<p>The Saros series scheme for numbering eclipses is based on the <strong>Saros–Inex Panorama</strong>.</p>
<p>The Key idea is this:</p>
<ul>
<li>After one Saros (223 months) the moon returns nearly exactly to the same node</li>
<li>After one Inex (358 months) the moon returns very nearly exactly to the opposite node.</li>
<li>Any other eclipse period (or month interval) can be expressed as a linear combination of the Saros and the Inex. This is because the numbers 223 and 358 are relatively prime.</li>
</ul>
<p>We can express the relation as follows:
<span class="math-container">$$T = 223n + 358m $$</span>
Where <span class="math-container">$T$</span> is some interval of months between eclipses, and <span class="math-container">$n$</span> and <span class="math-container">$m$</span> represents the number of Saros or Inex we have to jump forwards or backwards (positive or negative) to arrive at our chosen interval.</p>
<p>This equation is what is referred to in mathematics as a <em>Linear Diophantine Equation</em>, and its can be solved quite easily for <span class="math-container">$n$</span> and <span class="math-container">$m$</span> using the <em>Extended Euclidean Algorithm</em>. We can make use of <a href="https://www.math.uwaterloo.ca/%7Esnburris/cgi-bin/linear-query" rel="nofollow noreferrer">an online calculator</a> which implements this algorithm.</p>
<p>Some example solutions are:
<span class="math-container">$$6 = 223(-8) + 358(5)$$</span>
<span class="math-container">$$5 = 223(53) + 358(-33)$$</span>
<span class="math-container">$$1 = 223(-61) + 358(38) $$</span></p>
<p>These are the solutions for 6-month, 5-month, and 1-month intervals respectively.</p>
<p>If two eclipses are separated by an interval <span class="math-container">$T$</span>, the <strong>Saros number</strong> is defined in such a way that the Saros number for the next eclipse is given by adding the value of <span class="math-container">$m$</span> to the current Saros number.</p>
<p>So, if the next eclipse occurs 6 months after the current eclipse, we add <span class="math-container">$m=5$</span> to find the current Saros number. For example, the lunar eclipse of April 24,2005 is numbered 141 in the Saros scheme. The next eclipse, of October 17,2005, is separated from it by a 6-month interval, so its Saros number is <span class="math-container">$141 + 5 = 146$</span>. If they were separated by a 5-month interval instead then we use <span class="math-container">$m =-33$</span>, or for 1-month interval, <span class="math-container">$m=38$</span> (according to the solutions).</p>
<p>Now, regarding your question: Why does the particular pattern appear between eclipses separated by Saros number 40.</p>
<p>The first thing to note is that we can combine different intervals to arrive at new solutions. In particular we can take the 6-month interval and multiply all values by <span class="math-container">$8$</span>. This will give us a new solution:
<span class="math-container">$$48 = 223(-64) + 358(40)$$</span></p>
<p>This immediately gives us an explanation for why the eclipses separated by <span class="math-container">$40$</span> Saros are found to be ~1418 days apart, because an interval of 48 months is given by <span class="math-container">$48\times29.53 \approx 1417.44$</span> days <span class="math-container">$\approx 3.88$</span> years, which in turn corresponds to a value <span class="math-container">$m=40$</span></p>
<p>The next thing to note is that the value <span class="math-container">$n=-64$</span>. Now <span class="math-container">$n$</span> indicates the amount we jump backwards or forwards between eclipses in the same Saros series. However, eclipses in different saros series do not all start at the same Inex.  This can be easily seen if we take a look at a diagram of the Saros–Inex Panorama.</p>
