# What makes the right angle special enough to be distinguished in the French metric system?

When introducting the metric system, the French tried to decimalise the degrees used for angles. They defined the right angle to contain 100 gradians.

Why was the right angle chosen? A somewhat equivalent question: Out of all possible angles, why is the right angle particularly special?

The only idea that comes to my mind is that the right angle is the "average" angle, in the sense that it is the arithmetic mean of all angles between 0 degrees and 180 degrees. A problem with this explanation is that an angle is something that can be an arbitrary number of degrees: it certainly does make sense to talk about 321 degrees, as well as minus 123 (-123) degrees. One can argue that any talk of such angles can be paraphrased into talk of an angle in the interval [0,180] degrees. If my argument holds, the right angle is therefore the "average", "quintessential" angle.

I will probably not accept an explanation along the lines of: "Well, it is called right...". Such an explanation is as good as saying that the golden ratio is of utmost importance in mathematics, which is not the case.

• This seems hard to answer since right angles have existed outside of mathematics long before gradians were a thing. They're ubiquitous in woodworking or carpentry (for instance) to check whether parts or joints are square for various reasons - such as the fact that "horizontal" and "vertical" are 90 degrees apart, or that it's easy to reference positions off of a rectangular prism, or that two parts at right angles to another will be parallel to each other, or many other useful properties known intuitively to many - and are part of the common experience of people, not just pure math. Nov 2, 2021 at 16:26
• What's special about a square? about a rectangle? about an orthonormal basis? ... Nov 2, 2021 at 18:03
• Right angles are ubiquitous in everyday life. They are literally anywhere. And if you try to build any machine from scratch, the first thing you'll want to build is a level, and if you have a level and a line and weight you get a right angle, and can continue from there. Just like 100°C marks an important practical point (boiling), the right angle is immensely useful and practical in everyday life. Nov 2, 2021 at 23:36
• RIGHT ANGLE SPECIAL. IF WALL NOT BUILT AT RIGHT ANGLE TO GROUND, FALL DOWN. Nov 3, 2021 at 13:20
• There's a lot of right angles in a guillotine, and they must be accurate for it to work smoothly. Nov 4, 2021 at 0:09

"When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right" is one of the opening definitions of Euclid's Elements. Why did Euclid, or rather Pythagoreans he was systematizing, single it out? Well, it was not by averaging.

For one, the right angle is easy to define, with only two straight lines and equality as prerequisites. For comparison, one would need three straight lines to define $$60^\circ$$, and more for other alternatives. Dropping perpendiculars, that come out of this definition, is also a ubiquitous step in Euclidean demonstrations.

For two, as one can see from the opening propositions of the Elements, Pythagoreans were interested in angles, first and foremost, for their use in triangles. And the right angle produces the simplest form of the relation among the sides of a triangle having it, $$a^2+b^2=c^2$$, the crown jewel of Pythagorean geometry. For comparison, if the angle were $$60^\circ$$, say, it would have been
$$a^2+b^2+ab=c^2$$.

One could, perhaps, instead favor as the primitive what is sometimes called "straight angle" ($$180^\circ$$), definable with a single straight line. Today. But back then it was quite an unnatural "angle" to consider, as it happened before the advent of abstract mathematical formality that embraces extreme cases and extensions for the sake of generality. Indeed, Euclid and other ancients never talk of "straight angles", let alone greater ones, just as they never talk of zero and negative integers. Those creations came later. When Euclid needs to, as in the parallel postulate, he says "two right angles":

"If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

Also, bisection, needed to go from $$180^\circ$$ to $$90^\circ$$, is more complex than duplication, needed to go the other way. So if one intends to use both $$90^\circ$$ may still be easier to start with.

Regardless of whether one finds such arguments from simplicity compelling, by the time of the French revolution the force of Euclidean tradition was by itself more than enough to make the right angle "quintessential" and a natural choice for a primitive.

• I'd argue that today the best primitive would be the full angle (360˚), this does not make this answer any less valid though. Nov 4, 2021 at 14:45

The simplest answer could be that just as a metre is similar in length to a yard, a gram is similar in mass to a scruple, and a litre similar in volume to a quart, 100 gradians are similar in number to 90 degrees. That is, it's a new name for something that's close to what people are already familiar with. There doesn't necessarily have to be anything special about the right-angle itself.

