WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS

WHICH MEAN DO YOU MEAN? AN EXPOSITION ON MEANS A Thesis Submitted to the Graduate Faculty of the Louisiaa State Uiversity ad Agricultural ad Mechaical College i partial fulfillmet of the requiremets for the degree of Master of Sciece i The Departmet of Mathematics by Mabrouck K. Faradj B.S., L.S.U., 1986 M.P.A., L.S.U., 1997 August, 004

Ackowledgmets This work was motivated by a upublished paper writte by Dr. Madde i 000. This thesis would ot be possible without cotributios from may people. To every oe who cotributed to this project, my deepest gratitude. It is a pleasure to give special thaks to Professor James J. Madde for helpig me complete this work. This thesis is dedicated to my wife Mariaa for sacrificig so much of her self so that I may realize my dreams. It would ot have bee doe without her support. ii

Table of Cotets Ackowledgmets................................................................... List of Tables....................................................................... List of Figures...................................................................... Abstract............................................................................ ii iv v vi Chapter 1. Itroductio............................................................. 1 1.1 The Origis of the Term Mea............................ 1 1. Atique Meas.................................... 1 1.3 Geometric Iterpretatio of the Atique Meas................... 3 1.4 Atique Meas Iequality............................... 4 Chapter. Classical Meas.......................................................... 7.1 The History of Classical Meas............................ 7. The Developmet of Classical Meas Theory.................... 9.3 Nicomachus List of Meas.............................. 11.4 Pappus List of Meas................................ 13.5 A Moder Recostructio of the Classical Meas.................. 15.6 Other Meas of the Aciet Greeks......................... 16 Chapter 3. Biary Meas............................................................ 19 3.1 The Theory of Biary Meas............................. 19 3. Classical Meas as Biary Mea Fuctios..................... 0 3.3 Biary Power Meas................................. 3.4 The Logarithmic Biary Mea............................ 4 3.5 Represetatio of Liks betwee Biary Meas................... 6 3.6 Other Biary Meas.................................. 7 Chapter 4. -ary Meas............................................................. 3 4.1 Historical Overview.................................. 3 4. The Axiomatic Theory of -ary Meas........................ 36 4.3 Traslatio Ivariace Property of -ary Meas................... 43 4.4 Iequality Amog -ary Meas............................ 48 Chapter 5. Coclusio............................................................... 56 Refereces.......................................................................... 58 Vita................................................................................ 63 iii

List of Tables.1 Nicomachus Meas................................. 13. Pappus Equatios for Meas............................. 14 iv

List of Figures 1.1 Demostratio of Atique Meas usig a circle.................... 4 1. Proof Without Words: A Truly Algebraic Iequality.................. 5 3.3 Biary Meas as Parts of a Trapezoid........................ 6 3.4 Sigle Variable Fuctio Associated With Biary Meas.............. 9 v

Abstract The objective of this thesis is to give a brief expositio o the theory of meas. I Greek mathematics, meas are itermediate values betwee two extremes, while i moder mathematics, a mea is a measure of the cetral tedecy for a set of umbers. We begi by explorig the origi of the atique meas ad list the classical meas. Next, we preset a overview of the theories of biary meas ad -ary meas. We iclude a geeral discussio o axiomatic systems for meas ad preset theorems o properties that characterize the most commo types of meas. vi

Chapter 1. Itroductio I the this chapter we give a brief itroductio to the origis of the arithmetic, geometric, ad harmoic meas. 1.1 The Origis of the Term Mea Accordig to "Webster s New Uiversal Dictioary", the term mea is used to refer to a quatity that is betwee the values of two or more quatities. The term mea is derived from the Frech root word mie whose origi is the Lati word medius, a term used to refer to a place, time, quatity, value, kid, or quality which occupies a middle positio.the most commo usage of the term mea is to express the average of a set of values. The term average, from the Frech word averie, is itself rich i history ad has exteded usage. The term average was used i medieval Europe to refer to a taxig system levied by a liege lord o a vassal or a peasat. The word average is derived from the Arabic awariyah, which traslates as goods damaged i shippig. I the late middle ages, average was used i Frace ad Italy to refer to fiacial loss resultig from damaged goods, where it came to specify the portio of the loss bore by each of the may people who ivested i the ship or its cargo. I this usage, it is the amout idividually paid by each of the ivestors whe a loss is divided equally amog them. The otio of a average is very useful i commerce, sciece, ad legal pursuits; thus, it is ot surprisig that several possible kids of averages have bee iveted so that a wide array of choices of a itermediate value for a give set of values is available to the user to select from. 1. Atique Meas The earliest documeted usage of a mea was i coectio with arithmetic, geometry, ad music. I the 5th cetury B.C., the Greek mathematicia Archytas gave a defiitio of the three commoly used meas of his time i his treatise o music: we have the arithmetic mea whe, of three terms, the first exceeds the secod by the same amout as the secod exceeds the third; the geometric mea whe the first is to the 1

