1 It is reported that sunset(s) in Wallsend and Newcastle are four minutes apart. Can this be right?

a Find what distance you expect for a difference of four minutes in sunset on the equator.

b Find the distance you expect for a difference of four minutes in sunset at latitude 51º

c Taking the distance as five miles along an east-west line, find the time difference you would expect.

d I think the original statement should have said opposite ends of Hadrian's Wall. Do you agree at all?

2 Two other places on similar latitudes and in the same country are Chengdu [30.67ºN, 104.07º] and Hangzhou [30.25,120.17]. take these as being at 30ºN and the longitudes as 104 and 120º.

a Using an equator of 40 thousand kilometres, check that you think the distance between cities along the line of latitude is 1540km.

b Find the speed of the dawn line, the apparent surface speed at 30ºN.

c Find the time between sunsets for these two cities.

3 Britain moved to Summer Time (to ‘spring forward’ to UTC+1) on 29/3/2015, in line with most of Europe. New Zealand changed the other way (to ‘fall back’ to NZST, UTC+12) on April 5th. In winter the UK standard time, GMT, is at UTC+0 and in winter NZ is at NZST, UTC+12.

What is the time difference between UK and NZ on (i) March 1st (ii) April 1st (iii) May 1st ?

4 Calculation of the shortest distance between two points on a sphere (e.g. our planet) calls for understanding of a ‘great circle’. Two points are located by (latitude, longitude). Let these locations be (ϕ₁, λ₁), (ϕ₂, λ₂); then the angle between them at the centre, θ, has the relationship

cos θ = sin ϕ₁ sin ϕ₂ + cos ϕ₁ cos ϕ₂ cos (λ₂- λ₁) and the arc length is Rθ (in radians) or 2πRθ/360 in degrees.

a Use this formula to find the great circle distance between Chengdu and Hangzhou (Q2), which won’t be much different from the previous result because the latitudes are so similar.

b Find the great circle distance from Newcastle upon Tyne at (55, -1.5) to Newcastle, NSW at (-33N, 151.5).

c Find the centre angle for Qingdao (36, 120) and Buenos Aires (-34.5, -58.5).

d Find the centre angle for Stanley (-51.6, -58.0).and Mohe (52.4, 122.0). Mohe lies on the Amur river where it borders Russia and is the northernmost town in China. Stanley is the capital of the Falkland Islands, where the Chinese fleet may have reached. See *1421* by Gavin Menzies.

e A small (20km²) rocky island called Antipodes island belonging to New Zealand lies at (-49.7±0.2, 178.8±0.1). Its exact antipodes would be at (+49.7±0.2, 0.2±0.1). Find the largest town (if any exist) that falls inside this area.

DJS 20150329

The formula given in Q4 is unstable if used in software—the function inverse cosine is ‘ill-conditioned’—for small angles it is ‘unreliable’, meaning that small differences on one side of the function make for large differences on the other. A modern calculator may be immune to this effect. Users wishing to write a spreadsheet to play with this are directed to the Haversine formula, which is stable.

Challenge: using points within the conurbation of any large city, can you find:

i) Any well-known city at right angles to London (51.3ºN, 0º)

ii) two well-known cities mutually at right angles (at the centre of the earth)

iii) three well-known cities mutually at right angles. Can you do better than 400km error?

iv) three cities where any two are at the same centre angle. If this the same as the points being equidistant, that may prove an easier problem to solve.

v) the largest number of such places and suggest a solution set.

vi) Find four cities all around 12000km apart (quite hard)

vii) Find four smallish countries, islands or provinces (e.g. Goa, Fiji, Lesotho) each 12000 ±500 km from the others.

1a 40 million metres of equator rotating every 24 hours gives 463 m/sec. So 4 minutes => 111km

b the radius of the circle shrinks by a multiplier of cos51º. So the surface speed is 290 m/sec and four minutes represents 70km, more like Newcastle to Brampton.

c 5 miles is 8km, which at 290m/sec is 27 seconds. [That 4 minutes was wrong by a factor of ten].

Using Google, today sunset in Newcastle and Wallsend will be 19:36 and in Carlisle will be at 19:42.