• It's an interesting answer worth pondering about. Nov 2, 2021 at 13:48
• Related to idea of rounding to make it easier to use we also have NATO mils which is roughly a milli-radian, but rounded so that there 6400 mils for a complete circle, en.wikipedia.org/wiki/Milliradian Nov 2, 2021 at 15:27
• I think this is probably the real reason. Though I daresay the resulting similarity between degree and gradian would have probably caused more harm than good, because this makes it so much easier to mix up the units without it being noticed immediately. Nov 3, 2021 at 10:21
• @leftaroundabout, not only mix them up, in most common applications the gradians would be awkward to use. 30° and 60° for instance are very common angles in everyday life. People aren't going to start saying approximately 33.33 and 66.66 instead. I imagine that's a large factor in why this metric unit didn't catch on. Nov 3, 2021 at 12:51
• @RayButterworth IMO much the same could be said for all units. Base-12 is arguably superior to base-10 in almost every way, I wish the French Revolution had decided to abandon base-10 instead of abandoning the 360° system! Nov 3, 2021 at 15:10

I am answering the second question of yours: "Out of all possible angles, why is the right angle particularly special?"

An angle is something that exists on a plane which is a 2D system. In fact, for a system of defining points on a plane, we need two axes (defined by two unit vectors) which are nonparallel and thus inclined to each other by an angle. In the matter of coordinate systems, the Cartesian coordinate system is most popular -- where the axes are at right angles to each other. Right angle is special in this case because the axes become 'orthogonal' to each other i.e.

1. Moving a point on the plane in the direction of one axis does not make it move along the direction of the other axis.
2. A point's coordinate on one axis can be represented by that point's distance from the other axis (along with a sign for direction).

Both of the above won't be true for axes at an acute or obtuse angle to each other.

The 'dot product' of two vectors is zero if they are at right angle to each other. Dot product signifies interdependence in a sense – projection of one thing in the other's direction. Being independent is something of a trivial case, hence right angle is important here.

• Terrific answer! Nov 3, 2021 at 18:34
• @Servaes you're right. During writing I meant to say that moving a point in the 'direction' of one axis does not make it move along the direction of the other axis in the Cartesian system, but somehow I wrote 'coordinate' instead. I am editing to add a correct description. Nov 5, 2021 at 5:59

The unit of angle was tied to the unit of length. Just as a nautical mile is conceptually 1 arcminute (1/21600) of Earth's circumference, the kilometer would be 1 centigradian (1/40000) of it.

So it's possible that the metric system's designers thought that a revised “yard” (1 m, 39.37 in) would be a more convenient unit of length than a “cubit” (40 cm, 15.75 in) or “rod” (4 m, 13.12 ft).

• I think this is the correct answer too, and the person who decided on this was Jean Charles de Borda. Perhaps someone can find a better source. There was already a practice of defining distances in terms of the right angle from the north pole to the equator, de Borda wanted to do this decimally when defining the kilometer (both a decimal degree and a decimal minute). Nov 2, 2021 at 19:34

I don't know what the reasons were for the people who decided it, but one thing that makes the right angle special is that it is, in some sense, the "most" angle there is. If you have two intersecting lines, the farthest from parallel they can be is perpendicular. Once you make the angle between them more than a right angle, you're just making the other angle between them smaller. For instance, consider a stick. If it starts lying flat, and you increase the angle it makes with the ground, the greatest angle it can make is a right angle. Once it reach a right angle, rotating it further just brings it closer to the ground.

Another fact that might be relevant is that the metric system is also built around a right angle of the Earth: a meter was originally defined as 1/10,000,000 of the arc length from the equator to the North pole, or 100 gradians of arc length of the Earth. This makes one gradian of latitude equal to 100,000 meters (note that, again, 100 gradians is the maximum when it comes to latitude). While subdivisions of degrees have traditionally been done in powers of 60 (a minute is 1/60 of a degree, a second is 1/60 of a minute), the metric system would have subdivisions be based on powers of ten. So a meter could be described as 10 microgradians of latitude.

• I think your last paragraph is incorrect. It is the definition of the gradian that "implies" the definition of the metre, not the other way around. This is elaborated on in the accepted answer of "Why is one meter as long as it is?" Nov 3, 2021 at 4:17