secod as the secod is to the third; the harmoic mea whe the three terms are such that by what ever part of itself the first exceeds the secod, the secod exceeds the third by the same part of the third. (Thomas, 1939, p. 36) This ca be traslated to moder terms as follows. Let a ad b be two whole umbers such that a > b ad A, G, ad H are the arithmetic, geometric, ad harmoic meas of a ad b respectively. The (i) a A = A b = A = a+b, (ii) a G = G b = G = ab, a H a (iii) = H b b = H = a+b ab. The origis of the ames give to the atique meas are obscured by time. The first of these meas, ad probably the oldest, is the arithmetic mea. To the aciet Greeks, the term αριθµητικς refers to the art of coutig, ad so, fittigly, they referred to what we commoly call the average as the arithmetic mea sice it pertais to fidig a umber that is itermediate to a give pair of atural umbers. As for the ame give to the geometric mea, it appears that the Pythagorea school coied the term mea proportioal, i.e., the geometric mea, to refer to the measure of a altitude draw from the right agle to the hypoteuse of a right triagle. The measure of such a altitude is betwee the measures of the two segmets of the hypoteuse. The source of the ame give to the harmoic mea ca oly be foud i legeds. The Roma Boethius (circa 5 A.D.) tells us of a leged about Pythagoras who o passig a blacksmith shop was struck by the fact that the souds caused by the beatig of differet hammers o the avil formed a fairly musical whole. This observatio motivated Pythagoras to ivestigate the relatio betwee the legth of a vibratig strig ad the musical toe it produced. He observed that differet harmoic musical toes are produced by particular ratios of the legth of the vibratig strig to its whole. He cocluded, accordig to the leged, that the musical harmoy produced was to be foud i particular ratios of the legth of the vibratig strig. Thus to the Pythagoreas, who believed that all kowledge ca be reduced to relatios betwee umbers, musical harmoy

occurred because certai ratio of umbers that lie betwee two extremes are harmoic, ad thus the term harmoic mea was give to that value. Propositio 1. Suppose 0 < a b. Let A := a+b, G := ab, ad H := a+b ab. The a A b, a G b, ad a H b Proof. Sice 0 < a b, the a + b b; therefore, a+b b. Similarly, a a + b; therefore a a+b a+b. Therefore, a b. Thus, a A b. Hece, b 1 a+b 1 ab a. Therefore, a a+b b, ad a H b. If a b, a ab b ; therefore, a ab b. Hece a G b. 1.3 Geometric Iterpretatio of the Atique Meas Sice geometry is the aciet Greeks preferred veue of scietific ivestigatio, Greek mathematicias produced umerous geometric treatises that related the three atique meas to each other by usig straight edge ad compass costructio. A excellet example ca be foud i Schild (1974) ad reproduced here: Example 1.1. Suppose a ad b are two whole umbers. Let A, G, ad H be the arithmetic, geometric, ad harmoic meas respectively of a ad b. The by usig a straight edge ad compass we ca illustrate that A = a+b, G = ab, ad H = a+b ab. Draw the lie segmet LMN with LM = a ad MN = b (see figure 1.1). With LN as diameter, draw a semi circle with ceter O ad fix P o its circumferece. Draw MQ perpedicular to OP ad MP perpedicular to LN. The OP = A, MP = G, ad QP = H. To show this is true, we give the followig argumet. Sice OP is the radius of the circle whose diameter is LN, the OP = 1 (a + b) = A, ad sice (MP) = (LM)(MN) = ab, the MP = ab = G. Let α = POM. Observe that QMP = α ad POM is similar to PMQ; thus, PQ PM = PM PO. Therefore, PQ = (PM) PO = ab a+b = ab a+b = H. I figure 1.1, observe what happes if (a + b) remais fixed, i.e., segmet LN is fixed, ad M is allowed to move. As M moves toward N, both G ad H decrease. As M moves towards O, both G ad H icrease. If M coicides with O, i.e., a = b, the A = G = H. This may have bee the motivatio for ivestigatig the iequality betwee the three meas. 3

