The Train To Kushimoto: A JR West 283-series Kuroshio Express. The Sets Are Getting On A Bit, They Were

Pouancé

The small town of Pouancé is on a remarkable geographical "quadruple point", on the border of four départements! A peculiarity that dates back to the Middle Ages, when it was on the battlefront between France-affiliated Anjou and the still fiercely independent Duchy of Brittany. As such, Pouancé had a castle; its Breton counterpart was at Châteaubriant, and both towns were besieged at some point during the 15th century.

With a friend from Châteaubriant, we got to witness barriers being removed as what we guessed were maintenance or renovation works on the path around the castle were finished. The castle itself only opens during the summer, but at least we got to walk all around and get some good views of the castle, through the neighbouring park.

If you're driving into Pouancé from the West, this is how you know you've arrived:

Plan Incliné de Saint-Louis-Arzviller

This is going to be a rather long post as there is a lot to say about this thing! But the short version is: this is a boat lift.

Built in the 1960s, this "inclined plane" was designed to carry barges as part of the fluvial coal transportation industry. However, that trade declined pretty much during the edifice's construction, and today, it almost exclusively serves leisure boats. But if you're going to do a canal cruise, this thing gives it quite the difference!

Its function is that of a lock, taking boats from the lower water level to the higher level, or vice-versa, but it does this by technically being a lift or elevator. A caisson carries the boats and the water up and down, using counterweights to ease the travel.

In fact, the caisson will take on more or less water in order to be heavier or lighter than the counterweights. Though the total mass of the caisson and counterweights is enormous, the difference in mass between them isn't, so very little power is needed to get the system moving, and gravity does most of the work. Two relatively modest electric motors (centre of photo below, steps to the right for scale) start the movement and control the speed.

As such, the system uses comparatively little power, for impressive results. The boat lift was built to bypass a "ladder" of 17 locks which required a whole day to go through, while the travel time of the lift is just 4 minutes. The ride is seamless and very comfortable, effortless even, for reasons mentioned above but also because the effort is distributed across 5 times as many cables as physically required to hold everything together!

Water-tightness is also extremely important, not just for the caisson obviously, but also for the other doors, particularly the top door, which is holding back a whole length of canal. A serious incident in 2013 has led to further reinforcement of redundancies and the construction of an emergency dam closer to the lift in the event of major leaks.

With a lot of freight traffic in mind, the structure was actually designed for two caissons, side-by-side, as evidenced by a second gate hole visible at the top of the ramp (4th picture), and extra space at the bottom, visible in the final picture below. Doubling the caissons would have meant doubling the counterweights, and a second set of rails were laid for that scenario and are visible in the 4th picture. As mentioned earlier, demand dwindled as the lift was being built, so it never operated with two caissons.

For a long time, this place was a childhood memory, visited during a school trip. In my hiking spree after the 2020 and 2021 lockdowns, I sought this place out again and was glad to see it was still working. And just this week, I returned with my parents and rode the lift! It's without doubt one of my favourite pieces of engineering.

The fortifications of Wissembourg

From one fortified town to another, just South of the current French-German border: Wissembourg. In a region rife with conflict, between cities and lords, sometimes between a city and their own lord, protective walls, moats and towers around the town were a must.

This tower, called the Poudrière, was built in the 13th century, and served as gunpowder storage at some point, hence the name. The walls in front of it are more recent, dating back to the 16th century, featuring a dam system which would flood the moat if needed. This complex can be seen in the North-East corner of the town on this 1750 map.

Despite these protections, Wissembourg suffered massively between the 15th and 17th centuries. And in spite of all that, the town centre retains much of its original plan and many traditional buildings. I should go back on a nicer day to get better pictures...

Sangaku Sunday Bonus - what about the other problems?

In the sangaku series, we've solved two of the four problems on this tablet, the middle two, which I believe were the easiest to work on in terms of geometric arguments - we hardly ever used more than Pythagoras's theorem, though the second one needed some more advanced algebra to finish off.

Here's a quick look at the problems at each end of the tablet, and the main ideas I had to solve them.

