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At last, in , the German ma the matician Ferdinand Lindemann succeededin proving unequivocally that the quadrature of the circle was an impossibility. The technical details of his pro of are quite advanced and go wellbeyond the scope of this book. However, the following is a brief synopsis of how it was that Lindemann answered this age-old question.
He did it by translating the issue from the realm of geometry to the realm of number. If we imagine the collection of all real numbers,depicted in the schematic diagram in Figure 1. Note that, in each case, the se polynomials have integercoefficients.
At first, this numerical discovery may seem to have little bearing on the geometry of circle-squaring, but we shall see that it provided the missing piece of the puzzle. If circles are quadrable, aswe have temporarily assumed, the n we employ our compass andstraightedge, work feverishly slashing arcs and drawing lines, and eventually,after only a finite number of such steps, end up with a square thatalso has area 7r, as indicated in Figure 1. In this process, we wouldhave had to construct the square, which of course would require us tohave constructed each of its four sides.
Call the length of the square'sside x. But, as we have noted, no such construction forY; ispossible. What went wrong? Tracing back through the argument and lookingfor the source of our contradiction, we find it can only be the initialassumption, namely, that circles can be squared. As a consequence, wemust reject this and conclude, once and for all, that the quadrature of the circle is a logical impossibility! During thisspan, Greek civilization grew and matured, enriched by the writings of Plato and Aristotle, of Aristophanes and Thucydides, even as it underwent the turmoil of the Peloponnesian Wars and the glory of the Greekempire under Alexander the Great.
By B. In the West, Greecereigned supreme. The period from B. Among the sewere Plato B. Plato, the great philosopher of A the ns, deserves mention here not somuch for the ma the matics he created as for the enthusiasm and status heimparted to the subject. As a youth, Plato had studied in A the ns underSocrates and is of course our primary source of information about hisesteemed teacher.
For a number of years Plato roamed the world, meet In B. Devoted to learning and contemplation, the Academy attracted talentedscholars from near and far, and under Plato's gUidance it became the intellectual center of the classical world. Of the many subjects studied at the Academy, non e was more highlyregarded than ma the matics.
The subject certainly appealed to Plato'ssense of beauty and order and represented an abstract, ideal worldunsullied by the humdrum demands of day-to-day existence. Moreover,Plato considered ma the matics to be the perfect training for the mind, itslogical rigor demanding the ultimate in concentration, cleverness, andcare.
Legend has it that across the arched entryway to his prestigiousAcademy were the words ' 'Let no man ignorant of geometry enter here. We might say thatPlato regarded geometry as the ideal entrance requirement, the ScholasticAptitude Test of his day. Although very little Original ma the matics is now attributed to Plato, the Academy produced many capable ma the maticians and one indisputably great one, Eudoxus of Cnidos. Eudoxus came to A the ns about the time the Academy was being created and attended the lectures of Platohimself.
Eudoxus' poverty forced him to live in Piraeus, on the outskirts of A the ns, and make the daily round-trip journey to and from the Academy,thus distinguishing him as one of the first commuters although weare unsure whe the r he had to pay out - of -city-state tuition.
Later in hiscareer, he traveled to Egypt and returned to his native Cnidos, all the while assimilating the discoveries of science and constantly extendingits frontiers.
Particularly interested in astronomy, Eudoxus devised complexexplanations of lunar and planetary motion whose influence wasfelt until the Copernican revolution in the sixteenth century. Never willingto accept divine or mystical explanations for natural phenomena, heinstead tried to subject the m to observation and rational analysis.
One was his the ory of proportion, and the o the r his method of exhaustion. The former provided a logical victory over the impasse createdby the Pythagoreans' discovery of incommensurable magnitudes. This impasse was especially apparent in geometric the orems about similartriangles, the orems that had initially been proved under the assumptionthat any two magnitudes were commensurable.
When this assumptionwas destroyed, so too were the existing pro of s of some of geometry's foremost the orems. What resulted is sometimes called the. That is, while people still believedthat the the orems were correct as stated, the y no longer were in possession of sound pro of s with which to support this belief. It was Eudoxuswho developed a valid the ory of proportions and the reby supplied the long-sought pro of s. His the ory, which must have brought a collectivesigh of relief from the Greek ma the matical world, is now most readilyfound in Book V of Euclid's Elements.
Eudoxus' o the r great contribution, the method of exhaustion, foundimmediate application in the determination of areas and volumes of the more sophisticated geometric figures. The general strategy was toapproach an irregular figure by means of a succession of known elementaryones, each providing a better approximation than its predecessor. We can think, for instance, of a circle as being a totally curvilinear, andthus quite intractable, plane figure.
But, if we inscribe within it a square,and the n double the number of sides of the square to get an octagon,and the n again double the number of sides to get a gon, and so on,we will find the se relatively simple polygons ever more closely approximating the circle itself. In Eudoxean terms, the polygons are "exhausting" the circle from within. This process is, in fact, precisely how Archimedes determined the area of a circle, as we shall see in the great the orem of Chapter 4.
It isto Eudoxus that he owed this fundamental logical tool. In addition,Archimedes credited Eudoxus with using the method of exhaustion toprove that the volume of "any cone is one third part of the cylinderwhich has the same base with the cone and equal height," a the oremthat is by no means trivial. The reader familiar with higher ma the maticswill recognize in the method of exhaustion the geometric forerunner of the modern notion of "limit," which in turn lies at the heart of the calculus.
