THE NUMBER CONCEPT IN GREEK MATHEMATICS SPRING 2009 0.1. VII, Definition 1. A unit is tht by virtue of which ech of the things tht exist is clled one. 0.2. VII, Definition 2. A number is multitude composed of units: 2 = 1 + 1, 3 = 1 + 1 + 1, etc. 1
2 SPRING 2009 0.3. (1) (Congruence clsses of) Line segments replce rel numbers. These cn be dded nd subtrcted. (2) The product of two line segments is the rectngle formed by these. (3) The product of three line segments is the rectngulr box formed by these. b b + b b b b c bc
EUCLID 3 0.4. Existence of line segments is never n issue. h p q
4 SPRING 2009 0.5. The Archimeden Axiom nd = mb + r, 0 r < b. b 1. Book I 1.1. Euclid s Common Notions or Axioms (1) Things which re equl to the sme thing re lso equl to one nother. (2) If equls be dded to equls, the wholes re equl. (3) If equls be subtrcted from equls, the reminders re equl. (4) Things which coincide with one nother re equl to one nother. (5) The whole is greter thn the prt.
EUCLID 5 2. Book II 2.1. Proposition 1. If there re two stright lines, nd one of them is cut into ny number of segments whtever, then the rectngle contined by the two stright lines equls the sum of the rectngles contined by the uncut stright line nd ech of the segments.[if y = y 1 + y 2 +... + y n, then xy = xy 1 + xy 2 +... + xy n.] 2.2. Proposition 4. If stright line is cut t rndom, the squre on the whole equls the squres on the segments plus twice the rectngle contined by the segments. [(y + z) 2 = y 2 + z 2 + 2yz.] b b
6 SPRING 2009 2.3. Proposition 14. To construct squre equl to given rectilinel figure. h p q
EUCLID 7 3. Book V This book contins the theory of proportions due to Eudoxus nd the lgebr of line segments. Definition 3.1. (1) A mgnitude is prt of mgnitude, the less of the greter, when it mesures the greter. (2) The greter is multitude of the less when it is mesured by the less. (3) A rtio is sort of reltion in respect of size between two mgnitudes of the sme kind. : b = /b. (4) Mgnitudes re sid to hve rtio to one nother which re cpble when multiplied, of exceeding one nother.
8 SPRING 2009 (5) Mgnitudes re sid to be in the sme rtio, the first to the second nd the third to the fourth, when, if ny equimultiples whtever be tken of the first nd third, nd ny equimultiples whtever of the second nd the fourth, the former equimultiples like exceed, re like equl to, or like fll short of, the ltter equimultiples respectively tken in corresponding order. (6) Let mgnitudes which hve the sme rtio be clled proportionl. [The re of circle is proportionl to its rdius squred.]
EUCLID 9 Definition 4. For ny integrl multiple m there is n integrl multiple nb such tht nb > m nd conversely. Definition 4 is n xiom for mgnitudes rther thn definition. This definition lso sys tht the rtio : b cn be pproximted to ny degree of precision by rtionl numbers: Choose m nd then choose n such tht (n 1)b < m nb. Then (n 1)/m < /b < n/m nd n/m (n 1)/m = 1/m.
10 SPRING 2009 Definition 5 then sys when two rtios : nd b : b re equl in terms of rtionls: / = b/b, when for ll numbers n nd m it is the cse tht if n is greter, equl, or less thn m, then nb is greter, equl, or less thn mb, respectively, tht is, n > = < m nb > = < mb. equivlently, for every rtionl m/n, we hve > > / = m/n b/b = < < m/n. This is vlid criterion for the equlity of the rel numbers / nd b/b.
EUCLID 11 3.2. Proposition 1. m + mb + mc + = m( + b + c +...). 3.3. Proposition 2. m+n+p+ = (m + n + p +...). 3.4. Proposition 3. n(m) = (nm). 3.5. Proposition 4. If : b = c : d then m : nb = mc : nd. 3.6. Proposition 5. (m) (nb) = (m n)b. There re 25 propositions of this nture ltogether.
12 SPRING 2009 4. Book VI The results of this book which dels with similrity contins very useful nd importnt results. Definition 4.1. Similr rectilinel figures re such s hve their ngles severlly equl nd the sides bout the equl ngles proportionl. 4.2. Proposition 1. Tringles nd prllelogrms which re under the sme height re to one nother s their bses. Are( (ABC)) : Are( (AB C)) = AB : AB. C A B B
EUCLID 13 The following result is the bsic result on similrity. 4.3. Proposition 2. Let CAB be cut by trnsversl prllel to BC in the points B, C where the nottion is chosen so tht B AB nd C AC. Then AB : BB = AC : CC iff BC B C. Proof. (AB C) = (ABC ) nd AB AB = (ABC) (AB C), (ABC) (ABC ) = AC AC C C A B B The next four proposition re similrity theorems in nlogy to the congruence theorems.
14 SPRING 2009 5. Book X Book X does not mke esy reding (B. vn der Werden, Science Awkening, p. 172.) It dels vi geometry nd geometric lgebr with wht we cll tody rtionl nd irrtionl numbers. In fct, 13 different kinds of irrtionlities re distinguished. Definition 5.1. (1) Those mgnitudes re sid to be commensurble which re mesured by the sme mesure, nd those incommensurble which cnnot hve ny common mesure. (2) Stright lines re commensurble in squre when the squres on them re mesured by the sme re, otherwise they re incommensurble in squre.
EUCLID 15 5.2. Proposition 2. If, when the less of two unequl mgnitudes is continully subtrcted in turn from the greter tht which is left never mesures the one before it, then the two mgnitudes re incommensurble. 36 r >, 2 > r, 1 < r/ < 2 r 1 = 1 = r r/ = r 1 / 1 =... r 36 72 36
16 SPRING 2009 5.3. Proposition 3. To find the gretest common mesure of two given commensurble mgnitudes. 5.4. Corollry. If mgnitude mesures two mgnitudes, then it lso mesures their gretest common mesure.
EUCLID 17 The book contins 115 propositions none of which is recognizble t first sight. There is generl greement tht the difficulty nd the limittions of geometric lgebr contributed to the decy of Greek mthemtics (Vn der Werden, Science Awkening, p.265.) Author like Archimedes nd Apollonius were too difficult to red. However, Vn der Werden disputes tht it ws lck of understnding of irrtionlity which drove the Greek mthemticins into the ded-end street of geometric lgebr. Rther it ws the discovery of irrtionlity, e.g. the digonl of squre is incommensurble with the side of the squre, nd strict, logicl concept of number which ws the root cuse.