Dear Marcus, I have finally got around to forming some thoughts about Angles � quads and milliquads.
Thanks to you and to John Ward for the stimulus to continue this discussion. Cheers, Pat Naughtin LCAMS Geelong, Australia Pat Naughtin is the editor of the free online newsletter, 'Metrication matters'. You can subscribe by sending an email containing the words subscribe Metrication matters to [EMAIL PROTECTED] ** History of the quadrant As you know, it is my firm belief that the first unit of the metric system was the quadrant. Following debate about the pendulum versus the quadrant as the basis for the metre, the metric founders decided on the quadrant. Because of this decision the quadrant was, historically, the founding unit for the whole system. The elaboration of the quadrant into metres, and subsequently into millimetres and kilometres, came after the acceptance of the quadrant as the basis for these later divisions and multiplications. Need for a simple angle measure I firmly believe that the world needs a better � and by better I mean simpler � method of measuring and comprehending angles and their measurement. That is why I proposed the quad and its decimal division using SI prefixes such as the milliquad. I am suitably vague about multiplication of the quad using SI prefixes, as I don't have any real feeling for the meaning of kiloquads or megaquads. The Conf�rence G�n�rale de Poids et Mesures (CGPM) has long recognised the need for the world to have a distinct unit for plane angle. To this end they long supported the unit radian, even giving it the status of a 'Supplementary Unit' for many years. [For those unfamiliar with unit classification a 'Supplementary Unit' was a sort of half-way house between 'Base Units' and 'Derived Units'. The only 'Supplementary units', ever, were the radian and the steradian, and both of these were 'downgraded' to dimensionless 'Derived units' in 1980.] Present muddle Our current angle measures include at least these descriptors: angular per cent degrees with decimal degrees degrees with minutes of degree and decimal minutes of degree degrees with minutes of degree, seconds of degree and decimal seconds of degree descriptors such as right angles, straight angles, and revolutions glide ratios such as metres per kilometre grads, grades, or gons (100 grads = 100 grads = 100 gons = 1 right angle = 1 quadrant) mils of angle (1000 mils = 1 quadrant) nautical measures from an octant, a quadrant, or a sextant pitch of a roof points of a compass radians revolutions seconds of Right Ascension or hours of RA slope ratios such as metres per kilometre Prefixes I like John Wards idea that 'that the original killer feature of the metric system was to extend each base unit by the application of a prefix to make a new unit that is 10^N the size of the base unit'. This feature would be particularly appropriate for angle measurements, in building, navigation, and civil engineering. John Ward says, 'If we simply drop minutes and seconds of arc and instead use centidegrees, millidegrees, etc. then 99% of our objections to degrees go away'. What John is saying here is that we should choose as our base unit for plane angle a fraction (1/90) of another angle � the quadrant. This sounds overly complex and confusing to me. John goes on to say, 'Furthermore, the second killer advantage of SI is to have a single, unambiguous global standard. Degrees are exactly that'. I agree that degrees are a defacto global standard, but I repeat that they hold this status because they are defined as part (1/90) of a quadrant. It is the quadrant that is the real standard underlying the definition of the degree. It is also true that the quadrant is the real standard underlying the definition of the grad, the grade, and the gon. John is correct in raising the issue of the need for 'unambiguous global standard' for angle measures. We are all constantly impeded financially, and in terms of our use of time. As Marcus says. 'We're unfortunately wasting countless money by sticking to a mediocre system, even if EVERYBODY uses it!' Calculators Most scientific calculators have an option to flip between degrees, radians, and grades (DEG RAD GRAD on the Radio Shack EC-4021 that I have on my desk). I imagine that if the quad ever became an accepted unit for plane angle, I would simply think of any values of grads, grades, or gons as 10 milliquads (and Marcus could think of them as centiquads) and we could continue to use the same calculators. In time the only alteration needed internally in the calculator is the order of size of entry (as a quad, decimal fraction of a quad, or as a milliquad) and the change of the word, GRAD, on the display to (say) mq for milliquads. Education The first and by far the most common form of angle education is about right angles. When children first draw squares and rectangles the angles they draw are either 'right angles' or they are 'wrong angles'. I doubt that children get much past their first school year without experiencing right angles and how to draw them. Next, most children are taught about right angles being divided into 90 degrees and circles being divided into 360 degrees. This information is then used as the basis for dividing right angles into fractional parts 1/2 = 45�, 1/3 = 30� and 2/3 = 60� being the most common. When I reflect on this, I wonder whether these are taught more as part of the