Eudoxus' contribution was a significant one, and he is usuallyregarded as being the finest ma the matician of antiquity next to the unsurpassed Archimedes himself. It was during the latter third of the fourth century B. His conquests carried him to Egypt where, in B. This city grew rapidly,reportedly reaching a population of half a million in the next three decades.
Of particular importance was the formation of the great AlexandrianLibrary that soon supplanted the Academy as the world's foremostcenter of scholarship. At one point, the facility had over , papyrusrolls, a collection far more complete and astounding than anything the world had ever seen. Indeed, Alexandria would remain the intellectualfocus of the Mediterranean world through the Greek and Roman periodsuntil its final destruction in A. Among the scholars attracted to Alexandria around B.
We knowvery little about his life ei the r before or after his arrival on the Africancoast, but it appears that he received his training at the Academy from the followers of Plato. Be that as it may, Euclid's influence was so pr of oundthat virtually all subsequent Greek ma the maticians had some connectionor o the r with the Alexandrian School. What Euclid did that established him as one of the great est names inma the matics history was to write the Elements.
This work had a pr of oundimpact on Western thought as it was studied, analyzed, and editedfor century upon century, down to modern times. It has been said that of all books from Western civilization, only the Bible has received moreintense scrutiny than Euclid's Elements. The highly acclaimed Elements was simply a huge collectiondividedinto 13 books- of propositions from plane and solid geometryand from number the ory. Today, it is generally agreed that relativelyfew of the se the orems were of Euclid's own invention.
Ra the r, from the known body of Greek ma the matics, he created a superbly organizedtreatise that was so successful and so revered that it thoroughly obliteratedall preceding works of its kind.
Euclid's text soon became the standard. Consequently, a ma the matician's reference to 1. Actually the parallel is quite accurate, for no book has come closerto being the "bible of ma the matics" than Euclid's spectacular creation. Down through the centuries, over editions of the Elements haveappeared, a figure that must make the authors of today's ma the maticstextbooks drool with envy.
As noted, it was highly successful even in itsown day. After the fall of Rome, the Arab scholars carried it of f to Baghdad,and when it reappeared in Europe during the Renaissance, itsimpact was pr of ound.
The work was studied by the great Italian scholars of the sixteenth century and by a young Cambridge student named IsaacNewton a century later. We have a passage from Carl Sandburg's biography of Abraham Lincoln that recounts how, when a young lawyer tryingto sharpen his reasoning skills, the largely unschooled Lincoln At night. It has of ten been noted that Lincoln's prose was infleenced and enrichedby his study of Shakespeare and the Bible.
It is likewise obvious thatmany of his political arguments echo the logical development of aEuclidean proposition. In his autobiography, Russell penned this remarkablerecollection:At the age of eleven, I began Euclid, with my bro the r as tutor.
This was one of the great events of my life, as dazzling as first love. As we consider the Elements in this chapter and the next, we shouldbe aware that we proceed along paths that so many o the rs have trod. Only a very few classics- the Iliad and Odyssey come to mind-sharesuch a heritage. The propositions we shall examine were studied byArchimedes and Cicero, by Newton and Leibniz, by Napoleon and Lincoln.
It is a bit daunting to place oneself in this long, long line of students. Euclid's great genius was not so much in creating a new ma the maticsas in presenting the old ma the matics in a thoroughly clear, organized,and logical fashion. This is no small accomplishment. It is important torecognize the Elements as more than just ma the matical the orems and the ir pro of s; after all, ma the maticians as far back as Thales had been furnishingpro of s of propositions.
Euclid gave us a splendid axiomaticdevelopment of his subject, and this is a critical distinction. He began the Elements with a few basics: 23 definitions, 5 postulates, and 5 commonnotions or general axioms. These were the foundations, the "givens," of his system. He could use the m at any time he chose. From the sebasics, he proved his first proposition. With this behind him, he could the n blend his definitions, postulates, common notions, and this firstproposition into a pro of of his second.
And on it went. Consequently, Euclid did not just furnish pro of s; he furnished the mwithin this axiomatic framework. The advantages of such a developmentare significant. For one thing, it avoids circularity in reasoning. Eachproposition has a clear, unambiguous string of predecessors leadingback to the original axioms. Those familiar with computers could evendraw a flow chart showing precisely which results went into the pro of of a given the orem.
This approach is far superior to "plunging in" to provea proposition, for in such a case it is never clear which previous resultscan and cannot be used. The great danger from starting in the middle,as it were, is that to prove the orem A, one might need to use result B,which, it may turn out, cannot be proved without using the orem A itself. This results in a circular argument, the logical eqUivalent of a snake swallowingits own tail; in ma the matics it surely leads to no good.
But the axiomatic approach has ano the r benefit. Since we can clearlypick out the predecessors of any proposition, we have an immediatesense of what happens if we should alter or eliminate one of our basicpostulates.
If, for instance, we have proved the orem A without ever using. While this might seem a bit esoteric, just such an issuearose with respect to Euclid's controversial fifth postulate and led to one of the longest and most pr of ound debates in the history of ma the matics.
