Sarreguemines Station

Sangaku Sunday #11

We're almost there! We have three relations between our unknowns, the radii p, q and r. Actually, let's write them in the general setting, with any height.

Set SO = h and ON = k (so the number b in the problem so far has been equal to k/h). Repeating what we've done in previous steps, and substituting q and r in the final equation so that we get an equation with just p (I've done it so you don't have to), this is what we're solving:

The plan is simple: get p with the last equation, deduce q then r with the first two. The execution of the plan... not so simple. That last equation is messy. Let's tidy it up a bit by noting that it is actually a polynomial equation of the variable x=squareroot(2p):

There are formulas for the solutions to an equation like this, but if we can avoid using them, we'll be happy.

Here's what I did - and you can do too: a numerical test. Let's take the simplest dimensions for a right triangle, h = 4 and k = 3. Replace in the last equation and notice an obvious solution. Deduce p, then q, then r. Jubilate - until you realise something is very, very wrong...

C'est avec grand plaisir que je présenterai le mardi 16 avril à la Maison Universitaire France-Japon de Strasbourg une conférence sur la géométrie pendant la période d'Edo, avec en support le sangaku de Kashihara. Entre grande Histoire et petits calculs. Lien vers les détails 4月16日(火)、ストラスブール市の日仏大学会館に江戸時代の算額についてコンファレンスをします。楽しみにしています！ Looking forward to giving a conference on Edo-period geometry on 16 April at Strasbourg's French-Japanese Institute. Expect a few posts about Kashihara around then. Has it really been 6 years?...

Sangaku Saturday #12

Having mentioned previously how mathematical schools were organised during the Edo period in Japan, we can briefly talk about how mathematicians of the time worked. This was a time of near-perfect isolation, but some information from the outside did reach Japanese scholars via the Dutch outpost near Nagasaki. In fact, a whole field of work became known as "Dutch studies" or rangaku.

One such example was Fujioka Yûichi (藤岡雄市, a.k.a. Arisada), a surveyor from Matsue. I have only been able to find extra information on him on Kotobank: lived 1820-1850, described first as a wasanka (practitioner of Japanese mathematics), who also worked in astronomy, geography and "Dutch studies". The Matsue City History Museum displays some of the tools he would have used in his day: ruler, compass and chain, and counting sticks to perform calculations on the fly.

No doubt that those who had access to European knowledge would have seen the calculus revolution that was going on at the time. Some instances of differential and integral calculus can be found in Japan, but the theory was never formalised, owing to the secretive and clannish culture of the day.

That said, let's have a look at where our "three circles in a triangle" problem stands.

The crucial step is to solve this equation,

and I suggested that we start with a test case, setting the sizes of the triangle SON as SO = h = 4 and ON = k = 3. Therefore, simply, the square root of h is 2, and h²+k² = 16+9 = 25 = 5², and our equation is

x = 1 is an obvious solution, because 32+64 = 96 = 48+48. This means we can deduce a solution to our problem:

Hooray! We did it!

What do you mean, "six"? The triangle is 4x3, that last radius makes the third circle way larger...

Okay, looking back at how the problem was formulated, one has to admit that this is a solution: the third circle is tangent to the first two, and to two sides of the triangle SNN' - you just need to extend the side NN' to see it.

But evidently, we're not done.

Wow, I basically forgot the Nishi-Kyushu Shinkansen in my run-down at the end! But yes, it is a complicated situation, with a stretch of line isolated from the rest of the network (there's that gauge difference explained above), and Saga-ken disagreeing on how to build the connection to the main Kyushu line. No clear plan yet apparently. :(

60 Years of the Tokaido Shinkansen!

On 1 October 1964, a railway line like no other opened. Connecting Tôkyô and Ôsaka, paralleling an existing main line, the Tôkaidô New Trunk Line had minimal curves, lots of bridges, zero level crossings. Striking white and blue electric multiple units, with noses shaped like bullets some would say, started zooming between the two cities as at the unheard-of speed of 210 km/h.

