Dear Marcus and All, A little over a year ago, Marcus and I openly discussed ways that we might improve the way that SI measures plane angles.
Generally, we agree that, currently, the SI is fundamentally flawed in that it does not have a base unit of plane angle and that this is a significant flaw of the SI. In 2002, July, Marcus challenged me to support a suggestion that I had made that a new unit, quad, should be created to measure the quantity of plane angle. It has taken me this long to get around to it. This posting is in four parts: 1 Some definitions 2 Arguments for quads 3 Arguments for grads, grades, and gons 4 Some reflections on mindsets Definitions Quad The suggested unit, quad, is derived from an abbreviation and a jargon word for quadrant; quad is simply a word that, in English, is a short form of quadrant, which in turn means a right angle. The word quad also fits well with the SI concept of using prefixes as it is a short monosyllabic word. In other European languages the word quadrant translates to: French = quart de cercle German = Quadrant Italian = quadrante Portugese = quadrante Spanish = cuadrante It is clear that quad could be an acceptable unit in these languages. Grad, grade Grade or grad comes from the Latin word, gradus, that means a 'step'. Grade has always had a connotation of dividing a larger quantity or unit. In this way grade has had the same meaning as degree. The Oxford English Dictionary cites a 1593 definition when it defines 'Grade' as: 'Math. A degree; the 90th part of a right angle or quadrant'. Gon Gon comes a Greek suffix that means 'angled'. It appears in words such as pentagon (5 angled), octagon (8 angled) and polygon (many angled). Arguments for quad 1 I think that it would be more appropriate to introduce the 'new' unit quad rather than re�adopting, and trying again to re�introduce, the grade, grad, or gon. We should recognise that grads, grade, and gons have all failed, and as the riddle goes: Q What would you be doing if you were a sadist, a masochist, and a necrophile all at the same time? A You'd be flogging a dead horse. 2 Grads and grades (like degrees) are products of the late 18th century where they were common words to refer to parts of larger units. In this context I suspect that the word grad is simply a short term for gradation and that grad only later became a slang or jargon term. Grads and grades of plane angle were simply the marks that divided up a quadrant into parts. 3 In the late 16th century the sub�divisions grade and degree were regarded as synonymous and they were defined as 1/90th of a quadrant or right angle. 4 In the late 17th and all of the 18th centuries it was clear that the natural unit of plane angle was the quadrant; which was in turn divided by gradations. It sounds to me like the idea of using prefixes such as in centiquadrants had not yet been developed as we now know them. By the time that grads had established themselves as an alternative to centiquadrants (or centiquads as I would say today) the grad or the grade was well established as an alternative word, but it was still not any part of a rational prefixed system. 5 The grad, grade, or gon does not exist without reference to a quadrant; as a challenge, could you define grads, grades, or gons for me without referring � in any way � to a quadrant (and circles of four quadrants I will regard as a copout). 6 There is a very interesting 'problem' with the grade in the fact that the kilometre accuracy would mean the use of the defunct unit 'centigrade', which is still often confused with degrees Celsius. 7 For aircraft navigation 1 milliquad would allow for an aircraft navigating, on their own, to within 10 kilometres of an airfield and then being navigated by the air traffic controller. 8 Using microquads as a small unit would describe a distance of 10 metres on the surface of the Earth � this measure, within the wingspan of most aircraft, should be sufficiently precise to allow for all aviation, nautical and for most cartographic purposes. 9 Shortcuts and rules of thumb can and will be devised to suit any system. For example, I could say that 100 milliquads of angle at the centre of the Earth is equivalent to 1 km on the surface. And in many ways I find this a superior correlation, as it does not contain the possibility of directly confusing plane angle with distance since one is (in numerical terms) 100 times the other. 10 The infrastructure for using quads is already in place in educational institutions at all levels; every right angle need only be given the additional measure of 'one quad'. Right angles (say in a furniture building class) would still be called right angles as well as being measured as one quad; the angles are 'right' because any angle smaller or bigger than '1 quad' is a wrong angle. Information about the quadrant is also taught in all primary, secondary, and tertiary schools - constantly, and in my view quite effectively. The only problem is that it is not now measured as a quad, it is called a right angle. Children draw angles � of 1 quad � in every square and rectangle they use, they cut angles of a quad out of paper and fabric. Children are taught through chanting 'The square on the hypotenuse of a right triangle are equal to the sum of the squares on the two adjacent sides'. Apprentices in all trades are totally trained in the uses of angles of a quad on every job that they do. There is a vast difference between having 'already been exposed to it at one point in their lives' (as some of us have to grads, grades, and gons) and the constant and necessary understanding that comes with constant daily contact in everything you do. Even every classroom the student sits in was designed and built using quads between all the walls, floors, and ceilings. 11 The size of right angles, and therefore the size of one quad, already enjoys widespread public awareness. 12 A centigon is a 100 angled figure if we use the linguistic models of hexagon (= 2 angled) and nonagon (= 9 angled). 