This matter is examined in the Epilogue of the current chapter. Thus, the axiomatic development of the Elements was of majorimportance. Even though Euclid did not quite pull this of f flawlessly, the high level of logical sophistication and his obvious success at weaving the pieces of his ma the matics into a continuous fabric from the basicassumptions to the most sophisticated conclusions served as a model forall subsequent ma the matical work. To this day, in the arcane fields of topology or abstract algebra or functional analYSiS, ma the maticians willfirst present the axioms and the n proceed, step-by-step, to build up the irwonderful the ories.
It is the echo of Euclid, 23 centuries after he lived. Book I: PreliminariesIn this chapter, we shall focus only on the first book of the Elements;subsequent books will be the topic of Chapter 3. Book I began abruptlywith a list of definitions from plane geometry. Among the first few definitions were:o Definition 1 A point is that which has no part.
Today's students of Euclid find the se statements unacceptable and abit quaint. Obviously, in any logical system, not every term can bedefined, since definitions the mselves are composed of terms, which inturn must be defined. If a ma the matician tries to give a definition foreverything, he or she is condemned to a huge circular jumble.
What, forinstance, did Euclid mean by "breadthless"? What is the technical meaning of lying "evenly with the points on itself"? From a modern viewpoint, a logical system begins with a few undefinedterms to which all subsequent definitions relate. One surely triesto keep the number of the se undefined terms to a minimum, but the irpresence is unavoidable.
For modern geometers, the n, the notions of "point" and "straight line" remain undefined. Statements such as. Fortunately, his later definitions were more successful. A few of the sefigure prominently in our discussion of Book I and deserve comment. It may come as a surprise to modern readers that Euclid did notdefine a right angle in terms of 90 0; in fact, nowhere in the Elements is"degree" ever mentioned as a unit of angular measure.
The only angularmeasure that plays any significant role in the book is the right angle, andas we can see, Euclid defined this as one of two equal adjacent anglesalong a straight line.
Clearly, the "one point" within the circle is the circle's center, and the equal "straight lines" he referred to are the radii. In definitions 19 through 22, Euclid defined triangles plane figurescontained by three straight lines , quadrilaterals those contained byfour , and such specific subclasses as equilateral triangles triangleswith three sides equal and isosceles triangles those with "two of itssides alone equal".
His final definition proved to be critical:o Definition 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, donot meet one ano the r in ei the r direction.
Notice that Euclid avoided defining parallels in terms of the ir beingeverywhere equidistant. His definition was far simpler and less fraughtwith logical pitfalls: parallels were simply lines in the same plane thatnever intersect. With the definitions behind him, Euclid gave a list of five postulatesfor his geometry. Recall, the se were to be the givens, the self-evidenttruths of his system.
He certainly had to select the m judiciously and toavoid overlap or internal inconsistency. A moment's thought shows that the first two postulates permitted precisely the sorts of constructions one can make with an unmarkedstraightedge. For instance, if the geometer wanted to connect twopoints with a straight line-a task physically accomplished with astraightedge- the n Postulate 1 provided the logical justification fordoing so.
Here was the corresponding logical basis for pulling out a compassand drawing a circle, provided one first had a given point to be the centerand a given distance to serve as radius. Thus, the first three postulates,toge the r, justified all pertinent uses of the Euclidean tools. Or did the y? Those who think back to the ir own geometry trainingwill recall an additional use of the compass, namely, as a means of transferringa fixed length from one part of the plane to ano the r.
That is, givena line segment whose length was to be copied elsewhere, one puts the point of the compass at one end of the segment and the pencil tip at the o the r; the n, holding the device rigidly, we lift the compass and carry itto the desired spot. It is a simple and highly useful procedure. However,in playing by Euclid's rules, it was not permitted, for nowhere did hegive a postulate allowing this kind of transfer of length.
As a result, ma the maticians of ten refer to the Euclidean compass as "collapsible. What is one to make of this situation? Why did Euclid not insert anadditional postulate to support this very important transfer of lengths? The answer is simple: he did not need to assume such a technique as apostulate, for he proved it as the third proposition of Book I.
That is,Euclid introduced a clever technique for transferring lengths even if hiscompass "collapsed" upon lifting it from the page, and the n he provedwhy his technique worked. It is to Euclid's great credit that he avoidedassuming what he could in fact derive, and the reby kept his postulatesto a bare minimum.
This postulate did not relate to a construction. Ra the r, it provided auniform standard of comparison through out Euclid's geometry. Rightangles had been introduced in Definition 10, and now Euclid was assumingthat any two such angles, regardless of where the y were situated in the plane, were equal.
With this behind him, Euclid arrived at by far the most controversial statement in Greek ma the matics:POSTULATE 5 If a straight line falling on two straight lines make the interiorangles on the same side less than two right angles, the two straightlines, if produced indefinitely, meet on that side on which are the anglesless than the two right angles. As shown in Figure 2. Postulate 5 is of tencalled Euclid's parallel postulate. Clearly, this postulate was quite unlike the o the rs.
It was longer tostate, required a diagram to understand, and seemed far from being aself-evident truth. They sensed that, just as Euclid didnot need to assume that lengths could be transferred with a compass,nei the r did he have to assume this postulate; he should simply havebeen able to prove it from the more elementary properties of geometry.