P Q α α L O M N a b FIGURE 1.1. Demostratio of Atique Meas usig a circle. 1.4 Atique Meas Iequality I this sectio we will preset several proofs of the iequality: H G A (1.1) Of the umerous useful iequalities i mathematics, the arithmetic-geometric mea iequality occupies a special positio, ot oly from a historical stadpoit, but also o accout of its frequet usage i differet mathematical proofs. We will give a more i-depth discussio about this iequality i Chapters 3 ad 4. At this poit, it suffices to say that there have bee umerous proofs give for the above iequality over the ceturies. We begi our discussio by presetig a iformal argumet of the iequality. Referrig back to Figure 1.1, we ote that siα 1. From OPM, we have siα = MP OP, ad from PQM, we have siα = QP MP. Therefore MP = OPsiα G = Asiα. Hece G A, (1.) ad QP = MPsiα H = Gsiα. Hece H G. (1.3) From 1. ad 1.3, we get 1.1. However, sice the above argumet uses trigoometry, it does ot reflect the spirit of the aciet proofs for this iequality. I Figure 1., we preset a illustratio that captures the fudametal 4

character of this iequality i mathematics, which may have motivated the aciet mathematicias to establish proofs of the arithmetic-geometric mea iequality (Gallat 1977). The iequality as illustrated by Figure 1. requires oly rudimetary kowledge of geometry to prove. Now we give a more moder algebraic proof for the geometric-arithmetic mea iequality. ab a b FIGURE 1.. Proof Without Words: A Truly Algebraic Iequality. Theorem 1. For ay oegative umbers a ad b, ab a+b, with equality holdig if ad oly if a = b. Proof. Let a = c ad b = d. The a+b ab becomes c +d cd, or equivaletly, c +d cd 0. This is equivalet to c cd + d 0 which is i tur equivalet to (c d) 0. Sice the square of ay real umber is oegative, we see that the iequality stated i the theorem is ideed true. Equality holds if ad oly if c d = 0, that is c = d, or equivaletly, if ad oly if, a = b. We use the result from theorem 1 to establish a iequality betwee the harmoic ad geometric meas of ay two oegative umbers. Corollary 1.. For ay oegative umbers a ad b, ab a+b ab, with equality holdig if ad oly if a = b. Proof. Sice ab a+b, the ab (a + b). Therefore, ab (a + b) ab, ad ab a+b ab. 5

From theorem 1 ad corollary 1., we have H G A. (1.4) 6

Chapter. Classical Meas I this chapter we will explore the origis of the theory of biary meas. The chapter icludes two lists of the classical biary meas as give by Greek mathematicias. The followig list gives the ames of Greek mathematicia ad the approximate dates of their work o meas. It is helpful to the uderstadig of the historical developmet of the theory meas i the aciet Greek world (Smith 1951). Thales, 600 B.C. Pythagoras, 540 B.C. Archytas, 400 B.C. Plato, 380 B.C. Eudoxus, 370 B.C. Eudemus, 335 B.C. Euclid, 300 B.C. Archimedes, 30 B.C. Hero, 50 A.D. Nicomachus, 100 A.D. Theo, 15 A.D. Porphyrius, 75 A.D. Pappus, 300 A.D. Iamblichus, 35 A.D. Proclus, 460 A.D. Boethius, 510 A.D..1 The History of Classical Meas I this sectio we will give a brief discussio o what motivated Greek mathematicias to study ad develop a doctrie for meas by presetig the ratioale give by promiet Greek mathematicias who touched o the history of the theory of meas i their work ad the opiios of Greek mathematics scholars o this matter. Accordig to Gow (193), by Plato s time umbers were grouped ito two geeral categories. First, as sigle umbers categorized by their attributes such as odd, eve, triagular, perfect, excessive, defective, amicable etc. Secod, umbers were viewed as groups comprised of umbers that are either i series or proportios. The aciet Greeks viewed meas as a special case of proportios (Allema 1877, Thomas 1939, Gow 193). Smith (1951) writes, " Early [Greek] writers spoke of a arithmetic proportio, meaig b a = d c as i,3,4,5, ad of geometric proportio, meaig a : b = c : d as i, 4, 5, 10, ad a harmoic proportio, meaig 1 b a 1 = d 1 1 c as i 1, 1 3, 4 1; 1 5." I his commets o paradigms of aciet Greek mathematics, Allema (1877) says, "whe two quatities were compared [i Greek mathematics], the basis for the compariso seems to be either how much the oe is greater tha the other, i.e., a arithmetic 7

