frac{1-r_j^2}{r_j^2}{m}_j^2$ (27)

$\gamma =\frac{1}{R}$ (28)

In other words, we study how the determistic relationship (22), which describes a noiseless channel, is transformed into the experimental relationship of the noisy channel, because now $m_j\ne 1$ and $\left|r_{jk}\right|<1$.

In conclusion, we can study a single scatterplot with the tools of the general theory, therefore the signal-to-noise ratio $\gamma$ is still a single index that synthetically describes the relationship between $x$ and $y$.

In the next section we study the four channels mentioned above.

Table 5 reports, for example, Γ in the S-channels. Appendices B and C report Γ for the other three channels. Table 5 is interpreted as follows. The author/text in the first row is the reference author/text, i.e., the channel input author/text $Y_k$ of the theory; the author/text in the first column is the channel dependent output author/text $Y_j$. For example, if Aristides is the input and Demosthenes is the output, then $\text{Γ}=22.59$ dB ( $\gamma =181.55$); viceversa, if Demosthenes is the input and Aristides is the output, then $\text{Γ}=23.16$ ( $\gamma =207.01$), a small asymmetry always found in linguistic channels [3] . In other words, for the same number of words, the number of sentences in Aristides is transformed into the number of sentences, for the same number of words in Demosthenes with a high Γ, and viceversa. This finding means the two texts share a common style very much, as far as sentences are concerned. The channel is little noisy, the regression line that relates $n_S$ of Demosthenes (dependent variable) to $n_S$ of Aristides (independent variable) has $m_{jk}=1.0392$ and $r_{jk}=0.9982$. Now, in this example since Aristides lived before Demosthenes the large Γ may indicate that Aristides influenced Demosthenes’ style. In any case, the two texts are much correlated in the S-Channel.

The red and blue colours in Table 5 highlight the channels with $\text{Γ}\ge 15$ dB ( $\gamma \ge 31.62$), with the following meaning: blue indicates not only that the number of sentences of the input and output texts are much correlated but also that the input author might have influenced the output author because he lived before. Red indicates a large correlation, as in the blue cases, but no likely influence can be supposed because the input author lived after the output author. Similar observations can be made for the other authors/texts and linguistic channels (see Appendix B).

Figure 2 syntesizes the results of the four channels by showing the average Γ calculated by considering the input author (left panel, arithmetic average of the values reported in the corresponding column of Table 5 ) or the output author

(right panel, arithmetic average of the values reported in the corresponding row). The asymmetry typical of linguisic channels is clearly evident.

For example, Aristides (no. 3) has large Γ both when he is the input author (left panel) and when he is the output author (right panel). The authors who are very uncorrerated, along with all others, are Plato (no. 9) and Thucydides (no. 13).

From Figure 2 , we can conclude that:

1) C-Channels (green line) give large Γ for all authors, in any case. These large values are just saying that all authors use the same language because $C_p$ changes little from author to author. The minimum is found with Aristotle (no. 4) which is not a historian or geographer like the other authors. These channels are not very apt to distinguish or assess large differences between texts or authors [11] .

2) S-Channels (red line) and WI-Channels (magenta line) are the most similar. This may be due to the fact that both are linked to the E-STM capacity (see Section 8).

3) I-Channels (blue line) give Γ just smaller than that of C-Channels. I-Channels deal with $I_P$, therefore the word interval used by all authors is not very different (see Table 3 and Section 8).

Table 6 shows the results in the S-channel for Greek-2, Appendix C reports Γ for the other three channels. We can notice that the cases of similarity or likely dependence are very few. Sofocles may be influenced by Aeschylus, and Pindarus by the writer of Odissey therefore confirming their closeness in Figure 1 .

Notice that Iliad and Odissey have significant different Γ in the three channels able to distinguish better authors/texts. They are also distant in Figure 1 . Now, modern scholars generally agree that Homer composed the Iliad most likely relying on oral traditions, and at least inspired the composition of the Odyssey but did not write it [59] .