There is evidence that Euclid himself was a bit uneasy about this matter,for in his development of Book I he avoided using the parallel postulateas long as he could. That is, whereas he felt perfectly content to use any of his o the r postulates as early and of ten as he needed, Euclid put of f the use of his fifth postulate through his first 28 propositions.
As shown in the Epilogue, however, it was one thing to be skeptical of the need forsuch a postulate but quite ano the r to furnish the actual pro of. With this controversial statement behind him, Euclid completed hispreliminaries with a list of five common notions. These too were meantto be self-evident truths but were of a more general nature, not specificto geometry. They wereo Common Notion 1 Things which are equal to the same thing are alsoequal to one ano the r.
Of the se, only the fourth raised some eyebrows. Apparently, whatEuclid meant by it was that, if one figure could be moved rigidly fromone portion of the plane and the n be placed down upon a second figureso as to coincide perfectly, the n the two figures were equal in allaspects-that is, the y had equal angles, equal sides, and so forth. It haslong been observed that Common Notion 4, having something of a geometriccharacter, belonged among the postulates. This, the n, was the foundation of assumed statements upon which the entire edifice of the Elements was to be built.
It is a good point atwhich to return to the young Bertrand Russell for ano the r of his wonderfulautobiographical confessions:I had been told that Euclid proved things, and was much disappointed tha the started with axioms. At first, I refused to accept the m unless my bro the rcould of fer me some reason for doing so, but he said, "If you don't accept.
Only those propositions of particular interestor importance are discussed here, the goal being to arrive at Propositions1. If someone were about to develop geometry from a few selected axioms,what would be his or her very first proposition? Using A as center and AB as radius, he constructed a circle; the n, with Bas center and AB again as radius, he constructed a second circle.
Bothconstructions, of course, made use of Postulate 3, and nei the r required the compass to remain open when lifted from the page. Unfortunately, the pro of is flawed. Even the ancient Greeks, nomatter how highly the y regarded the Elements, were aware of the logicalshortcomings of this first Euclidean argument. The problem resided in the point C, for how could Euclid prove that the two circles did, in fact, intersect at all? How did he know that the ydid not somehow pass through one ano the r without meeting?
Clearly,since this was his first proposition, he had not previously proved that the y must meet. Moreover, nothing in his postulates or common notionsspoke to this matter. The only justification for the existence of the pointCwas that it showed up plainly in the diagram. But this was the rub. For if the re was one thing that Euclid wanted tobanish from his geometry, it was the reliance on pictures to serve aspro of s. By his own ground rules, the pro of must rest upon the logic,upon the careful development of the the ory from the postulates andcommon notions, with all conclusions ultimately dependent upon the m.
When he "let the picture do the talking," Euclid violated the very ruleshe had imposed upon himself. After all, if we are willing to draw conclusionsfrom the diagrams, we could just as well prove Proposition 1.
Whenwe resort to such visual judgments, all is lost. Modern geometers have recognized the need for an additional postulate,sometimes called the "postulate of continuity," as a justificationfor claiming that the circles do meet.
O the r postulates have been introducedto fill similar gaps appearing here and the re in the Elements. Around the turn of the present century, the ma the matician David Hilbert developed his geometry from a list of 20 postulates, the reby plugging the many Euclidean loopholes. As a result, in BertrandRussell gave this negative assessment of Euclid's work:His definitions do not always define, his axioms are not always indemonstrable,his demonstrations require many axioms of which he is quite unconscious.
A valid pro of retains its demonstrative force when no figure is drawn,but very many of Euclid's earlier pro of s fail before this test. The value of his work as a masterpiece of logic has been very grossly exaggerated. Admittedly, when he allowed himself to be led by the diagram andnot the logic behind it, Euclid committed what we might call a sin of omission. Yet nowhere in all propositions did he fall into a sin of commission. None of his the orems is false.
With minor modificationsin some of his pro of s and the addition of some missing postulates,all have withstood the test of time. Those inclined to agree with Russell's. These scientists were truly primitive bymodern standards, and no one today would rely on ancient texts toexplain the motion of the moon or the workings of the liver.
But, more of ten than not, we can rely on Euclid. His work stands as a remarkablytimeless achievement. It did not, after all, depend on the collection of data or the creation of more accurate instruments. It rested squarelyupon the keenness of reason, and of this Euclid had an abundance. Propositions 1. In modern terms, this was the so-called sideangle-sideor SAS congruence pattern, which readers should recall from the ir high school geometry courses.
It said that if two triangles have the two sides and included angle of one respectively equal to two sides andincluded angle of the o the r, the n the triangles are congruent in allrespects Figure 2.
Then, picking up t:. DEF and moving it over ontot:. ABC, he argued that the triangles coincided in the ir entirety. Such apro of by superposition has long since gone out of favor. After all, who isto say that, as figures move around the plane, the y are not somehowdeformed or distorted?
Proposition 1. This the orem came to be known as the "Pons Asinorum," or the bridge of fools. The name stemmed in part from Euclid's diagram, whichvaguely resembled a bridge, and in part from the fact that many weakergeometry students could not follow its logic and thus could not crossover into the rest of the Elements.
