analysis and synthesis (euclid's elements, thomas heath, ed)

7/29/2019 Analysis and Synthesis (Euclid's Elements, Thomas Heath, Ed)

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CH. IX. 6] OTHER TECHNICAL TERMS 137calIed reductiones ad absurdum. For it is the function of this methodto upset something admitted as clear 1. "8. Analysis and Synthesis.

I t will be seen from the note on Euc!. XIII. 1 that the MSS. of theElements contain definitions of A nalysis and Sytzthesis followed byalternative proofs of XIII. I -5 after that method. The definitions andalternative proofs are interpolated, but they have great historicalinterest because of the possibility that they represent an ancientmethod of dealing with these propositions, anterior to Eudid. Thepropositions give properties of a line cut "in extreme and mean ratio,"and theyare preliminary to the construction and comparison of thefive regular solids. N ow Pappus, in the section of his Collectz'ott dealingwith the latter subjecf'l, says that he will give the comparisons betweenthe five figtues, the pyramid, cube, octahedron, dodecahedron andicosahedron, which have equal surfaces, " not by means of the so-calledanalyticalinquiry, by which sorne of the ancients worked out the proofs,but by the synthetical method 3 .... The conjecture of Bretschneiderthat the matter interpolated in EucJ. XII!. is a survival of investigations due to Eudoxus has at first sight much to commend it4 In thefirst place, we are told by Proclus that Eudoxus "greatly added tothe number of the theorems which Plato originated regarding thesec#on, and employed in them the method of analysis lS." I t is obviousthat "the section" was sorne, particular section which by the time ofPlato had assumed great importance; and the one section of whichthis can safely be said is that which was called the "golden section,"namely, the divison of a straight line in extreme and mean ratiowhich appears in Eucl. n. 11 and is therefore most probably Pythagorean. Secondly, as Cantor points out S, Eudoxus was the founderof the theory of proportions in the form in which we find it in Euclidv., VI., and it was no doubt through meeting, in the course of hisinvestigations, with proportions not expressible by whole numbersthat he carne to realise the necessity for a new theory of proportionswhich should be applicable to incommensurable as well as commensurable magnitudes. The" golden section" would f!Jrnish such a case.And it is even mentioned by Proclus in this connexion. He isexplaining'l that it is onIy in arithmetic that all quantities bear

~ ' r a t i o J 1 a l " raHos (p1}7ot;; 'A"l0") to one another, while in geometry thereare l, irrational" ones (app1J7ot;;) as wel!. ' H Theorems about sectionslike those in Euc1id's second Book are common to both [arithmeticand geometry] except that in which the straight line is cut in extremeand mean ratios."

1 Produs, p. 255, 8--26.2 Pappu, v. p .po sqq. 3 ibid. pp. 410 , 17-4I2 , 2.4 Bretschneider, p. 168. See however Heiberg's recent suggestion (ParalipolIlena zu

Euklid in Hermes, XXXVIII., 1903) that the author was Heron. The suggestion is basedon a ~ o m p a r i s o n with the remarks on analysis and synthesis quoted from Heron byan-Nairizi(ed. Curtze, p. 89) at the beginning oC his commentary on Euc1. Book I l . On the whole,this suggeston commends itself to me more than that of Bretschneider 6 Proclus, p. 67, 6. 6 Cantor, Gesch. d. Malh. Is ' p. 141.7 ProcIus, p. 60, 7-9 , 8 ibid. p. 60, 16 -19 .


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INTRODUCTION [CH. IX. 6The definitions of Ana/ysis and Synthesis interpolated in Eucl.

XIII. are as follows (1 adopt the reading of B and V ~ the only intelligible one, for the second)."Analysis is an assumption of that which is sought as if it wereadmitted < and the passage > through its consequences to somethingadmitted (to be) true." Synthesis is an assumption of that which is admitted < and thepassage > through its consequences to the finishing or attainment ofwhat is sought."The language is by no means clear and has, at the best, to befilled out.