Figure 3 syntesizes the results of the four channels of Greek-2. We notice that these channels are less correlated that those of Greek-1, therefore texts are significantly different, therefore the literary gerne affects these channels (details are reported in Appendix C). C-Channels (green line) give the largest Γ, in the same range of Greek-1 because the authors use the same language.

In this sub-section, we show how to apply the single scatterplot theory (Section 5.3) by considering, as an example, the translation of Iliad from Greek to Italian in the classical translation done by Ippolito Pindemopnte (1753-1828), an Italian poet. Specifically, we compare the number of words per Book, 24 samples. Figure 4 . Shows the scatterplot, the linear best fit, with $r_j=0.9836$ and $m_j=1.2215$, therefore $\text{Γ}=10.03$ dB, a rather poor value due equally to the correlation ( $R_r=0.0502$) and regression ( $R_m=0.0491$) noise.

We can notice that the number of words in Italian is always larger than in Greek, therefore the translator seems to need more words to explain the original meaning, with an average 22.15% per Book. The total number of words in Italian is 136,050 against 111,878 in Greek (21.6% increase).

In the next section, we will estimate the readability of these authors by considering a universal readability index.

Figure 7 shows $\langle I_P\rangle$ versus $\langle P_F\rangle$ and the non-linear best-fit regression curves for Greek-1, Greek-2 and NT. As we have already established in modern languages and Latin [1] [2] [4] , if $\langle P_F\rangle$ increases $\langle I_p\rangle$ tends to approach a horizontal asymptote. In other words, even if a sentence gets longer, $\langle I_p\rangle$ cannot become larger than about the upper limit of 7 ± 2 Miller’s law (namely about 9), because of the constraints imposed by the E-STM capacity of readers and writers.

The coincidence of $\langle I_P\rangle$ with the bounds of Miller’s law is clearly evident in Figure 6 , just like in modern languages as the best-fit curves found in Italian and English novels [9] , in modern languages [4] —also drawn in Figure 6 —clearly show.

From Figure 6 , we can draw the following conclusions.

1) There is a marked distinction between the regression curves concerning Greek-1, Greek-2 and NT.

2) The regression curves of Italian and English, which refer only to novels, agree very well with the regression curves of Greek-2 and NT.

3) Greek-1 is clearly mathematically different of Greek-2. The difference between novels and other types of writings, such as essays, was clearly found also in Italian writers as well [1] .

Figure 8 shows the scatterplot between $M_F$ and $I_P$ for the samples of the entire data bank used to calculate the statistical means of Figure 6 . The horizontal green line reports the unconditional statistical mean $\langle M_F\rangle$, the black line reports the conditional mean versus $I_P$.

Now, the correlation coefficient between $I_P$ and $M_F$ in Figure 8 is practically zero (namely 0.03). The probability density of $I_P$ samples ( Figure 9 , left panel) and $M_F$ samples ( Figure 9 , right panel) can be modelled with a three-parameter log-normal density function—because $I_P\ge 1$, $M_F\ge 1$—as in Italian and English [9] . Since a bivariate log-normal density function can be a sufficiently good model for the joint density of $\text{log}\left(I_P\right)$ and $\text{log}\left(M_F\right)$, at least in the central part of the marginal distributions, it follows that, if the correlation coefficient is zero, $\text{log}\left(I_P\right)$ and $\text{log}\left(M_F\right)$ are not only uncorrelated but also independent in the Gaussian case. Therefore, $I_P$ and $M_F$ are also independent and the two processing units of the E-STM work sufficiently independently, as with modern readers.

The size of the second E-STM buffer is in the same range of modern languages, as the bulk of the data in Figure 8 is the range from $M_F\approx 2$ to $M_F\approx 6$ word intervals per sentence.

In conclusion, these texts were conjecturally processed by a two-unit E-STM very similar to the E-STM of modern readers, even though these ancient readers were more accustomed to memorize oral information than modern ones. The specific size of the two buffers required in reading a text depended only on the genre, as for modern readers.