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identify the keywords in this sentence that relate to geometry | 6 FINAL OBSERVATIONS ABOUT THE ELEMENTS Our study of geometry begins with an examination of the historical origins of the axiomatic method in geometry. | axiomatic, geometry, elements |
identify the keywords in this sentence that relate to geometry | While the material in this chapter is not a math- ematical prerequisite for what comes later, an appreciation of the historical roots of axiomatic thinking is essential to an understanding of why the foundations of geometry are systematized as they are. | axiomatic, geometry |
identify the keywords in this sentence that relate to geometry | 1 GEOMETRY BEFORE EUCLID Geometry is an ancient subject. | euclid, geometry |
identify the keywords in this sentence that relate to geometry | Its roots go back thousands of years and geometric ideas of one kind or another are found in nearly every human culture. | geometric, ideas |
identify the keywords in this sentence that relate to geometry | The beauty of geometric patterns is universally appreciated and often investigated in a systematic way. | geometric, patterns |
identify the keywords in this sentence that relate to geometry | The study of geometry as we know it emerged more than 4000 years ago in Mesopotamia, Egypt, India, and China. | geometry |
identify the keywords in this sentence that relate to geometry | Because the Nile River annually flooded vast areas of land and obliterated property lines, surveying and measuring were important to the ancient Egyptians. | lines |
identify the keywords in this sentence that relate to geometry | This practical interest may have motivated their study of geometry. | geometry |
identify the keywords in this sentence that relate to geometry | Egyptian geometry was mostly an empirical science, consisting of many rule-of-thumb procedures that were arrived at through experimentation, observation, and trial and error. | geometry |
identify the keywords in this sentence that relate to geometry | Formulas were approximate ones that appeared to work, or at least gave answers that were close enough for practical purposes. | formulas |
identify the keywords in this sentence that relate to geometry | But the ancient Egyptians were also aware of more general principles, such as special cases of the Pythagorean Theorem and formulas for volumes. | pythagorean theorem, formulas, volume, case |
identify the keywords in this sentence that relate to geometry | The ancient Mesopotamians, or Babylonians, had an even more advanced under- standing of geometry. | geometry |
identify the keywords in this sentence that relate to geometry | They knew the Pythagorean Theorem long before Pythagoras. | pythagorean, theorem |
identify the keywords in this sentence that relate to geometry | They discovered some of the area-based proofs of the theorem that will be discussed in Chapter 7, and knew a general method that generates all triples of integers that are lengths of sides of right triangles. | triangles, theorem, proofs, sides |
identify the keywords in this sentence that relate to geometry | In India, ancient texts apply the Pythagorean Theorem to geometric problems associated with the design of structures. | pythagorean theorem, geometric problems |
identify the keywords in this sentence that relate to geometry | The Pythagorean Theorem was also discovered in China at roughly the same time. | pythagorean, theorem |
identify the keywords in this sentence that relate to geometry | About 2500 years ago there was a profound change in the way geometry was practiced: Greek mathematicians introduced abstraction, logical deduction, and proof into geometry. | geometry, proof |
identify the keywords in this sentence that relate to geometry | They insisted that geometric results be based on logical reasoning from first principles. | geometric, results |
identify the keywords in this sentence that relate to geometry | In theory this made the results of geometry exact, certain, and undeniable, rather than just likely or approximate. | geometry |
identify the keywords in this sentence that relate to geometry | It also took geometry out of the realm of everyday experience and made it a subject that studies abstract entities. | geometry, real |
identify the keywords in this sentence that relate to geometry | Since the purpose of this 1 For use by the Norwegian University of Science and Technology Copyright Pearson. | |
identify the keywords in this sentence that relate to geometry | All Rights Reserved. | |
identify the keywords in this sentence that relate to geometry | 2 Chapter 1 Prologue: Euclid’s Elements course is to study the logical foundations of geometry, it is natural that we should start with the geometry of the ancient Greeks. | geometry, elements, euclid |
identify the keywords in this sentence that relate to geometry | The process of introducing logic into geometry apparently began with Thales of Miletus around 600 B. | geometry |
identify the keywords in this sentence that relate to geometry | and culminated in the work of Euclid of Alexandria in approximately 300 B. | euclid |
identify the keywords in this sentence that relate to geometry | Euclid is the most famous of the Greek geometers and his name is still universally associated with the geometry that is studied in schools today. | geometry, euclid |
identify the keywords in this sentence that relate to geometry | Most of the ideas that are included in what we call ‘‘Euclidean Geometry’’ probably did not originate with Euclid himself; rather, Euclid’s contribution was to organize and present the results of Greek geometry in a logical and coherent way. | euclidean, geometry |
identify the keywords in this sentence that relate to geometry | He published his results in a series of thirteen books known as his Elements. | elements |
identify the keywords in this sentence that relate to geometry | We begin our study of geometry by examining those Elements because they set the agenda for geometry for the next two millennia and more. | geometry, elements |
identify the keywords in this sentence that relate to geometry | 2 THE LOGICAL STRUCTURE OF EUCLID’S ELEMENTS Euclid’s Elements are organized according to strict logical rules. | elements, euclid |
identify the keywords in this sentence that relate to geometry | Euclid begins each book with a list of definitions of the technical terms he will use in that book. | definitions, euclid, terms |