ratio, or how may times is the oe cotaied i the other, i.e., their geometrical ratio." Allema (1877) claims that this type of compariso of ratios would aturally lead to the theory of meas because for ay three positive magitudes, be it lies or umbers, a, b, ad c, if a b = b c, the three magitudes are i arithmetical proportio, but if a : b :: b : c, they are i geometrical proportio. Allema s claim seems to be supported by the work of Nicomachus i "Itroductio to Arithmetic". I this work, Nicomachus bega his discourse o meas by givig the defiitio that distiguished a ratio from a proportio. He referred to the latter as the compositio of two ratios. He the stated that whe oe term appears o both sides of a proportio, as i b a = b c, the proportio is kow as a cotiued proportio. The proportio is called disjuct whe the middle terms are differet. The highest term i a cotiued proportio is called the cosequet, the least is called the atecedet, ad the middle term is the mea, µεστητες, which is medius whe traslated ito Lati ad from which the word mea is derived (Gow 193). As we have oted above, Greek mathematics viewed meas as a special proportio ivolvig three magitudes; therefore, it is appropriate that we begi our review of the history of developmet of meas by metioig that Proclus attributed to Thales the begiig of the doctrie of proportios (Allema 1877). Thales established the theorem that equiagular triagles have proportioal sides (Allema 1877). I "Itroductio to Arithmetic", Nicomachus writes, "the kowledge of proportios is particularly importat for the study of aciet mathematicias." This ca be take to mea that the doctrie of proportios played a importat role i the developmet of Greek mathematics. Maziarz (1968) commets o the atural developmet of the theory of proportioals i Greek mathematics by sayig, "If a poit is a uit i a positio, the a lie is made of poits. Cosequetly, the ratio of two give segmets is merely the ratio of the umber of poits i each. Moreover, because ay magitude ivolves a ratio betwee the umber of uits it cotais ad the uit itself, ad, thus, the compariso of two magitudes implies either or 4 ratios." By poits, Maziarz seems to imply the tick marks that would be made if the segmets were divided ito may small equal uits. 8

From the historical perspective, the aciet sources of Greek mathematics history that we have refereced do ot metio whe the arithmetic mea was first developed. However, they offer various explaatios as to whe the geometric ad harmoic meas were first itroduced. Allema (1877) states that aciet sources (Iamblichus, Nicomachus, Proclus) poit to Eudoxus as the oe who established the harmoic mea ad to Pythagoras as the oe who established the otio of a mea proportioal betwee two give lies. It is iterestig to ote that some facets of the theory of meas appear i various aciet Greek texts. Some of these were iteded as mathematics treatises, such as the collectio of books that costitute Euclid s work kow as the "Elemets", but others did ot have a apparet mathematical purpose. Oe such example, oted by Maziarz (1968), ca be foud i passages of "Timaeus" kow as "The Costructio the world-soul." I this sectio of the book, Plato attempts to costruct the arithmetical cotiuum usig two geometric progressios 1,,4,8 ad 1,3,9,7; the fillig i the itervals betwee these umbers with the arithmetic ad harmoic meas. By successive duplicatio of the two progressios ad fillig i with the appropriate combiatio of arithmetic ad harmoic meas, all umbers ca be geerated, but ot i their atural order. Aother example ca be foud i Aristotle s "Metaphysics". I this work, Aristotle describes Plato s otio of distributive justice as, " The just i this sese is a mea betwee two extremes that are disproportioate, sice the proportioate is a mea, ad the just is proportioate. This kid of proportio is termed by mathematicias geometrical proportio." From the above examples, oe gets the sese that to the aciet Greeks, the theory of meas ad proportios may ot have bee just a mere mathematical cocept sice some aspects of the theory of meas was also reflected i their literature, philosophy, ad religio.. The Developmet of Classical Meas Theory It appears that the classical meas were developed over a log period of time by the gradual additio of seve more meas to the first three (Heath 1963). I all his work, Euclid oly uses the three atique meas (Allema 1887, Gow 193). However, by first cetury A.D., we kow that 9