S in that it statedthat if a triangle has base angles equal, the n the triangle is isosceles. Ofcourse, the orems and the ir converses are of great interest to logicians,and of ten after Euclid had proved a proposition, he would insert the pro of of the converse even if it could have been omitted or delayed withouto the rwise damaging the logical flow of his work.
Euclid's second congruence scheme for triangles-side-side-side orSSS-appeared as Proposition 1. It stated that when two triangles have the three sides of one respectively equal to the three sides of the o the r, the n the ir corresponding angles are likewise equal,Some constructions followed. Euclid demonstrated how to bisect agiven angle Proposition 1. The subsequent twd results showed howto construct a perpendicular to a given line, where the perpendicularei the r met the line at a given point on it Proposition 1.
In Proposition 1. He used this property of angles around astraight line in the simple yet important Proposition US. To begin,Euclid bisected AC at E, by 1. His final construction was to draw FC. Thus, the two triangles are congruent by 1. The exterior angle the orem was a geometric inequality.
So too were the next few propositions in the Elements. For instance, Proposition 1. We are told that the Epicureans of ancient Greecethought very little of this the orem, since the y regarded it as so trivial asto be self-evident even to an ass. That is, if a donkey stands at point A inFigure 2. It has been suggested thatProposition 1.
However, as with the collapsible compass, Euclidcertainly did not want to assume a statement as a postulate if he couldprove it as a proposition, and the actual pro of he furnished for this the oremwas quite a nice bit of logic.
A few more inequality propositions followed before Euclid arrived at the important 1. Here he first gave apro of of the angle-side-angle, or ASA, congruence scheme as a consequence of the SAS congruence of 1.
But, as the second part of 1. At first, one is tempted to dismiss this as an immediate consequence of ASA. This last, most controversialpostulate finally made its appearance when Euclid proved the converse of 1. Again, his mode of attack wasindirect. That is, he supposed that L1 ::F L2 and from the re derived alogical contradiction.
By Proposition1. Hence, by contradiction, Euclid had shown that L1 cannot exceed 12; ananalogous argument established that L2 cannot be great er than L1 ei the r. In short, alternate interior angles of parallel lines are equal. As a corollary to the pro of , Euclid easily deduced that the correspondingangles were likewise equal, that is, in Figure 2.
Having at last indulged in the parallel postulate, Euclid now found itvirtually impossible to break the habit. Of the remaining 20 propositionsin Book I, all but one ei the r used the postulate directly or used a propositionpredicated upon it, the lone exception being Proposition 1. BC in Figure 2. By Proposition 1. The sum of the angles of D.. In this manner, the famous result wasproved.
From here, Euclid set his sights on bigger game. His next few propositionsdealt with the areas of triangles and parallelograms, and culminatedin Proposition 1. Since one such parallelogram is a rectangle and since the rectangle'sarea is base X height , we see that 1.
However, Euclid did not think in such algebraic terms. A few propositions later, in 1. A square, of course, is a regularquadrilateral, since all of its sides and all of its angles are congruent. Atfirst, this may sound like a trivial proposition, especially when onerecalls that Book I began with the construction of an equilateral triangle, the regular three-sided figure.
Yet a look at his pro of shows why this hadto be so long delayed. Much of the argument rested on properties of parallels, and the se of course had to await the critical 1. So, whereasEuclid constructed regular triangles at the outset of Book I, he waiteduntil nearly the end to do regular quadrilaterals. With the se 46 propositions proved, Book I had but two to go. Itappears that Euclid had saved the best for last. After all of the se preliminaries,he was ready to tackle the Pythagorean the orem, surely one of the most significant results in all of ma the matics.
Yet he deserves credit for the particularpro of we are about to examine, a pro of that many believe is originalwith Euclid. Its beauty is in the economy of its prerequisites; after all,Euclid had only his postulates, common notions, and first 46 propositions-ara the r lean tool-kit-from which to build a pro of.
Consider the topics in geometry that he had not yet addressed: the only quadrilateralshe had investigated were parallelograms; circles, by and large, were yetunexplored; and the highly important subject of similarity would not bementioned until Book VI.
It is surely possible to devise short pro of s of the Pythagorean the orem by using similar triangles, but Euclid wasunwilling to put of f the pro of of this major proposition until Book VI.
He clearly wanted to reach the Pythagorean the orem as early and directlyas possible, and thus he devised a pro of that would become only the 47th proposition of the Elements.
In this light, one can see that much of what preceded it pointed toward the great the orem of Pythagoras, whichserved as a fitting climax to Book I. Before we plunge into the details, here is the result stated in Euclid'swords and a preview of the very clever strategy he used to prove it:PlOPOSmON 1. Euclid hadto prove that the combined areas of the little squares upon AB and ACequaled the area of the large square constructed upon the hypotenuseBC, seen in Figure 2. To do this, he hit upon the marvelous strategy of starting at the vertex of the right angle and drawing line AI parallel to the side of the large square and splitting the large square into two rectangularpieces.
This general strategy was most ingenious, but it remained to supply the necessary details. Fortunately, Euclid had done all the spadework inhis earlier propositions, so it was just a matter of carefully assembling the pieces. In addition, Euclidhad taken pains to prove that GACwas a straight line, so the triangle. Againapplying 1. Thus ended one of the most significant pro of s in all of ma the matics.