TimeAndDate says 19:37 and 19:42. Sunrise-and-sunset agrees. 5 minutes (4.5 to 5.5 puts the distance between Newcastle and Carlisle as 78 and 96km which is about right.

The ends of Hadrians Wall are Bowness-on-Solway and Wallsend-on-Tyne, 114km and close to being east-west. This is very close to the 111km that 4 minutes would indicate - on the equator. At 51ºN, 4mins/cos51º => 6:20 minutes. To have any truth, someone has apporximated Hadrians Wall as Carlisle and Newcastle.

2a Distance between them along the line of latitude is (16/360)*40000*cos30º) = 1540 which checks well with mapcrow.info.

b The apparent surface speed is, at 30ºN, 463*cos30=401m/sec (40000cos30ºkm / 24 hours = 1433kmph = 401 m/sec)

c Since they are 16 degrees apart, you expect this to be 16/360 of a day, 64 minutes. Using 1540/401 we have the same answer to 3s.f., 3840 secs=64 minutes

3 +13 (0,+13), +12 (+1,+13), +11 (+1,+12) hours going from British time to NZ time. Look this up here. Read about time zone differences in Essay 97. So in practice NZ is 11 (or 13) hours different, but rarely only 12.

4a I get cos θ = 0.9708, θ=13.87º, arc length 40000*13.87/360 = 1541.4. Compare this with 16.1º at latitude 30.46º, 1542.0 to 4sf.

b The points given are within each conurbation. cos θ = -0.87475, θ=151º (less than I expected), arc length 40000*151/360 = 16780km to 4sf.

c 178º. Very nearly polar opposites. So, when the Chinese merchant fleet reached the Falklands, in around 1424, this was almost as far away from China as it could possibly be, as part d demonstrates.

d 179.2º This follows from looking where in the world (and on land) lie the antipodes to China.

e Le Havre lies at (49.5±.05, 0.2±0.2) on the Seine estuary. You are right; that is not in the UK. Rouen is bigger, at (49.4, 1.05). Spot on is a little place called Étretat, but that is not the answer I was looking for; "largest town in the area". Other places one might have heard of nearby are Deauville, Bayeux and Caen. But Le havre is the closest of these to the target position.

Challenge—I have no good idea as yet. My instant thought is that the upper limit is four, as if the vertices of a tetrahedron, so the common angle is around 110º (I think π - tan⁻¹√8). Practically, there might be many more, but if you were to set the test of 'large city' as being arguably of a million or more, or 'places that you have heard of', I think the possible solutions will be a low number. I'd like some correspondence on this, which might prompt a further question – with all due credit given, of course.

(i) One way to tackle this is to determine that a quarter of a great circle is 10,000km and look for cities that far from London, perhaps by using the ruler tool on Google Earth. Capetown is quite close to the target distance from London; you might settle for a 25km miss, so 10000±25. Lima is 10160km from London, so Lima looks a likely candidate, Lima to Barcelona is a good fit, well within 10km. Quito (Ecuador) to Yaounde (Cameroon) is close to 10000 km. London to Phnom Penh fits the 25km limit. Tokyo to Nantes, Poitiers, Monaco, Corsica, Messina (Sicily). Singapore to Malta, Rome, Venice, Leipzig, Rostock, Copenhagen (just), Gothenburg. It is raltively easy to find places in Europe to recognise if one starts from far enough away. from Capetown, Capetown is 9650km from London; so, working fromn Capetown, 10,000 km reaches Dublin (Galway is better), Douglas (Isle of Man) Ambleside, Durham, Newcastle, Copenhagen (almost), Dhaka (almost), Port Lincoln (near Adelaide), Hobart, Barcelona Venezuela, Martinique...

3. The best I've found to date shows that London, Capetown and Quito are all between 85 and 95º of each other. Lima, Barcelona and ...

London and Phnom Penh are 90º apart. Capetown is within 5º of right angles to both. Distances to 4 sig fig are 10020, 10400, 9640. There's a place called George on the RSA south coast south of Oudtshorn that fits better, 10040, 9730, Douglas (IoM)-Capetown-Phnom Penh 10020, 10180, 10400. I think this problem is not solvable unless we allow the 10 thousand km to be only 3 s.f. precise, i.e. up to 500km off.

