**MEMO FROM: Professor G Fractal, Mathematics Dept, University of the Highlands**

**TO: TAC**

**DATE: 16th October 1996**

**REF: Heights and Distances**

**Your approach to heights and distances indicates sloppy thinking, as well as an anglocentric bias. In TAC26 you imply that the shortest distance between two points is a "dead straight line", and blithely use distances without any mention of scale. Also, in your article "Talaidh-Ho", and elsewhere in the journal, you quibble about the exact height of hills while ignoring a major variable which is an order of magnitude more than these quibbles.**

**I will start with your misbelief that a straight line from, say, Melcome Regis to the Kyle of Tongue is the shortest distance. Over the surface of the globe a "great circle route" is the shortest distance between any two points. Of course, this applies at any scale, so that the shortest distance between any two hills is always a curved route - see FIG. 1. (Incidentally, our bodies instinctively know this, which is why in a whiteout we tend to walk round in circles: without external stimuli we over-compensate and exaggerate our naturally circular walk - in the same way that out natural circadian rhythm exaggerates the 24-hour day without external stimuli.)**

**Secondly, you are imprecise in your mention of distances relating to hills, and even imply that the concept has meaning! You must know the classic mathematical "coastline problem" - that there is no answer to the question "How long is a coastline?" The answer, of course, depends on the scale used: it will be longer when measured on a 1:10000 map than on a 1:50000 map. And the same applies to hills, as well as coasts, as I hope FIG. 2 shows:**

**This illustrates that, although the concepts of vertical and horizontal distance are valid (within the constraints discussed below), as these are theoretical mathematical constructs, there is no such thing as a fixed distance up a hill. A snow-covered hill with all the hollows filled in will provide a shorter journey than one where you have to walk up and down all the snow-free bumps. A short-legged dog will hug the ground profile even more, and will travel further over the same distance. Similarly, a Munro-bagging tardigrade may well be thwarted as it goes up and over every soil particle, and could get stuck going round and round a soil particle for ever - and hence travel an infinite distance to get up the hill.**

**But your most serious error probably relates to your attempts to define the heights of hills: heights are in fact defined as being the vertical distance above sea level, which in Britain is based at Newlyn of all places. Why not, for example, at Lothbeg? And we all know that the sea goes up and down. A hill will be higher at low tide than at high tide. Maybe borderline Munros and Marilyns could all be climbed at low tide, to give them their extra boost. It might be best, therefore, to choose your baseline sea level in an area with a high tidal range, such as the Bristol Channel; but this is, of course, a long way from the Scottish hills and could be construed as cheating. Clever, meteorologically-inclined baggers may boost their tick-list by climbing hills when the tide suits them most - in anticyclonic weather at low springs, with strong offshore winds.**

**Heights could be standardised, based on the nearest bit of sea. However, FIG. 3 shows the problem with hills down the spine of Scotland - for when the tide is out on the west coast it is in on the east coast.**

**Unscrupulous Munro-baggers, when climbing hills equidistant from both coasts, may choose the tide of most advantage to them! I think the heights of all hills should be expressed in two forms - the low-tide height (based on MLWS at the nearest bit of sea), and the high-tide height (based on MHWS). This would result in, for example, different lists for high-tide Munros (HTMs) and low-tide Munros (LTMs). Incidentally, I do not recommend the use of satellite-based height determining systems, for your readers may be surprised to know that these give your altitude based on the sea level in California.**