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Irrationals Or Incommensurables III: the Greek Solution

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Irrationals Or Incommensurables III: the Greek Solution Irrationals or Incommensurables III: The Greek Solution PH I L L I P S. J 0 N E S CBoTH THE PYTHAGOREAN number-theoretic with 17; but the fact that he treated only these proof of the irrationality of ;f2 (or of the incom­ cases indicates that he still did not have a general mensurability of the side and diagonal of a square) process.2 Theaetetus, on the other hand, is supposed and the successive subtraction process which we to have been the original source for such general applied to the square and the pentagon in earlier theorems as Euclid, Book X, Proposition 8: If two articles' required generalization before one could magnitudes have not to one another the ratio which a claim that mathematicians had a theory of number has to a number, the magnitudes will be in­ irrationals. Some generalizations developed quite commensurable; and Book X, 9: The squares on rapidly. We shall merely sketch in the steps in this straight lines commensurable in length have to one development and explain the finished product as it another the ratio which a square number has to a appeared in Euclid. square number; . .. ; and Squares which have not to The chief contributors to this development one another the ratio which a square number has to a were Theodorus of Cyrene (ca. 400 B.C.), Theae­ square number will not have their sides commensu­ tetus (ca. 375 B.C.), and Eudoxus of Cnidos (ca. rable in length either.3 40Q-347 B.C.). Theodorus showed the irrationality The use in these propositions of the words of the non-square integers from 3 to 17. Exactly "magnitude," "line," and "squares on lines" as dif­ how he did this is not certain, nor why he stopped ferent from "numbers" indicates that "number" was still "integer" to the' Greeks. It still had not occurred to them to extend the number system to include new numbers-irrational numbers-in or­ der that the ratios of all magnitudes could be represented by numbers. However, there is at this time explicit recognition of that which was a "logi­ cal scandal" to the Pythagoreans-the existence of magnitudes, areas or lines, which did not have ratios representable by integers. Eudoxus provided the definition of equal ra­ tios or proportion and the method for using the definition which for centuries and until relatively recent times was the tool for dealing with the Reprinted from Mathematics Teacher 49 (Apr., 1956): 282-85; with ratios of incommensurable quantities without the permission of the National Council ofTeachers of Mathematics. use of irrational numbers. This definition as stated 176 IRRATIONALS OR INCOMMENSURABLES: THE GREEK SOLUTION in Euclid's Book V, Definition 5, was: Magnitudes proved that if the base HC is in excess of the base are said to be in the same ratio, the first to the second CL, the triangle AHC is also in excess of the and the third to thefourth, when, ifany equimultiples triangle ALC; if equal, equal; and if less, less. whatever be taken ofthe first and third, and any equal Therefore as the base BC is to the base CD, so is multiples whatever ofthe second andfourth, theformer the triangle ABC to the triangle ACD."5 equimultiples alike exceed, are alike equal to, or alike The proof for parallelograms was similar. To­ fall short of, the latter equimultiples respectively taken day, as then, logically some recognition of the in corresponding order.4 This says that a:b = c:d if possible irrationality of the dimensions of geomet­ and only if ric figures must be made at the time of defining their areas or the ratios of areas. This may be done ka > mb implies kc > md in several ways. One procedure is to postulate a ka = nb implies kc = nd correspondence between the "lengths" of lines or ka < pb implies kc <pd. between points on lines and the real numbers {which ofcourse include irrationals) and to assume An illustration of the use of this definition is to the ability to operate with irrationals. A second be found in the proof of the theorem: Triangles procedure is to use a limit idea explicitly in getting and parallelograms which are under the same height the ratio of a figure, e.g., a rectangle, to a figure are to one another as their bases. In Figure 2, ABC defined to have a unit area. Ofcourse and unfortu­ and ACD are two triangles with the same height. nately, there has also been a tendency to ignore the Produce BD in both directions laying off any num­ whole question in secondary school instruction. ber of segments BG = GH = BC and DK = KL = Both