This was the start of the Shinkansen, inaugurating the age of high-speed rail.

The trains, with noses actually inspired by the aircraft of the time, originally didn't have a name, they were just "Shinkansen trains", as they couldn't mingle with other types anyway due to the difference in gauge between the Shinkansen (standard gauge, 1435 mm between rails) and the rest of the network (3'6" gauge, or 1067 mm between rails). The class would officially become the "0 Series" when new trains appeared in the 1980s, first the very similar 200 Series for the second new line, the Tôhoku Shinkansen, then the jet-age 100 Series. Yes, the 200 came first, as it was decided that trains heading North-East from Tôkyô would be given even first numbers, and trains heading West would have odd first numbers (0 is even, but never mind).

Hence the next new type to appear on the Tôkaidô Shinkansen was the 300 Series (second from left), designed by the privatised JR Tôkai to overcome some shortcomings of the line. Indeed, the curves on the Tôkaidô were still too pronounced to allow speeds to be increased, while all other new lines had been built ready for 300 km/h operations. But a revolution in train design allowed speeds to be raised from 220 km/h in the 80s to 285 km/h today, with lightweight construction (on the 300), active suspension (introduced on the 700 Series, left) and slight tilting (standard on the current N700 types).

Examples of five generations of train used on the Tôkaidô Shinkansen are preserved at JR Tôkai's museum, the SCMaglev & Railway Park, in Nagoya, with the N700 prototype lead car outdoors. It's striking to see how far high-speed train technology has come in Japan in 60 years. The network itself covers the country almost end-to-end, with a nearly continuous line from Kyûshû to Hokkaidô along the Pacific coast (no through trains at Tôkyô), and four branch lines inland and to the North coast, one of which recently got extended.

東海道新幹線、お誕生日おめでおう！

The things people write on clocks...

Craignez la dernière - Eglise Notre-Dame de Croaz-Batz, Roscoff

Literally, this means "fear the last". OK, but the last what? Well, you're looking at the sundial for the time, which here, in French, would be "12 heures", so the full saying is inferred to be "craignez la dernière heure" - "fear thy final hour". Reminding people of their own mortality was an important part of medieval-Renaissance Christian discourse, as we saw with the ages of Man passing before Death among the automatons on the Astronomical Clock in Strasbourg Cathedral, built around the same time as this church in Roscoff.

Die Zeit eilt, Die Zeit heilt - Rathaus St. Johann, Saarbrücken

I've only got a close-up of one, but the tower of the historic town hall in Saarbrücken has at least two clocks, each with a message. The meaning of "die Zeit eilt" is similar to "time flies", and could be linked to what we saw above: be aware of what time you may have left. The second clock cleverly adds one letter to that to make "die Zeit heilt": "time heals".

"How to use this sundial" - Piazza della Borsa, Trieste

In the ground in front of the Chamber of Commerce in Trieste is a sundial, but conspicuously, the hand is missing. That is because you are the hand! Taking astronomy into account, one should stand in a slightly different place depending on the season, and apply corrections to the minutes to get official time. It was cloudy when I saw this, so I'm afraid I don't have a nice picture of the sundial in action. The instructions are in Italian on the left side of the base.

Strasbourg's Christkindelsmärik

One of Europe's oldest Christmas market, and likely the most famous in France, is Strasbourg's. Its existence is attested as far back as 1570, appearing in the years following Protestant Reformation. Today it is a sprawling event, covering all the main squares of the central island of Strasbourg, and it's very busy, especially in the evenings and on weekends.

The traditional Alsatian name of the market is Christkindelsmärik, "the market of baby Jesus", while the city's more recent branding (since the 1990s) proclaims the town as "Capital of Christmas".

Pictures from 2018 - I haven't been to this year's market yet, but I plan to once my workload deflates - I get my annual stock of jams from the local producer's market!