Pappus has a fuller accountl :" The so-called (i,vaAvp,evo,; (' Treasury of Analysis ') is, to put itshortly, a spedal body of doctrine provided for the use m hose who,after finishing the ordinary Elements, are desirous of acquiring thepower of solvng problems which may be set them involving (theconstruction of) lines, and it is useful for this alone. It is the workof three men, Euclid the author of the Elements, Apollonius of Perga,and Aristaeus the elder, and proceeds by way of analysis and synthesis." Analysis then takes that which is sought as if it were admittedand passes from it through its successive consequences to somethingwhich is admitted as the result of synthesis: for in analysis we assumethat which is sought as if it were (already) done (ryeryovr;), and wenquire what it is from which this results, and again what is the antecedent cause of the Iatter, q,nd so on, until so retracing our stepswe come upon something already known or belonging to the class offirst principIes, and such -a method we call analysis as being soIutionbackwards (dV(L7raALV AVG'tV)." But in synthesis, reversing the process, we take as already donethat which was last arrived at in the analysis and, by arranging intheir natural order as consequences what were before antecedents,and successively connecting them one with another, we arrive finallyat the construction of what was sought; and this we call synthesis." Now analysis is of two kinds, the one directed to searching forthe truth and caBed theoretieal, the other directed to finding what weare told to find and called problematieal. (1) In t h ~ theoretieal kindwe assume what is sought as if it were existent and true, after whichwe pass through ts successive conseq uences, as if they too were trueand established by virtue of Qur hypothesis, to something admitted :then (a), if that something admitted is true, that which is sought willalso be true and the proof will correspond in the reverse order to theat;lalysis, hut (b), if we come upon something admittedly false, thatwhich is sought will also be falseo (2) In the problematical kind weassume that which is propounded as if it were known, after which wepass through its successive consequences, taking them as true, up tosomething admitted: if then (a) what is admitted is possible andobtainable, that is, what mathematicians caH given, what was originallyproposed will also be possible, and the proof will again corresponde in

1 Pappus, VII. pp. 634-6.


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CH. IX. 6] OTHER TECHNICAL TERMS 139reverse order to the analysis, but if (b) we come upon somethingadmittedly impossible, the problem will also be impossible."The ancient Analysis has been made the subject of careful studiesby several writers during the last half-century, the mos! completebeing those of HankeI, Duhamel and Zeuthen; others by Ofterdingerand Cantor should also be mentioned .The method is as follows. 1t is required, le t us say, to prove thata certain proposition A is true. We assume as a hypothesis that Ais true and, starting from this \Ve find that, if A is true, a certainother proposition B is true; if B is true, then C; and so on untilwe arrive at a proposition K which is admittedly true. The objectof the method is to enable us to infer, in the reverse order, that, sinceK is true, the proposition A originally assumed is true. NowAristotle had already made it clear that false hypotheses might leadto a conclusion which is true. There is therefore a possibility of errorunless a certain precaution is taken. While, for example, B may be anecessary consequence of A, it may happen that A is not a necessaryconsequence of B. Thus, in order that the reverse inference from thetruth of K that A is true may be logically justified, it is necessarythat each step in the chain of inferences should be unconditionallyconvertible. As a matter of fact, a very large number of theorems inelementary geometry are unconditionally convertible, so that in practicethe difficulty in securing that the successive steps shall be convertibleis not so great as might be 5upposed. But care is always necessary.For example, as Hankel says 2, a proposition may not be un conditionally convertible in the form in which it is generally quoted.Thus the proposition "The vertices of all triangles having a commonbase and constant vertical angle lie on a circle" cannot be con vertedinto the proposition that "All triangles with common base and verticeslying on a circle have a constant vertical angle"; for this is only trueif the further conditions are satisfied (1) that the circle passes throughthe extremities of the common base and (2) that only that part of thecircle is taken as the locus of the vertices which lies on 01te side of thebase. I f these conditions are added, the proposition is unconditionalIy. convertible. Or again, as Zeuthen remarks 3, K may be obtained bya series of inferences in which A or sorne other proposition in theseries is only apparently used; this would be the case e.g. when themethod of modern algebra is being employed and the expressions oneach side of the sign of equality have been inadvertently multipliedby sorne composite magnitude which is in reality equal to zero.Although the aboye extract from Pappus does not make it clearthat each step in the chain of argument must be convertible in tl).ecase taken, he almost implies this in the second part of the definitionof Analysis where, instead of speaking of the consequences B, C ...

1 Hankel, Zur GescMchte der Mathematik in A lterthum undMittdalter, 1874, pp. 137- 150;Duhamel, Des mthodes dans les sciences de raisonnement, Part l., 3 ed., Paris, 1885, pp. 39-68 ;Zeuthen, Geschichte der Mathelllatz"k im Altertum und Mittela/ter, 1896, pp. 92-104;Ofterdinger, Betriige zur Geschichte der gn"echischen Mathematik, VIm, 1860; Cantor,Geschichte" der Mathelllatik, 13 , pp. 220-2.2 Hankel, p. 139. a Zeuthen, p. 103.