<p>The following diagram of the Saros–Inex Panorama for solar eclipses from the <a href="https://eclipse.gsfc.nasa.gov/SEsaros/SEsaros.html" rel="nofollow noreferrer">NASA GSFC</a>. Each Saros series run along the vertical axis. We can see that each series starts or end at different points. In fact, the starting and end points appear to vary in a sawtooth wave-like pattern!
<a href="https://i.sstatic.net/pzHPs1Af.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pzHPs1Af.jpg" alt="Saros-Inex Panorama diagram of eclipses - from Fred Espenak, NASA GSFC" /></a></p>
<p>A saros series lasts on average 72 cycles; this means the value <span class="math-container">$n=-64$</span> should roughly correspond to jumping from the end of one Saros series, 64 steps backwards, to the beginning of another. There is variability however due to the different lengths of Saros Series and different starting and end points. So different saros series have different values for <span class="math-container">$n$</span> corresponding to the different start an finish times.</p>
<p>For example:</p>
<ul>
<li>between Lunar Saros 129, and Lunar Saros 89 there are 48 months ~ 4 years, corresponding to the value <span class="math-container">$n= 64$</span>.</li>
<li>between Lunar Saros 89 and 49 <span class="math-container">$-175$</span> months ~ <span class="math-container">$-14$</span> years, corresponding to <span class="math-container">$n=-62$</span>; here the Series overlaps.</li>
<li>between Lunar Saros 49 and Lunar Saros 9, there are <span class="math-container">$-621$</span> months ~ <span class="math-container">$-50$</span> years, corresponding to <span class="math-container">$n=-67$</span></li>
</ul>
<p>In general, we can check how different values of <span class="math-container">$n$</span> would affect the resulting interval of time between Saros Series separated by <span class="math-container">$40$</span> by substituting them in the expression, or equivalently, adding or subtracting 223 months from our initial solution with <span class="math-container">$n=-64$</span>. We get the following</p>
<ul>
<li><span class="math-container">$n=-63$</span> would give <span class="math-container">$T=271$</span> months or ~<span class="math-container">$22$</span> years.</li>
<li><span class="math-container">$n=-64$</span> would give <span class="math-container">$T=48$</span> months or ~<span class="math-container">$4$</span> years.</li>
<li><span class="math-container">$n=-65$</span> would give <span class="math-container">$T=-175$</span> months or ~<span class="math-container">$-14$</span> years, which would mean the series overlaps.</li>
<li>and the pattern continues…</li>
</ul>
<p>In the end, there is nothing particularly special about the number <span class="math-container">$40$</span> between Saros numbering here, other than the fact that <span class="math-container">$40$</span> Inexes ~ <span class="math-container">$64$</span> Saroses, and <span class="math-container">$64$</span> in turn is nearly the same as the length of an average Saros Series. In fact, we can find a better approximation in <span class="math-container">$43$</span> Inexes ~ <span class="math-container">$69$</span> Saroses. This corresponds to the solution:</p>
<p><span class="math-container">$$7 = 223(-69) + 358(43)$$</span></p>
<p>As for the apparent relation between 4 Metonic cycles and 4 Saros, this might just be a coincidence: <span class="math-container">$(235-223)\times 4 = 48$</span></p>