On the far left, we have two circles tangent to one another (with centres A and B), inside a larger circle (with centre O) so that their diameters add up to the diameter of the largest. The radii of these three circles, respectively p, q and p+q, are known. The unknown is the radius r of the circle with centre C, which must be tangent to all three original circles (it has a twin on the right-hand side with the same radius).

This is quite quick to solve. Remember that tangent circles mean that the distances between centres is equal to the sum of the radii, e.g. AC = p+r, BC = q+r... Al-Kashi's theorem, which is a general version of Pythagoras's theorem, links the lengths of three sides of a triangle with one of the triangle's angles, and the triangles CAO and CAB have an angle in common, which yields the equation for r by isolating this angle in each application of Al-Kashi's theorem. The result is:

The problem on the far right seems to start in a similar fashion: two circles with fixed radii are offset by a fixed distance. A third circle has its diameter equal to the remainder of the diameter of one of the large circles: this radius can be calculated with little difficulty. What we want to do next is construct circles which are tangent to the two large ones, and the one previously constructed.

The radius of the circle with centre C1 can be obtained as above, but this method does not seem to extend to the subsequent circles, as O, D and C1 are no longer aligned, and there no longer appears to be a common angle in the triangles we want to work with. So I went for a parametric approach, understanding the curves that contain points that are equidistant from two circles. The red curve (which looks like a circle but isn't one) is the set of points at equal distance from the two largest circles, and we seek to intersect this with the set of points that are at equal distance from one large circle and the smaller one, the green curve. The intersection is equidistant from all three circles, so it is the centre of the circle we want to construct. Rotate and repeat for subsequent circles.

The general formulas are horrible and not worth showing, but this is another problem where I have been able to read the results on the tablet. The large circles have radii 61 and 72, and the offset is 23. The radii of the smaller circles, starting with the one in the middle and working outwards are:

17, 15.55, 12.292, 8.832 and 6.038 (I see 八, but I'll give the authors the benefit of the doubt as the top of the character 六 may have been erased by time)

The results with our exact formulas are:

17, 15.58, 12.795, 9.076 and 6.444

Rather close! As with the "three circles in a triangle", I do not know how the authors originally solved this problem.

Sangaku Saturday #14 - the grand finale!

We are only a few steps of algebra away from solving the "three circles in a triangle" problem we set in episode 7. This method will also yield general formulas for the solutions (first with height 1 and base b; for any height h and half-base k, set b=k/h and multiply the results by h).

Before we do that, it's worth noting what the sangaku tablet says. Now I don't read classical Japanese (the tablet dates back to 1854 according to wasan.jp), but I can read numbers, and fishing for these in the text at least allows me to understand the result. The authors of the sangaku consider an equilateral triangle whose sides measure 60: boxed text on the right: 三角面六尺, sankaku-men roku shaku (probably rosshaku), in which 尺, shaku, is the ten marker. In their writing of numbers, each level has its own marker: 尺 shaku for ten, 寸 sun for units, 分 fun for tenths and 厘 rin for hundredths (毛 mô for thousandths also appear, which I will ignore for brevity). Their results are as follows:

甲径三尺八寸八分六厘: diameter of the top (甲 kou) circle 38.86

乙径一尺六寸四分二厘: diameter of the side (乙 otsu) circle 16.42

反径一尺二寸四分二厘: diameter of the bottom (反 han) circle 12.42

I repeat that I don't know classical Japanese (or much modern Japanese for that matter), so my readings may be off, not to mention that these are the only parts of the tablet that I understand, but the results seem clear enough. Let's see how they hold up to our final proof.

1: to prove the equality

simply expand the expression on the right, taking into account that

(s+b)(s-b) = s²-b² = 1+b²-b² = 1.