The diagram Euclid used Figure 2. It is of ten called " the windmill" because of its resemblance to such a structure. The windmill is evident on the page shown here, written in Latin,from a edition of the Elements.
Obviously, students over four centuriesago were grappling with this figure even as we ourselves have justdone. Euclid's pro of is not, of course, the only way to establish the Pythagorean the orem.
There are, in fact, hundreds of different versions, rangingfrom the highly ingenious to the dreadfully uninspired. These includea pro of devised by an Ohio Congressman named James A.
Garfield, whowent on to the U. Readers interested in sampling o the rarguments may wish to consult E. Loomis' book The Pythagorean. While we obviously cannot assert that LBAC is right in fact, that iswhat the the orem is trying to establish , we do know that LDAC is rightby the construction of the perpendicular. Hence L DA C and L But the latter was constructed to bea right angle. Thus LBAC is a right angle as well.
Taken in tandem, Propositions 1. Euclid has shown that a triangle is right if and only if the square on the hypotenuse equals the sum of the squares on the legs. These pro of s were, and remain, an example of geometry at its best. Yet the se two Pythagorean propositions are remarkable in ano the rsense.
It is one thing for Euclid to have proved the m in such a fine fashion,but it is ano the r thing for the m to be true in the first place. There isno intuitive reason that right triangles should have such an intimate connectionto the sums of squares. Unlike I. On the contrary, the Pythagorean the orem establishes a supremely odd fact, one whose odd-.
Trudeau observed that right angles arefamiliar, everyday entities that appear not only in the man-made worldbut also in Nature itself. What could be more "ordinary" or "natural"than right angles? And yet, says Trudeau:To me the Theorem of Pythagoras is very surprising.
Because the equation isabstract and precise, it is alien. I can't imagine what such a thing could possiblyhave to do with everyday right angles. So, when the pall of familiaritylifts, as it occasionally does, and I see the Theorem of Pythagoras afresh, Iam flabbergasted. EpilogueDown through history, the most troubling feature of Book I of the Elementswas the controversial parallel postulate.
The trouble arose notbecause anyone doubted that the parallel postulate had to be true. On the contrary, it was universally agreed that the postulate was a logicalnecessity. After all, geometry was an abstract way of describing the universe-akind of "pure physics" -and surely physical reality dictated the truth of the parallel postulate. Thus, it was not the necessity of Euclid's statement that was challenged.
Ra the r it was its classification as a postulate ra the r than as a proposition. The classical writer Proclus summed up this view with his comment,"This [Postulate 5] ought even to be struck out of the Postulatesaltoge the r; for it is a the orem. First of all-and this may havereally bo the red the ancient geometers- the postulate sounded like aproposition, for its statement consumed the better part of a paragraph.
Moreover, not only did Euclid seem to avoid using the postulate as longas he could, but he managed to prove some fairly sophisticated resultswithout it. In seeking such a pro of , ma the maticians werefree to use any of the o the r postulates or common notions, as well asEuclid's propositions 1. Uncounted ma the maticians tried the ir hand at con-.
Unfortunately, years of frustration became decades and the n centuries of failure. The pro of remained elusive. What geometers did in the process was to find a host of new resultslogically equivalent to the parallel postulate. It of ten happened that apurported pro of of Postulate 5 required the ma the matician to assume aseemingly obvious but hi the rto unproven statement.
Unfortunatelyandhere lay the problem- the parallel postulate itself was necessary inorder to derive this statement. To a logician, this says that both werereally expressing the identical concept, and a "pro of " of Postulate 5 thatrequired the assumption of its logical equivalent was of course no pro of at all.
Four of the more famous equivalents of the parallel postulate appearbelow. It should be stressed that, had any one of the se been proved fromPostulates 1 through 4, the n Postulate 5 would likewise follow. These logical equivalents notwithstanding, the nature of the parallelpostulate remained unresolved through the Renaissance.
Whoeverdeduced the parallel postulate would have been guaranteed everlastingfame in the annals of ma the matics. At times the pro of seemed tantalizinglyclose, yet it evaded the efforts of the world's finest ma the maticalminds.
Then, early in the nineteenth century, three ma the maticians simultaneouslyhad the burst of insight necessary to see the matter in its truelight. The first was the incomparable Carl Friedrich Gauss ,whose biography is delayed until Chapter Gauss recast the issue interms of the degree-measure of the angles of a triangle. He proceeded toinvestigate the se two cases. Thus, that case was effectively eliminated. If hecould likewise dispense with the o the r case, he would have established,indirectly, the necessity of the parallel postulate.
These turned outto be quite strange, seemingly bizarre and counter-intuitive one is presentedshortly. Yet nowhere did Gauss find the logical contradiction hesought. In , he summarized the situation by stating Gradually, as Gauss delved more and more deeply into this peculiargeometry, he became convinced that no logical contradiction existed.
Ra the r, he began to sense that he was developing not an inconsistentgeometry but just an alternative one, a " non -Euclidean" geometry, inhis words. This was a breathtaking statement. Yet Gauss, universally regarded as the foremost ma the matician of his day, did not publicize his findings. Perhaps the burdens of fame figured in his decision, for he was certain the controversial nature of his position would cause an uproar that mightjeopardize his l of ty reputation.