CC BY-SA 4.0
2024-11-08T12:52:48.817Z
2024-11-08T12:52:48.817Z
Ryan Cole
<p>It's been 3 years and I wondered if you found further answers on this.
It's sad the answers here don't relate to your question. I too wanted to find this out.</p>
<p>On the non overlapping, the 12 cycles are deducted from saros 120, so 83-12 = 71 eclipses or 70 cycles of 1260 years.
thus these same 12 eclipses will not be deducted within saros 80 since it's &quot;accounted&quot; for in saros 120.
For Saros 80, Saros 40 last till -71; so the first 2 eclipses in saros 80 is overlapped and not counted so it'll be 74-2 = 72 eclipses.</p>
<p>I wondered if there's a way to exclude/include &quot;overlapped periods&quot; between eclipses to make it constant at 70 cycles of 1260 years for each saros.</p>
<p>This is because whenever a saros series is only 71 eclipses/70 cycles of 1260 years, no overlapping occur, so it is sort of &quot;standalone&quot;/constant.</p>
<p>It still puzzles me why the saros numbering with interval of 40 able to lead to this linkage of saroses via 1418 days.
there must be an obscure cycle in the interval of rising of saros series.</p>

CC BY-SA 4.0
2021-09-07T19:20:01.193Z
Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/129366/discussion-on-answer-by-emotionallyattached-are-saros-series-separated-by-40-i).
called2voyage
CC BY-SA 4.0
2021-08-26T21:16:23.447Z
2021-08-26T21:16:23.447Z
emotionallyattached
<p>Can anyone help with this question? My question arose from an eclipse related observation that I made while looking into lunar Saros series 129. It is the Saros series number for the lunar eclipse that occurred on this past July 27, 2018.</p>
<p>First I saw that this Saros 129 series was &quot;born&quot; June 10, 1351 and the series will end June 24, 2613 (70 saros cycles of 18 years = roughly 1260 years). So, I became curious to see if another Saros series had ended just before Saros 129 began. I saw that Saros 89 (less by 40 in series numbering) had ended just four years earlier on July 23, 1347. Investigating further, I learned that there are roughly 40 different Saros series that are &quot;active&quot; at any one given time. So then I looked back further to lunar Saros 49 and observed that there does indeed seem to be a pattern involving roughly &quot;40&quot; less in the numbering (i.e. yes indeed, series 49 also ended just as series 89 began). These observations do seem to me as more than mere coincidences.</p>
<p><strong>So, my question now is - Are these particular lunar Saros series (<em>separated by &quot;40&quot; in numbering</em>) related in any way? - series 49, 89, and 129. Is there another cycle or a &quot;greater&quot; cycle also at work here that I am unaware of?</strong></p>
<p>Please see the comments below if what I am asking seems unclear - my comments might rephrase the question for clarity.</p>
<blockquote>
<p><strong>Edit</strong> - I checked several other Saros series numbers separated by 40. I am adding additional information below - hoping that it might help find an answer to this question about relatedness of the &quot;40
apart&quot; series.</p>
</blockquote>
<p>Checking other series spaced by 40 numbering, these Saros series (that are numbered 40 apart – i.e. 40, 80, 120 etc.) always contain eclipses that are separated by <strong>~1418 days</strong> between eclipses in each respective series (i.e. exactly 4 lunar years/48 lunar months).</p>
<p><strong>Small Saros series</strong> (those containing fewer cycles - i.e. 70-74 saros cycles in the series) contain saros cycles that <strong>overlap minimally</strong> (Saros 129 which is only <strong>70 eclipse cycles in length has no overlap</strong> and began just after Saros 89 came to an end <strong>1418 days earlier</strong>). Note - zero eclipse cycles between them overlap.</p>
<p><strong>Large Saros series</strong> (those containing more cycles – i.e. 80-84 saros cycles in the series) contain saros cycles that <strong>overlap accordingly</strong> (Saros 120 which is 83 cycles long thus began 13 cycles before Saros 80 came to an end with the overlapping eclipses each gapped <strong>by 1418 days</strong>). Note - 13 eclipse cycles overlap accordingly - again about <strong>70 eclipse cycles with no overlap</strong> between these series.</p>
<p>It seems like about 70 eclipse cycles (constituting ~1262 solar years or ~1300 lunar years) is the standard length of each adjacent Saros series – considering the <strong>non-overlapping portion only</strong> – each non-overlapping portion separated by 4 lunar years.</p>
<blockquote>
<p><strong>Edit</strong> - Thanks to anyone who can shed any additional light on this.</p>
<p><strong>Additional Edit</strong> - In response to the given answer concerning the &quot;puzzling&quot; 1418 days observed between every &quot;standard saros series of
70 cycles&quot;, I have also noted that the observed &quot;1418 day difference&quot;
between them appears to be the <em>same difference</em> that exists between
every &quot;4 Saros cycles of approx 18.03 years <em>and</em> 4 Metonic cycles of
approx 19.00 years&quot; - i.e (4 x 19.00 x 365.25 days) - (4 x 18.03 x
365.25 days) =  1417.17 days - equaling four lunar years difference (<em>which is the actual basis for reconciling the solar and lunar
cycles</em>). I suspect that these two distinct observations are therefore
somehow related.</p>
</blockquote>