fractions teaching rather than the angles teaching. I accept from Marcus that grads, grades, and/or gons are taught in schools. However, I am unaware of this being a general case. In my experience, the only places that grads, grades, and/or gons were mentioned is in relatively senior mathematics and physics courses in upper secondary schools � and this is a place where many (most) people choose not to go. Having said that, I agree with John Ward when he says, 'Perhaps they were mentioned and most of us just forget about them since they are so seldom used'. Generally, in structuring an education program about angles, we teach our students that they only need to learn about right angles (and occasionally straight angles, because other angles (as Marcus put it) 'self-repeat or collapse into these ones!' Thirds and two-thirds We are quite comfortable with a third of a kilometre � if we ever mention it. We are quite comfortable with a two-thirds of a kilogram � if we ever think of it or use it to order meat at the butchers. I can see no real reason to keep and to maintain our current faulty angle measures simply to preserve a couple of ratios that give convenient calculations in some limited circumstances. Time The issue of the 24 hour clock obviously arises in this discussion as it is (sort of) based on the idea of neat twelfths of a 360 degree circle. This issue needs to be reconsidered, but I won't explore it here. I lean a little toward 'beats' as a concept that might lead us forward. Maybe people such as navigators, cartographers, time-clock and calendar makers could use the opportunity of developing a plane angle unit to rethink all of their priorities and practices. Circles and angles At a basic level, circles are not angles. When a first grade student draws circles, squares, and triangles, they are introduced to angles as being different to circles. It requires a major mind-set shift for people to believe that a circle is a special kind of angle. People need a lot of academic mathematical training to make this mind-set change. I'm still not sure whether I've got over my grade 1 experiences yet. Navigation Modern navigators are introduced to the jargon of their trade during training. They learn that the minute of arc on a grand circle route is equivalent to a nautical mile, but no-one ever tells them that one of the main purposes of the development of the metric system was to make their lives, as navigators, easier. Remember that the great political motivator of the late 18th century was the development of 'empire', and the best, most modern, technology for achieving 'empire' was sailing ships that were constantly being lost because of faulty navigation. In this context, I am rather attracted to Marcus' idea for a reference world globe with a circumference of exactly 40 megametres. Cartography I don't think that the cartographers now have a choice about changing to a new, more rational, way of describing the world � the presence of Global Positioning Systems have put paid to their quaint old ways. Cartographers are being forced into the modern world, and they know it. There are already many moves happening around the world for national coordination of various cartographic systems. I only hope that they can also coordinate these efforts on a true international basis. This need to reorganise the Global Positioning Systems in cartography could also provide an opportunity to rethink the way that cartographers describe angle measurements. Conversion to quads One of the issues to consider here is the way that we (as a whole world group of people) change to new ways of measuring. I think that one of the reasons for the success of the original metric system was that it was sufficiently different from the units used in the past to be differentiated from the older ways. Let me consider some examples of trying to change while at the same time we try to give the illusion that we are remaining the same. The first of these was Napoleon's 'mesures usuelles' that were used to try to give the French people the illusion that things were staying the same while they were changing. In the USA, at about the same time, Thomas Jefferson (1743-1826) tried to redefine the pound, the foot, etc. in decimal terms � this was another failure. Marcus conjectures that, 'The problem though is that this calc seems to point towards "breaking even" in some 6 or 7 decades (or maybe even a century!...). And folks don't like that, so they prefer to put up with 60 minutes to an hour, 60 seconds to a minute and 24 hours to a day, for instance'. My observations of metrication programs lead me to believe that the length of time needed to make major changes to measurement methods is largely a matter of good planning � I have seen at first hand smooth and rapid metrication processes, and I am still living through and observing painful and slow metrication. Any attempt to redefine one of our current units (and to decimalise it in the process) is, in my opinion one of the slowest ways to go. I believe this, because of the evidence we have from the history of people trying to change something as fundamentally important as their measuring methods. I have written elsewhere that I believe that one of the reasons for the delay of metrication in the USA is the success in that nation of decimalisation. [Please let me know if you missed this posting and would like a copy] Pat Naughtin --