identify the keywords in this sentence that relate to geometry | In Book I he next states five ‘‘postulates’’ and five ‘‘common notions. | postulates |
identify the keywords in this sentence that relate to geometry | ’’ These are assumptions that are meant to be accepted without proof. | proof |
identify the keywords in this sentence that relate to geometry | Both the postulates and common notions are basic statements whose truth should be evident to any reasonable person. | postulates, statements |
identify the keywords in this sentence that relate to geometry | They are the starting point for what follows. | point |
identify the keywords in this sentence that relate to geometry | Euclid recognized that it is not possible to prove everything, that he had to start somewhere, and he attempted to be clear about exactly what his assumptions were. | euclid, prove |
identify the keywords in this sentence that relate to geometry | Most of Euclid’s postulates are simple statements of intuitively obvious and unde- niable facts about space. | postulates, statements, euclid |
identify the keywords in this sentence that relate to geometry | For example, Postulate I asserts that it is possible to draw a straight line through any two given points. | straight line, postulate, points |
identify the keywords in this sentence that relate to geometry | Postulate II says that a straight line segment can be extended to a longer segment. | straight line, line segment, postulate |
identify the keywords in this sentence that relate to geometry | Postulate III states that it is possible to construct a circle with any given center and radius. | postulate |
identify the keywords in this sentence that relate to geometry | Traditionally these first three postulates have been associated with the tools that are used to implement them on a piece of paper. | postulates |
identify the keywords in this sentence that relate to geometry | The first two postulates allow two different uses of a straight edge: A straight edge can be used to draw a line segment connecting any two points or to extend a given line segment to a longer one. | line segment, postulates, points |
identify the keywords in this sentence that relate to geometry | The third postulate affirms that a compass can be used to construct a circle with a given center and radius. | postulate, compass |
identify the keywords in this sentence that relate to geometry | Thus the first three postulates simply permit the familiar straightedge and compass constructions of high school geometry. | straightedge, postulates, geometry, compass |
identify the keywords in this sentence that relate to geometry | The fourth postulate asserts that all right angles are congruent (‘‘equal’’ in Euclid’s terminology). | right angle, postulate, congruent, euclid, angles, equal |
identify the keywords in this sentence that relate to geometry | The fifth postulate makes a more subtle and complicated assertion about two lines that are cut by a transversal. | fifth postulate, lines |
identify the keywords in this sentence that relate to geometry | These last two postulates are the two technical facts about geometry that Euclid needs in his proofs. | postulates, geometry, euclid, proofs |
identify the keywords in this sentence that relate to geometry | The common notions are also intuitively obvious facts that Euclid plans to use in his development of geometry. | geometry, euclid |
identify the keywords in this sentence that relate to geometry | The difference between the common notions and the postulates is that the common notions are not peculiar to geometry but are common to all branches of mathematics. | postulates, geometry, between |
identify the keywords in this sentence that relate to geometry | They are everyday, common-sense assumptions. | |
identify the keywords in this sentence that relate to geometry | Most spell out properties of equality, at least as Euclid used the term equal. | euclid, equal |
identify the keywords in this sentence that relate to geometry | The largest part of each Book of the Elements consists of propositions and proofs. | proposition, elements, proofs, part |
identify the keywords in this sentence that relate to geometry | These too are organized in a strict, logical progression. | |
identify the keywords in this sentence that relate to geometry | The first proposition is proved using only the postulates, Proposition 2 is proved using only the postulates and Proposition 1, and so on. | proposition, postulates, prove |
identify the keywords in this sentence that relate to geometry | Thus the entire edifice is built on just the postulates and common notions; once these are granted, everything else follows logically and inevitably from them. | postulates |
identify the keywords in this sentence that relate to geometry | What is astonishing is the number and variety of propositions that can be deduced from so few assumptions. | proposition, number |
identify the keywords in this sentence that relate to geometry | For use by the Norwegian University of Science and Technology Copyright Pearson. | |
identify the keywords in this sentence that relate to geometry | Section 1. | |
identify the keywords in this sentence that relate to geometry | 3 The historical significance of Euclid’s Elements 3 FIGURE 1. | elements, euclid |
identify the keywords in this sentence that relate to geometry | 1: Euclid’s tools: a compass and a straightedge 1. | straightedge, compass, euclid |
identify the keywords in this sentence that relate to geometry | 3 THE HISTORICAL SIGNIFICANCE OF EUCLID’S ELEMENTS It is nearly impossible to overstate the importance of Euclid’s Elements in the development of mathematics and human culture generally. | elements, euclid |
identify the keywords in this sentence that relate to geometry | From the time they were written, the Elements have been held up as the standard for the way in which careful thought ought to be organized. | elements |
identify the keywords in this sentence that relate to geometry | They became the model for the development of all scientific and philosophical theories. | model |
identify the keywords in this sentence that relate to geometry | What was especially admired about Euclid’s work was the way in which he clearly laid out his assumptions and then used pure logic to deduce an incredibly varied and extensive set of conclusions from them. | euclid |