Nagahama Castle Ruins

Like Hikone Sawayama, Nagahama Castle is not on the list of Japan's 100 Famous Castles, nor on the list of Japan's Next 100 Famous Castles. Like Hikone Sawayama, it was a pre-Edo period fort which was owned by clans which at some point landed on the wrong side of the unifiers of Japan. Little is left, and short posts mark where buildings would have been. It's a short climb from the base, and the view of Uchiura from this location is pretty good.

This part of Numazu, and the island in the middle of this shot, Awashima, is most famous for being featured in the anime Love Live! Sunshine!!, which I know nothing about but saw some hints of (train for another day). It's possible to see Mount Fuji from here on a good day. And... well, I could see some it...

It's a considerable distance from a train station, though there appears to be a bus stop at the base, but I was driven there by a friend so I don't know how frequent the buses are. We were also lucky to get on a quick boat tour out to Awashima and back, with the chance to see the castle from a the sea, against the hilly backdrop of Izu Peninsula.

Summer highlight reel: Hegau

Towering around 250 m above the industrial city of Singen, the Hohentwiel is one of many reminders of the geological history near the Bodensee, or Lake Constance. This whole area was once volcanic, then the softer rock was eroded by the ice age, leaving this type of steep structure, around the old volcanic tube, behind.

A mountain like this would not have been easy to climb, especially when laden with weapons and armour, so unsurprisingly, several summits have fortress ruins on them. Hohentwiel is one of them (more on that and the climb - oh, the climb! - in another post), and Hohenkrähen, that rock jutting out ahead, is another.

The views from the top of Hohentwiel are just splendid. Two more ex-volcanoes are visible here: the Hohenstoffeln to the left, and the Hohenhewen to the right - there are lots of 'hohens' in Hegau, but it just means they're higher than villages. The weather was something to admire that day too, with showers floating around, like this one over the Schaffhausen area.

Ruine Schauenburg

My hike in July took me to the South side of Oberkirch, while on the North side sit the ruins of Schauenburg castle. It's a short, but steep, climb from town centre (or you can drive up).

The castle was built at the end of the 11th century by the Duke of Swabia. It saw action mainly in disputes between local lords, most notably after much of the land around the castle was sold to the Margrave of Baden-Baden, around 35 km to the North. It became a ruin following a French invasion in 1689.

Today, a restaurant sits beside the ruin. When I was last there, there was a camera stand to take a clean selfie - you know, the one where you set a timer instead of holding at arm's length. That's where my photo on my professional website comes from.

As is the case from the South side, the castle has a good view of the Rhine plains, and Strasbourg cathedral sticks out. It would have stuck out even more back in the day, without the modern tower blocks. The lords of Schauenburg would have seen the massive gothic cathedral and its monumental spire being built... over the course of a few centuries.

The Montech water slope

At around the same time as the "Inclined Plane" of Saint-Louis-Arzviller, other types of boat lifts were being engineered in other places. One of these was the Montech water slope, situated in Southern France on the Canal Parallel to the Garonne (Canal Latéral à la Garonne), the canal from Toulouse to Bordeaux which most people would probably refer to as the Canal du Midi - a better-known term though strictly speaking, the Canal du Midi is only the section from Toulouse to the Mediterranean.

From what I can garner, the two-headed vehicle used on this slope is a conjoining of two Diesel railcars built by Soulé in the early 70s, running on tyres and featuring a daunting-looking shield. This shield would be lowered behind a boat, and the machine would travel up and down the slope, carrying along the boat and the water it bathed in!

However, the performance of this system is less impressive than the Inclined Plane. It bypasses 5 locks, and saves 45 minutes of travel time. That's not nothing, but if you arrived at an end and just missed the train, then going straight to the locks wasn't going to be much longer than waiting for the next one. Also, far more power is required to make this work (I'm reading 1000 hp motors, versus 125 hp for the Inclined Plane), and it's Diesel.

These photos were taken in late 2017, 8 years after the water slope closed. It was only serving leisure boats by then, and, as I've insinuated, it wasn't very economical to run. As far as I can tell, the 5 locks beside it never closed, and are in use today.

The slope and its tractor have since been renovated, and the site's current state can be seen in a Tim Traveller video published in 2021.