Eudoxus' definition of proportion and CD. Complete the triangles with these segments his "method ofexhaustion" are illustrated by Euclid as bases and with vertex at A. Therefore triangle XII, 2, which says: Circles are to one another as AHC is the same multiple ofABC that base HC is squares on the diameters. The proof is an indirect of BC. proof in which the desired conclusion is reached Similarly triangle ADL is the same multiple of by showing that assuming the ratio of the circles to ACD as DL is of CD. be different from the ratio of the squares of the "Thus there being four magnitudes, two bases diameters leads to a contradiction. BC, CD, and two triangles ABC and ACD, The essential steps in the proof are shown on equimultiples have been taken pf the base BC and the next page with Figure 3 to illustrate them. the triangle ABC, namely the base HC and the (1) If circle BXARD BD2 triangleAHC; and of the base CD and the triangle '* -- ADC other, chance, equimultiples, namely, the circle FKENH FH base LC and the triangle ALC; and it has been then 2 FIGURE 2 BD circle BXARD A FH S // ~ where Sis some circle other than FKENH // v, (2) S is either greater or less than circle / / ,, // FKENH Assume it is less. (Here another type of // \ '- indirect proofis being used. We will follow through / / / \ ', only one-half of it, the "less than" part.) ,L _ _L_ - ~-~ (3) If EFGHis a square inscribed in FKENHit H G B c D K L is more than one-half of FKENH because the in- 177 PART III/HUMAN IMPACT AND THE SOCIETAL STRUCTURING OF MATHEMATICAL A FIGURE 3 E B D F H 0 G scribed square is one-half of a circumscribed square Note that Eudoxus' "method of exhaustion" and the circle is less than its circumscribed square. makes use of Euclid X, 1: Two unequal magnitudes (4) Use the midpoints KNMP of the arcs as being set out, iffrom the greater there be subtracted a vertices of triangles based on the sides of the in­ magnitude greater than its half, andfrom that which scribed square. These triangles are together more is left a magnitude greater than its half, and if this than one-half of the area between the inscribed process be repeated continually, there will be left some square and the circle. magnitude which will be less than the lesser magni­ (5) By repeatedly inscribing triangles in smaller tude set out. 6 In XII, 2 the lesser magnitude was the and smaller circular arcs one can eventually arrive at difference between circle FKENH and S. an inscribed polygon which differs in area from circle This proposition was also used to prove X, 2: If FKENH by an amount less than the difference be­ when the less of two unequal magnitudes is continu­ tween circle FKENH and S. In other words, there is ally subtracted in turn from the greater, that which is a polygon in FKENH which is greater than S. left over never measures the one before it, the magni­ (6) Now inscribe a similar polygon in BXARD. tudes will be incommensurable. This is a general Then: theorem on incommensurables, and like XII, 2, area of polygon in BXARD also has embodied in its proof the subtractive ap­ area of polygon in FKENH proach which we discussed earlier. The subtractive approach coupled with X, 1 BD2 circle BXARD (which can be derived from what is today called FI-P s the Axiom of Archimedes) comes close to modern (7) But circle BXARD is greater than the poly­ limit ideas. In Greek times they were closely re­ gon in BXARD, hence the proportion shows that lated to the philosophical-physical problem of S is greater than the polygon in FKENH whether matter or magnitude is infinitely divis­ But this conclusion contradicts the conclusion ible. Were we to tell the entire story ofincommen­ of step (5), and hence the assumption of step (2) surables and all the related questions of Greek that Sis less than circle FKENH is false. mathematics, we would have to discuss Democritus A similar procedure would show that S cannot (470?-370? B.C.), the first atomist who wrote two be larger than circle FKENH and thus the original books on irrational lines, Zeno (ca. 450 B.C.) and theorem is established. his paradoxes, the ideas of Anaxagoras (ca. 440 178 IRRATIONALS OR INCOMMENSURABLES: THE GREEK SOLUTION B.c.) on infinite divisibility, as well as Antiphon therefore logical necessity, not the mere delight in (ca. 430 B.c.), Bryson (ca. 450 B.c.), and Archytas the visible, which compelled the Pythagoreans to (ca. 400 B.c.), who had some conception of the transmute their algebra into a geometric form."7 method of exhaustion before Eudoxus. However, we will close our discussion of incommensurables through the Greek period by NOTES quoting B.
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