CC BY-SA 4.0
2018-07-26T16:42:19.050Z
Saros cycles are the same length of time. Saros series are not. What governs the fact that there are 40 numbered series "active" at any given time? How frequently are new Saros series "born".? Is there a pattern that governs the new ones? I actually went back to Saros 9 for further comparison.  These are not just random Saros numbers.
user22542
CC BY-SA 4.0
2018-07-26T21:02:06.330Z
Another way to say it is this: - consider - 40 new Saros series are born during the lifespan of the first Saros series. Each new Saros series appears to begin with a wide range of spacing between them, but that is about one new series beginning (on average) every 31.5 years. Each Saros cycle within a series is always 18 yrs  and 11.3 days.
user22542
CC BY-SA 4.0
2018-07-26T21:12:08.320Z
Rephrased question ------ So then, what BEGINNING criteria lead to the birth of a new SERIES, and are the BEGINNING criteria somehow linked to the ENDING criteria of another series (40 earlier) in the numbering system?
user22542
CC BY-SA 4.0
2018-07-29T19:52:04.060Z
Not 40-year pattern - repeat - Not 40-year pattern ---I am asking about 40 greater-numbering difference pattern.  The End of one SERIES possibly leads to the Beginning of another SERIES (which has a 40 greater number value for the NEW SERIES - roughly 1260 years later). ARE THEY RELATED----hypothetically---by the cosmic conditions that ended the first and gave rise to the second?
user22542
CC BY-SA 4.0
2018-07-30T18:59:00.313Z
Related: [Spacing (2422 days) between solar eclipses - unrelated saros numbers](https://astronomy.stackexchange.com/questions/25749/spacing-2422-days-between-solar-eclipses-unrelated-saros-numbers)
Mike G
CC BY-SA 4.0
2018-07-30T21:51:29.653Z
Hi Mike. Thanks for your previous answer. I'm not sure, but I don't think this question is related though. This is about lunar eclipses Saros series mainly.  Just another question that came up along the way. These cycles are fascinating, but I really can't visualize what is happening when one Saros series is ending while another is beginning. Why exactly 40 Saros series active at one particular time? The series charts don't give a clue about the actual state of affairs in the solar system etc.. Is my question unreasonable?
user22542
<the-moon><solar-system><observational-astronomy><solar-eclipse><lunar-eclipse>
Are Saros series separated by "40" in numbering related in any way?
user22542
# Are Saros series separated by "40" in numbering related in any way?

(Asked by: username_0 on 2018-07-26)

Can anyone help with this question? My question arose from an eclipse related observation that I made while looking into lunar Saros series 129. It is the Saros series number for the lunar eclipse that occurred on this past July 27, 2018.

First I saw that this Saros 129 series was "born" June 10, 1351 and the series will end June 24, 2613 (70 saros cycles of 18 years = roughly 1260 years). So, I became curious to see if another Saros series had ended just before Saros 129 began. I saw that Saros 89 (less by 40 in series numbering) had ended just four years earlier on July 23, 1347. Investigating further, I learned that there are roughly 40 different Saros series that are "active" at any one given time. So then I looked back further to lunar Saros 49 and observed that there does indeed seem to be a pattern involving roughly "40" less in the numbering (i.e. yes indeed, series 49 also ended just as series 89 began). These observations do seem to me as more than mere coincidences.

**So, my question now is - Are these particular lunar Saros series (*separated by "40" in numbering*) related in any way? - series 49, 89, and 129. Is there another cycle or a "greater" cycle also at work here that I am unaware of?**

Please see the comments below if what I am asking seems unclear - my comments might rephrase the question for clarity.

> **Edit** - I checked several other Saros series numbers separated by 40. I am adding additional information below - hoping that it might help find an answer to this question about relatedness of the "40
> apart" series.

Checking other series spaced by 40 numbering, these Saros series (that are numbered 40 apart – i.e. 40, 80, 120 etc.) always contain eclipses that are separated by **~1418 days** between eclipses in each respective series (i.e. exactly 4 lunar years/48 lunar months).

**Small Saros series** (those containing fewer cycles - i.e. 70-74 saros cycles in the series) contain saros cycles that **overlap minimally** (Saros 129 which is only **70 eclipse cycles in length has no overlap** and began just after Saros 89 came to an end **1418 days earlier**). Note - zero eclipse cycles between them overlap.

**Large Saros series** (those containing more cycles – i.e. 80-84 saros cycles in the series) contain saros cycles that **overlap accordingly** (Saros 120 which is 83 cycles long thus began 13 cycles before Saros 80 came to an end with the overlapping eclipses each gapped **by 1418 days**). Note - 13 eclipse cycles overlap accordingly - again about **70 eclipse cycles with no overlap** between these series.

It seems like about 70 eclipse cycles (constituting ~1262 solar years or ~1300 lunar years) is the standard length of each adjacent Saros series – considering the **non-overlapping portion only** – each non-overlapping portion separated by 4 lunar years.