Greek mathematicias referred to te meas. All the sources reviewed (Allema 1887, Bema 1910, Heath 191, Gow 193, Thomas 1939, Smith 1951) suggest that Greek mathematicias geerated these meas by cosiderig three quatities a, b, ad c, such that a > b > c. They assumed b to be the mea ad formed three positive differeces with the a, b, ad c: (a b), (b c), ad (a c). The they formed a proportio by equatig a ratio of two of these differeces to a ratio of two of the origial magitudes, a, b, ad c. For example, b is the harmoic mea of a ad c whe a b b c = a c. Nicomachus i "Itroductio to Arithmetic" (Gow 193) goes o to say: "Pythagoras, Plato, ad Aristotle kew oly six kids of [cotiued] proportios: the arithmetic, geometric, ad harmoic meas, ad their subcotraries, which have o ames. Later writers added four more." Greek mathematicias referred to certai classical meas as cotrary ad subcotrary meas because these meas were see to be i a cotrary (opposite) order from the arithmetic mea whe compared to the geometric or harmoic meas (Oxford Eglish Dictioary 004). I his work "I Nicomachus" (Heath 191), Iamblichus says, "the first three [atique meas] oly were kow to Pythagoras, the secod three were iveted by Eudoxus." The remaiig four, Iamblichus attributed to the later Pythagoreas. He adds that all te were treated i the Euclidea maer by Pappus. Gow (193) states that the umber of cotiued proportios was raised to te ad kept at that umber because the umber te was held by the aciet Greek mathematicias to be the most perfect umber. He adds, "how else ca we explai the fact that the golde mea, which Nicomachus calls the most perfect ad embracig of all proportios, was left out from the list of meas." All these testimoies poit to the coclusio that the theory of meas i Greek mathematics was well established by the First Cetury. Our mai complete source for aciet Greek mathematics theory of meas is Boethius commetary o the works of Pappus ad Nicomachus. I this work, 10

Boethius credits Nicomachus ad Pappus as the mai Greek mathematicias who dealt with meas from a theoretical perspective (Smith 1951)..3 Nicomachus List of Meas The earliest kow treatmet of classical meas as a idepedet body of kowledge was give by Nicomachus i "Itroductio to Arithmetic" (Allema 1887, Heath 191, Gow 193, Thomas 1939, Smith 1951). Allema, Gow, Heath, ad Thomas cocluded (seemigly idepedet of each other) that Nicomachus proceeded to develop his list as follows: He bega his list by commetig o the cotiued arithmetical proportio a b = b c. This suggests that a b : b c :: a : a, which allows us to make a coectio to other meas. Gow (193) remarks, "I a cotiued geometric proportio, a : b :: b : c, he otices that a b : b c :: a : b. Fially, the three magitudes, a, b, c, are i harmoic proportio if a b : b c :: a : c." A similar approach was used by Archytas (as cited by Porphyrius i his commetary o Ptolemy s "Harmoics") whe discussig the three atique meas i terms of three magitudes i cotiued arithmetic, geometric, ad harmoic proportios (Thomas 1939). Gow (193) also poits out that Nicomachus failed to metio that the arithmetic, geometric, ad harmoic meas of two umbers are i geometric proportio: a+b : ab : a+b ab. I Thomas traslatio of Nicomachus "Itroductio to Arithmetic" (Thomas 1939), Nicomachus itroduces the seve other meas usig the same treatmet as the oe metioed above. (The reader may wish to refer to Table.3 for a compact summary of the followig.) The fourth mea, which is also called the subcotrary by reaso of its beig reciprocal ad atithetical to the harmoic, comes about whe of the three terms the greatest bears the same ratio to the least as the differece of the lesser terms bears to the differece of the greater, as i the case of 3, 5; 6 (Thomas, 1939, p. 119). Nicomachus itroduces the fifth mea as the subcotrary mea to the geometric mea, The fifth [mea] exists whe of the three terms, the middle bears to the least the same ratio as their differece bears to the differece betwee the greatest ad the middle 11