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INTRODUCTION [CH. IX. 6successively following from A, he suddenly changes the expressionand says that we inquire what it t"s (B)from which Afollows (A beingthus the con sequen ce of B, instead of the reverse), and then what(viz. C) is the antecedent cause of B; and in practice the Greekssecured what was wantep by a l w ~ y s insisting'on the analysis beingconfirmed ~ y subsequent sYlzthesis, that is, they laboriously workedbackwards the whole way from K to A, reversing the order of theanalysis, which process would undoubtedly bring to light any flawwhich had erept into the argument through the a.ccidental neglect of-the necessary precautions.Reductio ad absurdum a variety of analysis.In the process of analysis starting from the hypothesis that aproposition A is true and passing through B, C ... as successive consequences we may arrive at a proposition K which, instead of beingadmittedly true, is either admittedly false or the contradictory of theoriginal hypothesis A or of sorne one or more of the propositions B, C ..intermediatc between A and K. Now correct inference from a trueproposition cannot lead to a false proposition; and in this case therefore we may at once conelude, without any inquiry whether thevarious steps in the argument are convertible or not, that the hypothesis A is faIse, for, if it were true, all the consequences correctlyinferred from it would be true and no ineompatibility could arise.This method of proving that a given hypothesis is false furnishes anindirect method of proving that a given hypothesis A is true, since wehave only to take the contl'adictory of A and to prove that it is falseoThis is the method of reductio ad absurdum, which is therefore a varietyof analysis. The contradictory of A, or not-A, will general1y ineludemore than one case and, in order to prove its falsity, each of the casesmust l1e separate ly disposed of: e.g., if it is desired to prove that acertain" part of a figure is equal to sorne other part, we take separatelythe hypotheses < ) that it is greater, (2) that it is less, and provethat each of these hypotheses leads to a conel usion either admittedlyfalse or contradictory to the hypothesis itself or to sorne one of itsconsequenees.Analysis as applied to problems.

I t is in relation to problems that the ancient analysis has thegreatest significance, because it was the one general method whichthe Greeks used for solving all ,< the more abstruse problems" ('TaaCTacf>CT'Tpa 'TWV Trpo/3"ATJp,'T(JJV )1.We have, let us suppose, to construct a figure satisfying a certainset of conditions. I f we are to proceed at all methodically and notby mere guesswork, it is first necessary to "analyse" those conditions.To enable this to be done we must get them clearly in our minds,which is only possible by assuming aH the eonditions to be aetuaHyfulfilled, in other words, by supposing the problem solved. Then wehave to transform those conditions, by all the means which practice insueh cases has taught us to employ, into other eonditions which arenecessarily fulfilled if the original conditions are, and to continue this1 Proclus. p. '24'2, 16, 17.


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CH. IX. 6] OTHER TECHNICAL TERMStransforrnation until we at length arrive at conditions which weare in a position to satisfyI. In other words, we must arrive atsorne relation which enables us to construct a particular part ofthe figure which, it is true, has been hypothetically assumed andeven drawn, but which nevertheless really requires to be found inorder that the problern may be solved. Frorn that rnornent theparticular part of the figure becomes one of the data, and a freshrelation _has to be found which enables a fresh part of the figureto be determined by means of the original data and the new onetogether. When tl'lis is done, the second new part of the figure alsobelongs to the data; and we proceed in this way until all the partsof the required figure are found 2. The first part of the analysisdown to the point of discovery of a relation which enablesus to say that a certain new part of the figure not belongingto the original data is given, Hankel calls the transforma#on; thesecond part, in which it is proved that all the rernaining parts ofthe figure are "given," he calls the resolution. Then follows thesynthesis, which also consists of two parts, (1) the COllstruction, inthe order in which it has to be actually carried out, and in generalfollowing the course of the second part of the analysis, the resolution;(2) the demOltstration that the figure obtained does satisfy all the givenconditions, which follows the steps of the first part of the analysis,the tra11.ifonnation, but in the reverse order. The second part ofthe analysis, the resolutioll,- would be rnuch facilitated and shortenedby the existence of a systernatic collection of Data such as Euc1id'sbook bearing that title, consisting of propositions proving that, ifin a figure certain parts or relations are give1t, other parts or relationsare also give1l. As regards the first part of the analysis, the trallsformation, the usual rule applies that every step in. the chain mustbe unconditionally convertible; and any failure to observe thiscondition will be brought to light by the subsequent synthesis.The second part, the resolutitJ1l, can be directly turned into theCOltstruction since that only is givell which can be constructed bythe rneans provided in the Elements.

I t would be difficult to find a better illustration of the aboye thanthe example chosen by Hankel from Pappus'l.Givell a circle ABe a1ld two points D, E extental to 'it, to drawstraight tines DB, EB from 'D, E to a puiut B Olt tlle circle such that,'tI DB, EB produced meet the circle again ll e, A, A e shall be para/lelto DE.Analysis.

S u p p o s ~ the problem solved and the tangent at A drawn, meetingED produced in F.(Part I. Transformati01l.)Then, since A e is parallel to DE, the angle at e is equal to theangle eDE. .But, since FA is a tangent, the angle at e is equal to the angle FAE.Therefore the angle FAE is equal to the angle eDE, whence A,

B, D, F are concyclic.1 Zeuthen, p. 93. 2 Hankel, p. 14 1 3 Pappus, VII. pp. 830-2.

analysis and synthesis (euclid's elements, thomas heath, ed)

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