> **Edit** - Thanks to anyone who can shed any additional light on this.
>
> **Additional Edit** - In response to the given answer concerning the "puzzling" 1418 days observed between every "standard saros series of
> 70 cycles", I have also noted that the observed "1418 day difference"
> between them appears to be the *same difference* that exists between
> every "4 Saros cycles of approx 18.03 years *and* 4 Metonic cycles of
> approx 19.00 years" - i.e (4 x 19.00 x 365.25 days) - (4 x 18.03 x
> 365.25 days) = 1417.17 days - equaling four lunar years difference (*which is the actual basis for reconciling the solar and lunar
> cycles*). I suspect that these two distinct observations are therefore
> somehow related.

**Tags:** the-moon, solar-system, observational-astronomy, solar-eclipse, lunar-eclipse

> **Comments:**
> **username_0**: Saros cycles are the same length of time. Saros series are not. What governs the fact that there are 40 numbered series "active" at any given time? How frequently are new Saros series "born".? Is there a pattern that governs the new ones? I actually went back to Saros 9 for further comparison.  These are not just random Saros numbers.  
> **username_0**: Another way to say it is this: - consider - 40 new Saros series are born during the lifespan of the first Saros series. Each new Saros series appears to begin with a wide range of spacing between them, but that is about one new series beginning (on average) every 31.5 years. Each Saros cycle within a series is always 18 yrs  and 11.3 days.  
> **username_0**: Rephrased question ------ So then, what BEGINNING criteria lead to the birth of a new SERIES, and are the BEGINNING criteria somehow linked to the ENDING criteria of another series (40 earlier) in the numbering system?  
> **username_0**: Not 40-year pattern - repeat - Not 40-year pattern ---I am asking about 40 greater-numbering difference pattern.  The End of one SERIES possibly leads to the Beginning of another SERIES (which has a 40 greater number value for the NEW SERIES - roughly 1260 years later). ARE THEY RELATED----hypothetically---by the cosmic conditions that ended the first and gave rise to the second?  
> **username_1**: Related: [Spacing (2422 days) between solar eclipses - unrelated saros numbers](https://astronomy.stackexchange.com/questions/25749/spacing-2422-days-between-solar-eclipses-unrelated-saros-numbers)  
> **username_0**: Hi username_1. Thanks for your previous answer. I'm not sure, but I don't think this question is related though. This is about lunar eclipses Saros series mainly.  Just another question that came up along the way. These cycles are fascinating, but I really can't visualize what is happening when one Saros series is ending while another is beginning. Why exactly 40 Saros series active at one particular time? The series charts don't give a clue about the actual state of affairs in the solar system etc.. Is my question unreasonable?  

---

## 2 answers

### Answer by username_3 on 2021-08-26. Score: 4 | Accepted answer

It's been 3 years and I wondered if you found further answers on this.
It's sad the answers here don't relate to your question. I too wanted to find this out.

On the non overlapping, the 12 cycles are deducted from saros 120, so 83-12 = 71 eclipses or 70 cycles of 1260 years.
thus these same 12 eclipses will not be deducted within saros 80 since it's "accounted" for in saros 120.
For Saros 80, Saros 40 last till -71; so the first 2 eclipses in saros 80 is overlapped and not counted so it'll be 74-2 = 72 eclipses.

I wondered if there's a way to exclude/include "overlapped periods" between eclipses to make it constant at 70 cycles of 1260 years for each saros.

This is because whenever a saros series is only 71 eclipses/70 cycles of 1260 years, no overlapping occur, so it is sort of "standalone"/constant.

It still puzzles me why the saros numbering with interval of 40 able to lead to this linkage of saroses via 1418 days.
there must be an obscure cycle in the interval of rising of saros series.

> **Comments:**
> **username_4**: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/129366/discussion-on-answer-by-username_3-are-saros-series-separated-by-40-i).  

---

### Answer by username_2 on 2024-11-08. Score: 1

The Saros series scheme for numbering eclipses is based on the **Saros–Inex Panorama**.

The Key idea is this:

* After one Saros (223 months) the moon returns nearly exactly to the same node
* After one Inex (358 months) the moon returns very nearly exactly to the opposite node.
* Any other eclipse period (or month interval) can be expressed as a linear combination of the Saros and the Inex. This is because the numbers 223 and 358 are relatively prime.

We can express the relation as follows:
$$T = 223n + 358m $$
Where $T$ is some interval of months between eclipses, and $n$ and $m$ represents the number of Saros or Inex we have to jump forwards or backwards (positive or negative) to arrive at our chosen interval.