13 The claim that grads, grades, and gons are currently known by the general community is, I believe, resting on rather shaky ground. When I first heard about grads, grades, and gons in senior secondary school mathematics or physics classes, others at my school (who didn't take these subjects) would never have heard of grads, grades, and gons in any of their courses. Secondly, I have never heard grads, grades, or gons ever referred to in any of the media. To the general public, grads, grades, and gons are simply non-existent and they always have been. On the other hand quadrants (a.k.a. right angles) surround us all constantly and most people are aware of them. 14 Using suitable prefixes with quads, milliquads or microquads for example, means that decimal numbers and decimal points might be avoided for almost all applications. Using quads, milliquads, microquads and nanoquads you might never require decimal points. You use the strengths of SI prefixes to choose the most appropriate (decimal point free) submultiple. Specifically, if one milliquad (equivalent to 10 kilometres on the Earth's surface) is a problem then use 100 microquad (equivalent to 1 kilometre) or one microquad (equivalent to 10 metres on the Earth's surface). I can't think of a practical use for a nanoquad so I won't explore it here! Grads, grades, or gons arguments This is my summary of grads, grades, or gons arguments put forward by Marcus Berger in mid 2002. These are the arguments that I reacted to in forming my opinions above. Marcus wrote: I think that one point to consider is that the unit grade *already* exist. I know it'd be two words and I know it didn't... "catch on" and all, but it would be easier to try to gather momentum for something that already IS there than for something "completely" new, even in concept (i.e. like the use of milli attached to it). Perhaps though we should, first of all, understand why the gon failed (if it really failed... from an educational point-of-view it actually didn't as we continue to teach the grade/gon(?) *everywhere* in this planet... We just didn't select to use it in most places, unfortunately...). I continue to believe that the "quad" has some nagging difficulties. It's just unfortunate that there are 40 Mm in the circumference of the earth, instead of 4 Mm, or 4 Gm, etc. I doubt it that people tie this to the fact that they have been drawing, cutting, etc, a (one) "quad". The right angle is being treated as an... *object*, not as a potential *measuring device* that has a unit value. When they draw any other angle they'd never associate it with its being some 1.4, 1.6, whatever, of the quad, but rather 90, 110, 180 degrees and so forth. Also, true, one could "introduce" the quad as actually being a unit, but then we could run into all sorts of difficulties like the fact that most angles are less than 1 hence generating decimal points which people appear to be naturally averse to. Sure, "use the milliquad", as you proposed, but then there is the meter question, the lack of an appropriate power of 10 for certain applications, etc... So... Wouldn't it perhaps be more prudent to revisit the grade/gon question, "for the first time"? The infrastructure is ALREADY in place! It's still taught all over the world. So the cost of having people learn it is practically zero! Suffice it for them to just refresh their memories and start using it! But this is also adequately addressed by the grade/gon, isn't it? In other words, the quad and the gon are actually the same thing but with a different face (quad = 1, gon = 100). If we consider this issue from technical, economical, practical, etc points-of-view I believe there would be more pros in favor of the gon than the quad (despite the "smell" of failure attached to the gon). However, if you disagree (and perhaps you do), let's please then consider what these are one by one and in the end let's see whether pros outweigh the cons in favor of any of these options. Joe, however, pointed out later on that there was another name for the grade, the gon, which would adequately address that problem. I, personally, must confess, I didn't know about it. On the other hand, if I, being an academic, didn't know about it, it may be fair to say that probably there would be many others out there who probably wouldn't either. Therefore, just this... "name" change from grade to gon could possibly invalidate one or two of my arguments in my list above... :-S (But, still, probably not enough yet to sway me from my preference in favor of the gon/grade. The... "breakers" to me are still the power of 10 issue, the use of decimal points with the added fact that we would be condemning one to use a prefixed unit, milliquad, most of the time, just like we do with the kilogram, this may ultimately lead people to argue about why not eliminating this by renaming the milliquad with some other name, just like we, here, do with the kilogram question... ;-) ) So far as I can tell this is a brief summary of the main *differences* and advantages of the gon over the quad. There could be others, but the above is what I could quickly come up with. Should you have a similar list of those in favor of the quad, please share it with us for our analysis. Mindsets Underlying a lot of the discussions between Marcus and myself is the issue of mindsets. In our two cases our main differences are between hundreds and thousands � and I'm not discussing fairy bread at a children's party. Marcus prefers to divide things into hundreds and I prefer to divide things into thousands. Marcus, based on his experience in Brasil (adopted metric measures in 1862) and Canada, prefers dividing many units into hundredths, such as centimetres and centilitres, and in the case of plane angles into grads, grades, or gons (1/100th of a quadrant). Whereas, based on my experience with the recent metrication in Australia, I favor dividing most units into thousands; I prefer millimetres, millilitres and, in the case of plane angles, milliquads. As I've pointed out previously I believe that this is largely a mindset issue. Marcus was brought up in Brazil where metric units have been normal for so long that the 'hundreds' of the original metric system are still the major mindset. On the other hand I was introduced to SI in Australia in the mid 20th century by which time such people as builders, engineers, architects, and many others had come to the realisation that, not only did division by 1000s make their work easier, it also made training and conversion from old metric systems much simpler and therefore much faster. As an example of this simplicity let me list a complete set of units for building a house in Australia: 1000 mm = 1 m 1000 m = 1 km 1000 mL = 1 L 1000 L = 1 m^3 1000 g = 1 kg 1000 kg = 1 t 1 m x 1 m = 1 m^2 As a challenge, you might compare this with the current set of units needed to build a house in Brasil or Canada. Cheers, Pat Naughtin LCAMS Geelong, Australia --