terms, as i the case of, 4; 5, for 4 is double, the middle term is double the least, ad is double 1, that is the differece of the least terms is double the differece of the greatest. What makes it subcotrary to the geometric mea is this property, that i the case of the geometric mea the middle term bears to the lesser the same ratio as the excess of the greater term over the middle bears to that of the middle term over the lesser, while i the case of this mea a cotrary relatio holds (Thomas, 1939, p. 11). Nicomachus itroduces the sixth mea as, The sixth mea comes about whe of the three terms the greatest bears the same ratio to the middle as the excess of the middle term over the least bears to the excess of the greatest term over the middle as i the case of 1, 4; 6, for i each case the ratio is sesquialter [3 : ]. No doubt, it is called subcotrary to the geometric mea because the ratios are reversed, as i the case of the fifth mea (Thomas, 1939, p. 11). Nicomachus itroduces the last 4 meas by sayig, By playig about with the terms ad their differeces certai me discovered four other meas which do ot fid a place i the writigs of the aciets, but which evertheless ca be treated briefly i some fashio, although they are superfluous refiemets, i order ot to appear igorat. The first of these, or the seveth i the complete list, exists whe the greatest term bears the same relatio to the least as their differece bears to the differece of the lesser terms, as i the case of 6, 8; 9, for the ratio of each is see by compoudig the terms to be the sesquialter. The eighth mea, or the secod of these, comes about whe the greatest term bears to the least the same ratio as the differece of the extreme bears to the differece of the greater terms, as i the case of 6, 7; 9, for here the two ratios are the sesquialter. The ith mea i the complete series, ad the third i the umber of those more recetly discovered, comes about whe there are three terms ad the middle bears to the least the same ratio as the differece betwee the extremes bears to the differece betwee the least terms, as 4, 6; 7. Fially, the teth i the 1

complete series, ad the fourth i the list set out by the moders, is see whe i three terms the middle term bears to the least the same ratio as the differece betwee the extremes bears to the differece of the greater terms, as i the case of 3, 5; 8, for the ratio i each couple is the super-bi-partiet [5 : 3] (Thomas, 1939, p. 11). TABLE.1. Nicomachus Meas Mea Proportio Numbers Exhibitig the Mea Arithmetic a b : b c :: a : a, 4, 6 Geometric a b : b c :: a : b 4,, 1 Harmoic a b : b c :: a : c 6, 3, Cot. Harmoic b c : a b :: a : c 3, 5, 6 Cot. Geometric b c : a b :: b : c, 4, 5 Subco. Geometric b c : a b :: a : b 1,4,6 Seveth a c : b c :: a : c 6, 8, 9 Eighth a c : a b :: a : c 6, 7, 9 Nith a c : b c :: b : c 4, 6, 7 Teth a c : a b :: b : c 3, 5, 8 (Thomas 1939).4 Pappus List of Meas Pappus used a differet approach tha Nicomachus whe presetig his list of meas (Heath 191, Thomas 1939). Both Heath ad Thomas state that the meas o Pappus list are similar to those preseted by Nicomachus, but i a differet order after the sixth mea. Meas umber 8, 9, ad 10 i Nicomachus list are respectively umbers 9, 10, ad 7 o Pappus list. Moreover, Pappus omits mea umber 7 o Nicomachus list ad gives as umber 8 a additioal mea equivalet to the proportio c : b :: c a : c b. Therefore, the two lists combied give five additioal meas to the first six. I Thomas traslatio (1939) of Pappus work kow as "Collectios III", Pappus itroduces his discussio o meas as a respose to a questio posed by a uiformed geometer. He 13

demostrates his aswer by the costructio of the three meas i a semicircle (see figure 1.1). Pappus shows, i a series of propositios, that give three terms α, β, ad γ i geometrical progressio (Heath 191 uses "i geometric proportio"), it is possible to form from them three other terms a, b, ad c which are itegral liear combiatio of α, β, ad γ such that b is oe of the classical meas. The solutios to Pappus s equatios are show i Table.. The liear (Heath 191, Thomas 1939) TABLE.. Pappus Equatios for Meas Mea a, b, c Numbers exhibitig the mea Arithmetic a = α + 3β + γ 6, 4, b = α + β + γ c = β + γ Geometric a = α + β + γ 4,, 1 b = β + γ c = γ Harmoic a = α + 3β + γ 6, 3, b = β + γ c = β + γ Subcotrary a = α + 3β + γ 6, 5, b = α + β + γ c = β + γ Fifth a = α + 3β + γ 5, 4, b = α + β + γ c = β + γ Sixth a = α + 3β + γ 6, 4, 1 b = α + β + γ c = α + β γ Seveth a = α + β + γ 3,, 1 b = β + γ c = γ Eighth a = α + 3β + γ 6, 4, 3 b = α + β + γ c = β + γ Nith a = α + β + γ 4, 3, b = α + β + γ c = β + γ Teth a = α + β + γ 3,, 1 b = β + γ c = γ equatios show i Table. are moder equivalets of the literal traslatio of the Greek versio of Pappus. For example (Thomas 1939), i the case of the geometric mea metioed i Table., the literal traslatio of Pappus words would be, "To form a take α oce, β twice, ad γ oce; ad to form b we have to take β oce ad γ oce; ad to form c we take γ oce." Notice also that the examples give by Pappus for the proportios formed by his equatios sometimes differ 14