This equation is what is referred to in mathematics as a *Linear Diophantine Equation*, and its can be solved quite easily for $n$ and $m$ using the *Extended Euclidean Algorithm*. We can make use of [an online calculator](https://www.math.uwaterloo.ca/%7Esnburris/cgi-bin/linear-query) which implements this algorithm.

Some example solutions are:
$$6 = 223(-8) + 358(5)$$
$$5 = 223(53) + 358(-33)$$
$$1 = 223(-61) + 358(38) $$

These are the solutions for 6-month, 5-month, and 1-month intervals respectively.

If two eclipses are separated by an interval $T$, the **Saros number** is defined in such a way that the Saros number for the next eclipse is given by adding the value of $m$ to the current Saros number.

So, if the next eclipse occurs 6 months after the current eclipse, we add $m=5$ to find the current Saros number. For example, the lunar eclipse of April 24,2005 is numbered 141 in the Saros scheme. The next eclipse, of October 17,2005, is separated from it by a 6-month interval, so its Saros number is $141 + 5 = 146$. If they were separated by a 5-month interval instead then we use $m =-33$, or for 1-month interval, $m=38$ (according to the solutions).

Now, regarding your question: Why does the particular pattern appear between eclipses separated by Saros number 40.

The first thing to note is that we can combine different intervals to arrive at new solutions. In particular we can take the 6-month interval and multiply all values by $8$. This will give us a new solution:
$$48 = 223(-64) + 358(40)$$

This immediately gives us an explanation for why the eclipses separated by $40$ Saros are found to be ~1418 days apart, because an interval of 48 months is given by $48\times29.53 \approx 1417.44$ days $\approx 3.88$ years, which in turn corresponds to a value $m=40$

The next thing to note is that the value $n=-64$. Now $n$ indicates the amount we jump backwards or forwards between eclipses in the same Saros series. However, eclipses in different saros series do not all start at the same Inex. This can be easily seen if we take a look at a diagram of the Saros–Inex Panorama.

The following diagram of the Saros–Inex Panorama for solar eclipses from the [NASA GSFC](https://eclipse.gsfc.nasa.gov/SEsaros/SEsaros.html). Each Saros series run along the vertical axis. We can see that each series starts or end at different points. In fact, the starting and end points appear to vary in a sawtooth wave-like pattern!
[![Saros-Inex Panorama diagram of eclipses - from Fred Espenak, NASA GSFC](https://i.sstatic.net/pzHPs1Af.jpg)](https://i.sstatic.net/pzHPs1Af.jpg)

A saros series lasts on average 72 cycles; this means the value $n=-64$ should roughly correspond to jumping from the end of one Saros series, 64 steps backwards, to the beginning of another. There is variability however due to the different lengths of Saros Series and different starting and end points. So different saros series have different values for $n$ corresponding to the different start an finish times.

For example:

* between Lunar Saros 129, and Lunar Saros 89 there are 48 months ~ 4 years, corresponding to the value $n= 64$.
* between Lunar Saros 89 and 49 $-175$ months ~ $-14$ years, corresponding to $n=-62$; here the Series overlaps.
* between Lunar Saros 49 and Lunar Saros 9, there are $-621$ months ~ $-50$ years, corresponding to $n=-67$

In general, we can check how different values of $n$ would affect the resulting interval of time between Saros Series separated by $40$ by substituting them in the expression, or equivalently, adding or subtracting 223 months from our initial solution with $n=-64$. We get the following

* $n=-63$ would give $T=271$ months or ~$22$ years.
* $n=-64$ would give $T=48$ months or ~$4$ years.
* $n=-65$ would give $T=-175$ months or ~$-14$ years, which would mean the series overlaps.
* and the pattern continues…

In the end, there is nothing particularly special about the number $40$ between Saros numbering here, other than the fact that $40$ Inexes ~ $64$ Saroses, and $64$ in turn is nearly the same as the length of an average Saros Series. In fact, we can find a better approximation in $43$ Inexes ~ $69$ Saroses. This corresponds to the solution:

$$7 = 223(-69) + 358(43)$$

As for the apparent relation between 4 Metonic cycles and 4 Saros, this might just be a coincidence: $(235-223)\times 4 = 48$


---

