Riddles in Mathematics by Eugene P Northrop

Author:Eugene P Northrop
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2014-04-15T04:00:00+00:00

FIG. 77

The hopeful geometer of 1834 set up the figure of diagram (b) to aid him in his proof. Through B he drew B Y parallel to CD, and constructed angles ABN, NBO, OBP, and PBQ each equal to angle YBA, He correctly argued that whatever the size of angle YBA, he could, by constructing enough angles ABN, NBO, . . . , finally arrive at one whose side (BQ in the figure) falls below the line BZ, Suppose there are n – 1 such angles (there are four in the figure). He then marked off n – 1 segments, CE, EG, GJ, . . . , each equal to BC, and through the points of division drew lines EF, GH, JK, . . . , each parallel to CD, So far so good. But at this point he began to compare infinite areas. He maintained, for example, that the infinite area bounded on two sides by the infinite lines BY and BA is equal to the infinite area bounded on two sides by BA and BN, and that the infinite area bounded on three sides by the line segment BC and the infinite lines B Y and CD is equal to the infinite area bounded on three sides by the line segment CE and the infinite lines CD and EF. Or, as he put it, the area YBA is equal to the area ABN, and the area YBCD is equal to the area DCEF.

Let us assume for the moment that this argument and similar arguments about infinite areas are valid. The rest of the proof then runs: Area YBLM is equal to n times area YBCD, and area YBQ is equal to n times area YBA. But area YBLM is only a part of area YBZ, while area YBZ is in turn only a part of area YBQ. Hence n times area YBCD is less than area YBZ, which in turn is less than n times area YBA. That is to say, n-(YBCD) is less than n-(YBA), or YBCD is less than YBA. But if such is the case, AB must meet CD. For if AB did not meet CD, then YBCD would be equal to the sum of YBA and ABCD, and so would be greater than YBA.

The catch in this innocent-looking proof lies, of course, in the fact that the areas involved are infinite. Of two finite areas, we can say that the first is less than, equal to, or greater than the second. But two infinite areas cannot be compared—we can say only that they are infinite.

* * *

The remainder of this section will be devoted for the most part to “limiting curves”—curves which are defined as the limit of an infinite sequence of polygons, that is, of an infinite sequence of figures made up of straight lines. The notion of a limiting curve is not new to any of us who have studied plane geometry. Let us recall briefly how as familiar a curve as the circle can be regarded as the limit of an infinite sequence of regular polygons.

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