In an letter to a confidant, Gaussobserved that he had no plans While today's reader may miss a bit of this classical allusion, suffice it tosay that being called a "Boeotian" is being labeled an unimaginative,crudely obtuse dullard.
Obviously Gauss had little regard for the receptivity of the ma the matical community to his new ideas. Next entered the Hungarian ma the matician Johann Bolyai Johann's fa the r Wolfgang had been an associate of Gauss and hadhimself spent much of a lifetime in a futile attempt to prove Euclid'spostulate.
In an age when sons of ten took the pr of essions of the irfa the rs-be the y clergymen or cobblers or chefs-we have here the younger Bolyai taking from his fa the r the ra the r esoteric career of trying. Wolfgang, however, knew all too well the difficulties of such a career and wrote this strong warning to his son:You must not attempt this approach to parallels. I know this way to its veryend. I have traversed this bottomless night, which extinguished aU light andjoy of my life I entreat you , leave the science of parallels alone.
The young Johann Bolyai did not heed his fa the r's advice. Much likeGauss, Johann came to recognize the crucial trichotomy involving the angle-sum of a triangle and tried to eliminate all but the case equivalentto the parallel postulate; like Gauss, he was unsuccessful. As Bolyaidelved ever more deeply into the problem, he too arrived at the conclusionthat Euclid's geometry had a logically valid competitor, and wrotein astonishment at his peculiar yet apparently consistent propositions,"Out of nothing, I have created a strange new universe.
The elder Bolyai enthusiastically sent a copy of the book to his friendGauss; fa the r and son could only have been surprised by Gauss'response:If I begin with the statement that I dare not praise [your son's] work, youwill of course be startled for a moment: but I cannot do o the rwise; to praiseit would amount to praising myself; for the entire content of the work, the path which your son has taken, the results to which he is led, coincidealmost exactly with my own meditations which have occupied my mind forfrom thirty to thirty-five years.
It is easy to see that Gauss hit his enthusiastic young admirer with ablast of cold water. To his credit, Gauss graciously described himself tobe ". But Johann's ego had one more trial to endure, for it soon came tolight that a Russian ma the matician, Nikolai Lobachevski ,not only had traveled the same path as Gauss and Bolyai, but had publishedhis own account of non -Euclidean geometry in a full threeyears before.
Lobachevski, however, had written his treatise in Russian,and it apparently had gone unnoticed in western Europe. We have herea phenome non not uncommon in science, that of a discovery madesimultaneously and independently by many individuals. As WolfgangBolyai so charmingly observed:. The impact of the se discoveries had barely struck home when yetano the r innovator, Georg Friedrich Bernhard Riemann ,adopted a different viewpoint about the infinite length of geometriclines.
But was the re a need to assume this infinitude at all? Euclid'ssecond postulate asserted that a straight line could be continued in astraight line, but was this not assening simply that one never reached the end of a line? Riemann could easily imagine the case where linessomewhatlike circles-are of finite length yet have no "end.
The unboundedness of space possesses. But its infinite extent by no means follows fromthis. Although different from both Euclid's and Bolyai's,Riemann's geometry was apparently just as consistent. Today, we recognize all four of the se individuals as the originators of non -Euclidean geometry. It seems fair that, as pioneers, the y shouldshare the glory.
But even the ir discoveries did not fully resolve the fundamentalissue of the parallel postulate. For, while the y had developed the ir geometries to a high level of sophistication, it was merely a feelingin the ir bones, not a logical argument on paper, that supponed the ir contentionthat the new geometries were valid alternatives to Euclid's. Thus, the final chapter of this age-old story was written in by the Italian Eugenio Beltrami , who unequivocally provedthat non -Euclidean geometry was as logically consistent as Euclid's own.
That is, if a contradiction lurked somewhere in the geometry of Gauss,Bolyai, and Lobachevski, or in that of Riemann, the n Beltrami showed. Sincevirtually everyone felt that Euclid's geometry was as consistent as couldbe, the conclusion was that non -Euclidean geometries were likewise asgood as gold. Put ano the r way, non -Euclidean geometry is not logicallyinferior to its older, Euclidean counterpart. The first involves ano the r look at the congruence of triangles.
The surprising development isthat, in Bolyai's geometry, the re is yet ano the r way to show congruence,namely "angle-angle-angle. They would have the same shape but need not be congruent; we could have, for example, atiny equilateral triangle and a large equilateral triangle, non -congruentfigures all of whose angles are equal. The non -Euclidean the orem thatfollows shows that no such thing is possible in this strange world.
If two of Bolyai's triangles have the same shape, the y must have the same size! We assert that sides AB and DE must have the same length. To prove this, suppose for the sake of an eventual contradictionthat the y differ in length, and without loss of generality we mightas well assume that AB the lower quadrilateral EFHG.
In short, the se sides areequal in length. Thismost fundamental property of Euclid's geometry-one that figuredprominently in so much geometric reasoning-must be discarded whenwe move to the non -Euclidean domain. For suppose we consider the two triangles shown in Figure 2. Now we assert thatcFA B D "' In short, knowing two angles of a non -Euclidean triangleis not sufficient to determine the third one. This result, and many o the rslike it, indicate what Bolyai meant when he described his "strange newuniverse" and why so many felt that, just over the horizon, a logical contradictionmust be waiting.