2: the equation 2x²-(s-b)x-1 = 0 can be solved via the discriminant

As this is positive (which isn't obvious as s>b, but it can be proved), the solutions of the equation are

x+ is clearly positive, while it can be proved the x- is negative. Given that x is defined as the square root of 2p in the set-up of the equation, x- is discarded. This yields the formulas for the solution of the geometry problem we've been looking for:

3: in the equilateral triangle, s=2b. Moreover, the height is fixed at 1, so b can be determined exactly: by Pythagoras's theorem in SON,

Replacing b with this value in the formulas for p, q and r, we get

Now we can compare our results with the tablet, all we need to do is multiply these by the height of the equilateral triangle whose sides measure 60. The height is obtained with the same Pythagoras's theorem as above, this time knowing SN = 60 and ON = 30, and we get h = SO = 30*sqrt(3). Bearing in mind that p, q and r are radii, while the tablet gives the diameters, here are our results:

diameter of the top circle: 2hp = 45*sqrt(3)/2 = 38.97 approx.

diameter of the side circle: 2hr = 10*sqrt(3) = 17.32 approx.

diameter of the bottom circle: 2hq = 15*sqrt(3)/2 = 12.99 approx.

We notice that the sangaku is off by up to nearly a whole unit. Whether they used the same geometric reasoning as us isn't clear (I can't read the rest of the tablet and I don't know if the method is even described), but if they did, the difference could be explained by some approximations they may have used, such as the square root of 3. Bear in mind they didn't have calculators in Edo period Japan.

With that, thank you very much for following the Sangaku Weekends series, hoping that you found at least some of it interesting.

Schirmeck castle

Overseeing the valley of Schirmeck, the castle, built for the Bishop of Strasbourg in the 13th century, is a short walk and climb from the town centre. As it was designed to protect a corner of the territory, that's apparently where we get the term: Schirm'eck. It was defeated by the Swedish during the Thirty-Years War, with some of the stone from the ruins being used to build other structures, such as the church.

Today, there's obviously not much of it left, though a square tower was restored and houses a small museum (closed when I visited). The Yoshi art was probably not part of the original episcopal aesthetic...

Hikone Sawayama: summit & castle

It's not about what is there today, as much as it's about what was there. Sawayama was the original location of Hikone Castle, and it is quite possibly the most important castle in Japan to have been completely lost, as it was the castle of Ishida Mitsunari, the leader of the Western Army which lost the battle to unite Japan following the death of Toyotomi Hideyoshi. There are so few traces of the castle, no obvious tell-tale structures... This small altar may trace its roots back to the days of the castle, or maybe not, but this is just about it.

Sawayama Castle was thoroughly dismanted after 1600 following the defeat of Ishida, as the new lord of the area, Ii Naomasa, appointed by the victorious Tokugawa clan, relocated the castle to a smaller hill closer to Lake Biwa. Hikone Castle, which still stands today, basically recycled the materials from Sawayama, and the view of the "new" castle complex and the lake is the main draw for hikers today.

The summit offers good views of the mountains on the other side too, with the industrial complexes near Maibara, most noticeably Fujitec and their 170 m-tall elevator test tower, in the foreground.

A few views of the Saar and Mosel rivers in Germany, which recently burst their banks due to heavy rain. The lower levels of the multi-lane motorway through Saarbrucken (second picture) were underwater, and the historic towns of Trier (top) and Cochem (below), which I have fond memories of, were flooded too.

Hoping that the communities can recover soon.

I promised more impressive views from the hills above Toba, and here they are. They're not very hard to reach: the Hiyoriyama circuit is only a couple of kilometres long around the station and involves climbing around 50 m. Hinoyama is further away, further South and a little higher.

The views of the coastline at Toba were good enough for Hiroshige to use in his Famous Views from the Sixty-Odd Provinces to illustrate Shima province (though there wasn't much else, I presume, Shima province was tiny, it was just Toba and the neighbouring town of Shima - also Shima is 志摩 and not 島 "island").

Beyond the islands near Toba, lies the mainland again, the Southern part of Aichi prefecture across the Ise Bay (Minamichita and Tahara), which the car ferry in the above picture traverses.

The Fog on the Rhine (is all mine, all mine)

After three weeks of marking, I finally managed to get out of my hole in late January. I was beckoned out by dense fog, seizing the chance to enjoy the misty atmosphere. When I reached the park that straddles the French-German border, I found it on the edge of a fog bank, with haze on one side of the footbridge and perfectly clear skies on the other.

While not among the most outstandingly beautiful parks, the Jardin des Deux Rives has things to offer on both sides of the border, and, just for that ability to hop over to another country, it ranks very high on the cool factor.

Not that the birds would know. They were just taking in the winter sunlight while they could.