4. To be more than three places, then if there were a set of four, they cannot be a rectangle, since the diagonals would be different in length from the sides. The only way for four points to be equidistant is for these to be the vertices of a tetrahedron common angle 109.47º, Which, I think, would require finding four places on land each **12165km** from the others. Call that 11900-12400 km, an 'error' of 500km. Call 100km=1Mm.

How about, for four points, one being the South Pole, the the other three are close to 20ºN and 120º apart in longitude. After quite a bit of searching I quite like this group (of places I've heard of): **R****i****yadh** (24.7N, 46.7E),11900km from **Wake** Island (19.5N 166.6E) and **Guantanamo** at (20.1N, 75.2W) 12000km from Wake, 12025km from Riyadh. Each of those is within 400km of 12500km from the **South** **pole**. [12230, 12720, 12140]. Pole, Riyadh, Guantanamo, Wake. You could swap Guantanamo for Port-au-Prince, on Haiti.

After a lot of searching, I offer **Bogota** (Colombia), **Wellington** (NZ), **Beira** (Mozambique) and the stuaries of the Novaya Zemla and Okhotsk rivers, which are quite close together. You might just replace that last with **Magadan**, a little more north-east along the coast (of the sea of Okhotsk).

Finding four smallish countries around 12000km apart.....

**New Zealand, Colombia, Mozambique, Cuba** follows from the last, but perhaps we can do better still.

Similarly one might just start at an equatorial city Bogota (4.3N ), Singapore (1.2N, ), Nairobi (1.2N, ) Kuala Lumpur,Kampala,Manaus,Libreville and look for the right distance longitudinally. That is not so straightforward.

Bogota; Fiji 12100, Wellington 12120km, Petropavlovsk 11920km, Nizhneavatovsk 12290km, Tblisi 11980km, Medina 12150 km, Jeddah 12200km, Kampala 11900km, Arusha 12340km, Llilongwe (Malawi) 12060km, Beira (Mozambique) 12170km, Maputo 11905km.

Bogota, Wellington 12120km, Okhotsk/Novaya Zemla estuaries11500km or Yakutsk 12150km, UlaanBator 11900, Lhasa 11600, Khatmandu 11950, Nagpur 12000, Goa 12040, Seychelles 12050, **Beira** 12090,

Bogota, Wellington, Beira, Okhotsk/NZ estuary, though you might settle for Magadan fits well enough, I think.

**Goa**: NZ, Fiji, Honolulu (13Mm), Nova Scotia, Bermuda (13.1Mm) Rio Grande Norte (Brazil),

**Hawaii**: South Pole, Cocos islands, Sri Lanka (12.8Mm), Punjab, Tadjikistan, Volgograd prefecture, Belarus, Czechia, Luxembourg, Azores, French Guiana (11.5Mm), Paraguay, Uruguay.

**Sri** **Lanka**: Cape Verde Islands 11.3Mm, Honolulu 12.8Mm, American Samoa 12.1Mm, Chatham Islands 11.6Mm, South Georgia Islands12.4Mm,

**Azores: Hawaii, ** Falklands (10.5Mm), Mauritius (10.8Mm), Malaysia (north), Hainan, Taiwan, Japan (Tokyo), French Polynesia (12.7Mm), St Pierre (Newfoundland) 12.4Mm, British Columbia (north),

**Luxembourg**: **Hawaii**, indonesia, Java, Palau, Guam, **Uruguay** (south), **South Georgia**, Kerguelen 12.5Mm, Cocos islands (11.3Mm),

**uruguay: Lux **11.3 at best, N**etherlands**, iceland, Shetland, Victoria Island, **Hawaii**, Fiji, Victoria (Aus, Adelaide), Seychelles, Djibouti, Eritrea,

Luxembourg, Hawaii, Uruguay, ...?

Hawaii, Cocos Islands, Luxembourg, ....?

London is at 105º to Jakarta, Santiago and Honolulu but their common angles don’t fit.