But as it turned out, the y were wrong. And where did the se nineteenth century discoveries leave Euclid? On the one hand, his geometry was displaced as the only logically consistentdescription of space. Much to the surprise of virtually everyone,it turned out that the parallel postulate was not mandated by logic.
Euclid assumed it, but the re was no ma the matical necessity to do so. Competing geometries, equally as valid, existed. Yet the net effect may be to enhance, not destroy, Euclid's reputation. For he, unlike so many who followed, did not fall into the trap of tryingto prove the parallel postulate from the o the r self-evident truths, anendeavor, we now know, that is utterly doomed to failure. Instead, hesimply laid out his assumption where it properly belonged, as a postulate.
Euclid certainly could not have known about the alternative geometriesthat would be discovered two millennia in the future. Yet somethingin his ma the matician's intuition must have told him that thisproperty was a separate, independent idea that needed its own postulate,no matter how wordy and complicated it sounded. Ma the maticians 22centuries later proved that Euclid had been right all along. ChapterEuclid and the Infinitude of Primes ca.
Because it comes first,this book is certainly the best known and most studied part of the Elements,but it is just one of 13 books into which the work was divided. Chapter 3 of fers a quick tour of the rest of this classic text.
Book II explored what we now call "geometric algebra. Of course, the notion of algebra wasforeign to the Greeks, and its appearance as a formal system lay centuriesin the future. We can get a sense of Book II by citing a representativeproposition, one whose statement at first glance appears ra the r convoluted,but which upon closer examination emerges as a simple and wellknownalgebraiC formula. DThis, of course, is a famous identity encountered in the first year of algebra.
Euclid approached it not as some algebraic expression but as a literalgeometric decomposition of the square upon AB into two smallersquares and two congruent rectangles. Yet the equivalence of his geometricstatement and its algebraic counterpart is clear. Much of Book IIwas of this nature. It concluded with Proposition The third book contained 37 propositions about circles.
Circles hadbeen used in the constructions of Book I but had not the mselves been the focus of the discussion. Proposition Of course, by Definition15, every circle has a center, but for a circle already drawn upon the page, it is not immediately clear how to find that central point.
Thus,Euclid provided the necessary construction. A few propositions later, we find the important result, "Ina circle, angles in the same segment are equal to one ano the r. In modern terminology, we would saythat both intercept the same arc, namely, arc BD. Having proved this the orem, Euclid tackled the concept of a quadrilateralinscribed within a circle, a figure of ten called a "cyclic quadrilateral. A full treatment of the se latter topics wouldhave to await the arrival of Archimedes, as discussed in Chapter 4.
Euclid's fourth book dealt with inscribing and circumscribing certainkinds of geometric figures. As with all constructions in the Elements, hewas limited to his compass and unmarked straightedge.
These limitationsaside, he non e the less produced some fairly sophisticated results. For instance, Proposition IV. In the next proposition,he showed how to circumscribe a circle about a given triangle;this time, he located the center of the circle at the point where the perpendicularbisectors of the sides meet. From the re, Euclid considered the construction of regular polygons,all of whose sides are the same length and all of whose angles are congruent.
These are "perfect" polygons whose symmetry and beauty certainlyappealed to the Greek imagination. Recall that Euclid had begun the Elements with the construction of aregular, or "equilateral," triangle, and in Proposition! In Proposition IV. II, Euclidexpanded his repertoire by inscribing a regular pentagon in a circle, and.
The final constructionin this book was of the regular pentadecagon-that is, the regularIS-sided polygon-and his argument warrants a quick look. Within a given circle, Euclid inscribed both an equilateral trianglewith side AC and a regular pentagon with side AB, each sharing a vertexat A Figure 3. As Euclid observed, arc AC is a third of the circle'scircumference, while arc AB is a fifth of the same. If we bisect the chord from B to C and draw the perpendicular outwardfrom the chord's midpoint to point E on the circle, we shall havebisected arc Be.
Thus, arc BE is one-fifteenth of the circle, and so chordBE is the length of the side of a regular pentadecagon. Copying 15 of the se chords around the circle completes the construction. Euclid said no more about regular polygons in the Elements, but heclearly was aware that if one had constructed such a polygon, the bisectionprocedure outlined above would produce regular polygons withtwice as many sides. After constructing an equilateral triangle, Greekgeometers could the refore produce regular polygons of 6, 12, 24, 48, Sides; from the regular pentagon would emerge regular, , , Nowhere, forinstance, did Euclid mention constructing a regular 7-gon, or 9-gon, orgon, since the se did not fit the neat "doubling" patterns above.
Oneimagines that the Greeks put a lot of time and effort into trying to. In fact, while Euclid did not explicitly say so, most subsequent ma the maticiansassumed that his were the only constructible regular polygonsand that any o the rs were simply beyond the capability of compass andstraightedge.
It was thus a shock of monumental proportions when the teenagedCarl Friedrich Gauss discovered how to construct a regular heptadecagon gon in This discovery marked the young Gauss as a ma the matical genius of the first order.
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