In this part, “wretched matter” begins to assert itself.
If we want to measure length, we need a unit of length. Given such a unit, not only lengths, but areas and volumes as well, can be expressed in terms of it: foot, square foot, cubic foot. The founders of the new system desired a natural unit, one that would not depend upon an arbitrary choice, and that would be perpetually available in the natural world, so that it could readily be consulted in case of doubt. Neither criterion would be satisfied by (for example) the length of the king’s foot.
Where to find the natural unit? A first thought would be, “In geometry!”; for there we have the canonical description of the realm of lengths, areas, and volumes, established and known to all. But Euclidean geometry deals only in relative, not absolute, length. Given two straight line segments we can say that one is twice the length of another, or, more generally, speak of their ratio; but on the basis of geometry alone we have no way to describe the length of one such segment. Indeed, imagine trying to convey a length by telephone to someone who knows geometry and number. You can do it, if the two of you have a common material unit, say a foot ruler (in identical copies); but in the absence of such a unit, what could you say?
Yet geometry does afford one kind of absolute measure, that of angle. The whole circular angle (“360 degrees”) is determined by the notion of circle itself; hence that angle, or any part or multiple of it, can serve as a unit. Euclid uses a quarter of it, the right angle (he says, for example, that the sum of the interior angles of a triangle is two right angles); familiar to us is the degree, or 1/360 of the whole circular angle; modern mathematicians favor the radian, the angle at the center of a circle subtended by an arc of the circle equal in length to its radius—an angle which (one can show) is the same for all circles. This last, like the whole circular angle, can be considered especially “natural,” since its definition depends only upon characteristics of the circle, not upon an arbitrary choice such as that of the number 360. But any unit like these can be conveyed by telephone, without reference to a material measure.
Then although we have to refer to the physical world in order to specify a unit length, we may seek to legitimize our choice by deriving that length, in one way or another, from the circle and its angles. This is what the founders of the metric system tried to do, and in two ways. Both approaches are further “natural” in that they take as their point of physical reference the Earth itself, our given habitation.
The first approach proceeds by way of time. Rotating on its axis, the Earth accomplishes a whole circular motion in a day. Taking this as our unit of time, we ask for a simple and natural relation between time and length, from which to derive a corresponding unit of the latter. Nature does supply such a relation, and one which 18th-century scientists were very likely to notice, in the behavior of the pendulum. The time it takes a pendulum to swing from one side to the other is proportional to the square root of its length; thus to each time interval is “naturally” associated the length of the pendulum whose swing occupies that time interval. Hence if we construct a pendulum whose swing takes one day, its length will yield our unit measure. But recognizing that such a pendulum will be impractically long (some 4 1/2 million miles), we select a smaller unit of time, the second, or 1/86,400 of a day, and construct the pendulum whose swing takes one second. Its length proves to be a little more than three feet. This length we can use as our unit; or, admitting that there is nothing natural about the second, inasmuch as the number 86,400 has no natural basis, we can equally choose any fraction or multiple of the length we have found.
The second approach is directly by space. The surface of the spherical Earth contains distinguished circles, namely the so-called great circles, the largest that can be drawn upon it. Such are the equator and the meridians of longitude (the circles that pass through both poles). Corresponding to an angle at the center of one of these circles (which is the center of the Earth) is the arc that subtends it; choosing a unit angle, we can adopt as our unit of length the length of its subtending arc. To the whole circular angle will correspond the circumference of the Earth, about 25,000 miles, too long for ordinary purposes. A length similar to that of the pendulum beating seconds will be obtained by using a very small angle, about 1/40,000,000 of the whole angle. As with the choice of the second of time, an arbitrary number has here crept in. To be sure, we can try to excuse it by, first, calling the right angle natural, on the ground that it can be defined not only by dividing the circle into four parts, but also (as Euclid does) from the fundamental notion of perpendicularity; and then explaining that our unit angle is 1/10,000,000 of that—that is, a tenth of a tenth of a tenth … of it, in obedience to the decimal number system, which is based on our natural fingers.
Notes. 1. The constant of proportionality that enters into the pendulum relation will be considered later. 2. That relation is derived in elementary physics and calculus books; for example, in §5.2.4 of the book named in the “About” tab of this blog. In order for it to hold, the arc of swing must be small (strictly speaking, infinitesimal—the proportion is a limiting relation). 3. The number 86,400 is of course 24 × 60 × 60. The 24 = 2 × 12 may derive from the Zodiac; the 60s, from Babylonian sexagesimal enthusiasts. 4. In the first part of the 19th century—in fact, just possibly as early as 1794, five years before the metric units were finally constructed—the mathematician Gauss (aged 17 in 1794) observed that if the geometry of our space were Lobachevskian, with sufficiently great deviation from the Euclidean, one could pass directly from angle to length, without an intermediary such as the pendulum or the Earth’s surface. This condition is not fulfilled. 5. The reliance on the Earth as spherical will be criticized later.