from those give by Nicomachus. For example for the fourth mea, Nicomachus gave 3, 5, ad 6 as a example for a solutio, while Pappus gave, 5, ad 6 as a solutio. Pappus expositio o meas by usig equatios may be better uderstood from the perspective that proportios were used i those days to solve equatios. Usig Proclus commetary o Euclid as a referece, Klei (1966) states, " Greek mathematics usage of proportios ca be compared to the moder sese of costructio of a equatio, ad a equatio may be viewed as a solutio of a proportio. This may be due to the uderstadig of ratios, proportios, ad harmoy o the basis of a commo mathematical property." Bema (1910) claims that the mathematicias of Alexadria uderstood equatios of secod degree mostly i the form of proportios. If we express Pappus method i moder terms, Pappus is parmeterizig meas by quadratics ad, equivaletly, givig quadratic polyomials to illustrate the relatio amog terms i the various meas. For example, to calculate the harmoic mea, usig three quatities i geometric progressio is equivalet to usig α = 1, β = x, ad γ = x ; thus, give a = + 3x + x, b = x + x, ad c = x + x, we have ac a+c = (+3x+x )(x+x ) = x(x+1)(x +3x+) = x(x +3x+) +3x+x +x+x (x+1) x+1 = x(x + ) = x + x = b..5 A Moder Recostructio of the Classical Meas I this sectio, we will use a similar approach to the oe used by Nicomachus to geerate the classical meas by cosiderig three positive quatities a, b, ad c such that a > b > c, ad we wish to make b the mea of a ad c. We will form three positive differeces with these quatities: (a b), (b c), ad (a c). The we will form a proportio by equatig a ratio of two of these differeces to a ratio of two of the origial quatities (ot ecessarily distict). For example, if we set the ratio a b b c equal to the ratio a b, the result is b = ac, which represets the geometric mea. If you look at all the possible ways of doig this, several of them are automatically ruled out by the assumed iequality of a, b, ad c. The oes that are ot (ecessarily) ruled out are the eleve meas summarized below (Madde 000, Heath 1963): 1. (a b) (b c) = a a = b b = c c (a+c), we have the arithmetic mea b =. 15

. (a b) (b c) = b c = a b, we have the geometric mea b = ac. 3. 4. 5. 6. 7. 8. 9. 10. 11. (a b) (b c) = a c ; we have the harmoic mea b = 1 a +. 1 c (a b) (b c) = c a ; we have the cotra-harmoic mea b = a +c a+c. (a b) (b c) = c b ; we have the first cotra-geometric mea b = a c+ a ac+5c. (a b) (b c) = b a ; we have the secod cotra-geometric mea b = c a+ 5a ac+c. (b c) (a c) = a c; b = ac c a. This mea is o Nicomachus list but ot Pappus list. (b c) (a c) = c b ; b = c+ 4ac 3c. (a b) (a c) = c a ; b = a ac+c a. (a b) (a c) = b a ; b = a c a. This mea is o Pappus list but ot Nicomachus list. (a b) (a c) = c b ; b = a c. Note that some of these meas are ot very robust defiitios of meas. For example, if oe uses the 11 th mea o our list to fid the mea of 5 ad 4, the M(5,4) = 1, which is ot betwee 5 ad 4. Note also that usig the 5 th mea o our list to fid the mea of 1 ad, we obtai the celebrated golde umber Φ = 1.618... However, as we will show i the ext sectio, the above list does ot exhaust all the meas kow to the aciet Greek world..6 Other Meas of the Aciet Greeks I this sectio, we poit out that Greek mathematicias cotiued to develop ew meas which were ever icluded amog the classical meas. Nicomachus referred to a special mea obtaied by the divisio of a segmet ito what he called "the most perfect proportios". This mea, which we will call b, ca be expressed by the divisio of a segmet of magitude a ito two parts: A greater part, b, ad a lesser part, a b, i such a fashio that the ratio of a to b is equal to the ratio 16