What Magellan's Voyage Didn't Prove or Why the Earth Is Flat

When seeking to indict some belief as archaic, superstitious or ridiculous, the knowledgeable polemicist is likely to compare the position he opposes to the belief that the earth is flat. He will do this on the presumption that scarcely anything could serve as a better example of a falsehood which only the ignorant or the unthinking have accepted. Magellan's voyage proved that the earth is round. Therefore, so we are all prone to conclude, the earth isn't flat. Those who say it is are to be corrected if teachable, dismissed if not. In truth, however, the earth is flat, more or less. The conviction that it isn't affords a fine example of how easily and powerfully proof can be misconceived. A typical "demonstration" of the folly of believing in the earth's flatness will begin by pointing out the difference between something round, such as a tennis ball, and something flat, such as a table top. The question will then be asked which of the two things the earth is like. Since we know that the earth is a sphere and has no edges, we will say it is like the ball. After saying this we are apt to conclude that it, like the ball, is dissimilar to the table top, and so not flat. The obvious contrast between the ball and the table top appears to give us no other option. We are likely to feel certain that because the earth is round it just can't be flat. Persuasive as it is, a "demonstration" of this kind is faulty. Implicit in it is the assumption that flat and round are always contrasting features. Normally they are contrasting, and this explains our readiness to say they always are. Still, some objects are both round and flat. The earth is a case in point. To see that it is, one must bear in mind how in our practical affairs we use the word 'flat'. A horizontal surface we are pre- pared to call flat is one we recognize to be neither a dome nor a bowl. It is neither concave nor convex, but a level expanse. Our grounds for saying this have nothing to do with a measurement of zero curvature. Rather they pertain to the behaviour of liquids and solids placed on its surface. We call a table top flat, for instance, because when a spirit-level is placed on it a reading of zero inclination appears. It does not shed water drops and marbles like an umbrella, nor does it collect them like a bucket. Any surface which does collect water at its centre is a bowl; any surface which does shed water is a dome. By the conventions of language, then, a flat surface is a level expanse on which solids and liquids tend to remain stable. On these grounds there are plenty of regions on the earth's surface which are flat. Its regs, mesas, prairies, soda pans, lakes and oceans certainly come very close to being what any plain spoken man interested in having himself understood would call flat. One walks neither uphill nor downhill when treking across the Utah salt flats. They are, then, appropriately named. They match, too, the general curvature of the earth. They, and all similar areas, illustrate the point that being flat and being curved are compatible. Indeed, a little thought must show that a flat surface is necessarily a curved surface. What keeps a marble from rolling off a table is the uniformity of gravitational attraction at all points along the surface of the table-top. Since the force of gravity varies with the distance from the centre of gravitational attraction, all points on a horizontal, flat surface are equally distant from some one point. For us that point is approximately the centre of the earth. Because a surface can have all its points equally distant from some one point outside itself only if it is curved, flat surfaces must be curved. All their points must be equally distant from a point close to the earth's centre. Although variable, the earth's surface in general meets this condition. Its curvature acts as what might be called a natural paradigm of flatness. Surfaces whose curva- ture approximates that of a level expanse of the earth just are those we call flat. Any arc greater than the earth's curvature-such as that of a tennis ball-will result in a dome from which liquids and solids will flow, slide or roll. Any less an arc-such as that of a bucket-will result in a basin or bowl into which water and debris will collect. Flatness, so to speak, is a certain degree of curvature approximating that of the earth itself. If the right hand side of a perfectly flat table were to be extended, it would even- tually curve around the earth to join itself on the left. We do not make our tables this large, of course, and so they all have edges from which items will fall if pushed far enough. This explains why, when mentally comparing a tabletop to the earth, we tend to think of both as having edges. That is our true mistake, not the thought that the earth's surface is flat. In case this observation seems dubious, a thought experiment will confirm it. Imagine the earth with one of its polar regions scraped off to 5o degrees of latitude so that its surface there parallels a ray of light. Many people would be tempted to say the earth would then be flat in that region, just as many now say the poles are, in fact, comparatively flat because their curvature is less than what we find at the equator. The temptation to speak this way stems from our entertaining a faulty picture of a flat expanse as a surface of zero curvature. It is a temptation which experience would very quickly dissipate. If left standing in the centre of a scraped off polar region, we would note immediately that we were in a huge depression. Any walk we took would lead us further away from the earth's centre against the pull of gravity. Any walk would, then, be a climb, and no one who had to hike to the rim of the depression would consider the territory he crossed flat. At o degrees of latitude the slope would approach some 20 degrees of inclination. Clearly, a flat surface is one whose curvature approximates the gen- eral curvature of the earth. Those who think otherwise are guilty of the same mistake our medieval ancestors made. Some of the men sailing with Columbus feared they would sail off the earth, thinking it to be flat and with edges. We now discount those fears on the ground that the earth is round and without edges. The fear and its dismissal, however, rest on the shared error that what is flat cannot be round. The terror of the medieval sailor and the equanimity of mind of the modern seaman have in common a faulty picture of what it is for a surface to be flat. An increase in geographical knowledge has dispelled a groundless fear, but it has not corrected a faulty conception. Indeed, in one respect our knowledge has added to a store of confusion. From an essentially correct belief in the flatness of the earth the pre-Columbian sailor drew a wrong inference on the danger of falling into space, but at least there was no discrepancy between the truth, his usage and his conviction on what is flat. What he called flat is what he rightly believed to be flat. The modern sailor who erroneously denies that the world is flat faces, however, the embarrassment that his usage and his convictions are inconsistent. He is almost certain to call anything whose curvature matches that of the earth flat, even though on reflection he will deny that what he is prone to say is true. He is not in a position to "correct" his normal habits of speech either, for if he tried he would find that there is nothing he could call flat and still be understood. His offering a surface of zero curvature as an example of what he means would only puzzle others, for what is the point of calling a basin flat? From this example of how easily proof can be misconceived a lesson can be learned. It begins with the observation that a more or less spontaneous picture, or imaginative conception, can stand in the way of our seeing how we use a word. The picture of an uncurved surface gives us a wrong idea of our use of the word 'flat'. The lesson continues with the warning that in such a case we should not follow the advice of pragmatists and positivists on how to clarify our ideas by asking if reality corre- sponds to our picture. Imagining what would verify or refute the thesis that the earth is flat, for example, is apt to mire us deeper in a muddle, leaving our faulty picture intact but our convictions now fully at odds with our usage. For all of us in our unreflecting moments nothing could serve as a better example of a flat surface than a calm lake or salt deposit. Yet, when invited to think about the matter, we are prone to conclude that Magellan's voyage proved such expanses to be less than flat. The imagined "proof" is deceptive; it is the unreflecting usage which is right. Things, after all, are what we say they are. Clarity of mind requires that we honour our usage and resist the "proof" which only stengthens the force of a faulty picture.