I have a problem understanding something about the metric system and I was wondering if someone could help. Now, don’t get me wrong, the metric system looks great on paper. A decimal system is wonderful for mathematical renderings…it is really easy to compute and to understand. For example, to express 3mm all you need to do is present it as 3 x 10 to the -3rd meters. It is all nice and neat; straight forward. To tell the truth, the simplicity of the measure is somewhat attractive. Metric is so easy to remember: 10mm is 1cm, 10cm is 1dm, 10dm is 1 meter, etc. Metric is touted as being a wonderful scientific measure because of its exactness and its easy divisibility into units of 10.

Now, I have read many essays and editorials supporting the metric system on its ‘scientific’ basis. I have read several accounts of the history of the metric system. I have seen many references to how much better the metric system is because it is measured with divisions and multiples of 10. What I have not seen is HOW these 18th century scientists divided the meter into 10ths, 100ths, or 1000ths. Yes, the metric system is supposed to be ‘scientific’ and ‘exact’, but I have yet to hear a satisfactory explanation as to how one can divide a line segment into 10 or even 5 equal segments. As for as I can tell (and I have tried many times) it is impossible to divide a line segment into 5 or 10 equal segments using normal geometric techniques. The only ways I can think of are either using a ruler and measuring to the nearest 32nd of an inch or using a computer which is capable of using trial and error techniques until it gets the divisions evenly spaced.

Obviously, there must be a way other than the cheating methods I described. The 18th century French did not have computers; and the metric system was supposed to be separate from traditional measure so they would not have used a ruler. Now, I have been told that it is in fact possible to trisect an angle using a marked straight edge (not a ruler) and that it is possible to find 2/3 of a line segment. I have not heard whether it is possible to find 3/5 of a line segment or to ‘pentasect’ an angle.

Does anyone know a formula based geometric way (or another way accessible by a normal human) to divide a line segment into 5 or 10 equal segments? I would really like to know. For that matter, I would also like to know how to find out how to find 2/3 of a line segment now that I’ve been told its possible.

Thanks.

“Those who claim to discover everything, but produce no proofs of the same, may be confuted as having pretended to discover the impossible.” --Archimedes

As often happens when I ask a difficult question, I discovered the answer myself a mere 24 hours from asking. It is, in fact, possible to divide a line into any number of equal segments. The existence of this fact means that my essay question above and this addendum probably will not spark the debate that I had hoped but instead will merely serve as a useful reference to those interested in the geometric aspects of measurement. Since it is possible to ‘pentasect’ a line, my above argument really has no purpose other than the educational value of the question itself and this answer. Oh well. Here’s the method:

Example: Trisecting Line AB (for this exercise, you will need a compass and a straight edge; I recommend a foot long ruler… it has the right feel).

Given segment AB, draw a ray from A at any convenient acute angle. Open the compass to any convenient radius and construct (on that ray) a line of x segments where x is the number of segments you want line AB to be divided into (in this case, 3). Call the points on the 3 segments of the ray ‘A-C-D-E’. Connect points E and B to make a complete triangle. Construct points F and G on AB with FC and GD parallel to BE. A-F-G-B is your solution. [Note: to construct FC and GD in such a way that they are parallel with line BE, first construct the perpendicular bisector of line BE. Next, place the point of the compass on point D (or C, it doesn’t matter). With your compass at a convenient radius, draw an arc of a circle centered on point D (or C) that intersects the perpendicular bisector of BE at two points. Next, construct the perpendicular bisector of the segment created (by the two points of the arc) and you will have a line that intersects point D (or C) and that is parallel with line BE. Extend this perpendicular bisector to the point where it intersects line AB; this will be point G (or F, depending). Repeat this process with the point on line AE that you didn’t use the first time and you will have divided line AB into 3 equal segments.]

I have tested this method with 5 segments, as well, and it works. I suspect it will work with any number of line segments.

There is an advantage to having units that are divisible by 12. Using an ancient hand of 3 inches, the yard divides into 12 hands. The foot divides into 12 inches. 12 equal units can be arranged into a right triangle with sides of 3, 4, & 5 units respectively. Therefore, you can easily construct a right triangle with a perimeter equal to exactly a foot or a yard.

There isn’t much practical use for this in modern times. In the days before you could go to the hardware store and buy a right angle, however, you could make your own with a string of either a foot or a yard divided by knots spaced at either 1 or 3 inch intervals respectively.

And yes, you can do it with metric, but you can’t do it with an exact decimeter or an exact meter.

Anyhow, just an interesting piece of trivia I thought you might like…

Construct a line of n equal parts C0,C1...Cn disjoint from the line AB. Extend the lines AC0 and BCn to meet at D. Draw the lines DC1...DCn-1, cutting AB into n equal parts. QED.