identify the keywords in this sentence that relate to geometry | Up until the twentieth century, Euclid’s Elements were the textbook from which all students learned both geometry and logic. | geometry, elements, euclid |
identify the keywords in this sentence that relate to geometry | Even today the geometry in school textbooks is presented in a way that is remarkably close to that of Euclid. | geometry, euclid |
identify the keywords in this sentence that relate to geometry | Furthermore, much of what mathematicians did during the next two thousand years centered around tying up loose ends left by Euclid. | euclid |
identify the keywords in this sentence that relate to geometry | Countless mathematicians spent their careers trying to solve problems that were raised by Greek geometers of antiquity and trying to improve on Euclid’s treatment of the foundations. | euclid, prove |
identify the keywords in this sentence that relate to geometry | ℓ m α β FIGURE 1. | |
identify the keywords in this sentence that relate to geometry | 2: Euclid’s Postulate V. | postulate, euclid |
identify the keywords in this sentence that relate to geometry | If the sum of α and β is less than two right angles, then ℓand m must eventually intersect Most of the efforts at improvement focused on Euclid’s fifth postulate. | fifth postulate, right angle, euclid, angles, prove |
identify the keywords in this sentence that relate to geometry | Even though the statement does not explicitly mention parallel lines, this postulate is usually referred to as ‘‘Euclid’s parallel postulate. | parallel postulate, parallel lines, euclid |
identify the keywords in this sentence that relate to geometry | ’’ It asserts that two lines that are cut by a transversal must intersect on one side of the transversal if the interior angles on that side of the transversal sum to less than two right angles. | interior angles, right angle, lines, side |
identify the keywords in this sentence that relate to geometry | In particular, it proclaims that the condition on the an- gles formed by a transversal implies that the two given lines are not parallel. | parallel, lines, part, lies |
identify the keywords in this sentence that relate to geometry | Thus it is really a statement about nonparallel lines. | parallel lines, real |
identify the keywords in this sentence that relate to geometry | As we shall see later, the postulate can be reformulated in ways that make it more obviously and directly a statement about parallel lines. | parallel lines, postulate |
identify the keywords in this sentence that relate to geometry | A quick reading of the postulates (see the next section) reveals that Postulate V is noticeably different from the others. | postulates |
identify the keywords in this sentence that relate to geometry | For one thing, its statement is much longer than For use by the Norwegian University of Science and Technology Copyright Pearson. | |
identify the keywords in this sentence that relate to geometry | 4 Chapter 1 Prologue: Euclid’s Elements those of the other postulates. | postulates, elements, euclid |
identify the keywords in this sentence that relate to geometry | A more significant difference is the fact that it involves a fairly complicated arrangement of lines and also a certain amount of ambiguity in that the lines must be ‘‘produced indefinitely. | lines |
identify the keywords in this sentence that relate to geometry | ” It is not as intuitively obvious or self-evident as the other postulates; it has the look and feel of a proposition rather than a postulate. | proposition, postulates |
identify the keywords in this sentence that relate to geometry | For these reasons generations of mathematicians tried to improve on Euclid by attempting to prove that Postulate V is a logical consequence of the other postulates or, failing that, they tried at least to replace Postulate V with a simpler, more intuitively obvious postulate from which Postulate V could then be deduced... | postulates, euclid, prove |
identify the keywords in this sentence that relate to geometry | No one ever succeeded in proving the fifth postulate using just the first four postulates, but it was not until the nineteenth century that mathematicians fully understood why that was the case. | fifth postulate, postulates, case |
identify the keywords in this sentence that relate to geometry | It should be recognized that these efforts at improvement were not motivated by a perception that there was anything wrong with Euclid’s work. | euclid, prove |
identify the keywords in this sentence that relate to geometry | Quite the opposite: Thousands of mathematicians spent enormous amounts of time trying to improve on Euclid precisely because they thought so highly of Euclid’s accomplishments. | opposite, euclid, prove |
identify the keywords in this sentence that relate to geometry | They wanted to take what was universally regarded as the crown of theoretical thought and make it even more wonderful than it already was! Another important point is that efforts to rework Euclid’s treatment of geometry led indirectly to progress in mathematics that went far beyond mere improvements in the Elements the... | geometry, elements, euclid, point, prove |
identify the keywords in this sentence that relate to geometry | Attempts to prove Euclid’s Fifth Postulate eventually resulted in the realization that, in some kind of stroke of genius, Euclid somehow had the great insight to pinpoint one of the deepest properties that a geometry may have. | postulate, geometry, euclid, point, prove, real |
identify the keywords in this sentence that relate to geometry | Not only that, but it was discovered that there are alternative geometries in which Euclid’s Fifth Postulate fails to hold. | postulate, euclid |
identify the keywords in this sentence that relate to geometry | These discoveries were made in the early nineteenth century and had far-reaching implications for all of mathematics. | |
identify the keywords in this sentence that relate to geometry | They opened up whole new fields of mathematical study; they also produced a revolution in the conventional view of how mathematics relates to the real world and forced a new understanding of the nature of mathematical truth. | real |
identify the keywords in this sentence that relate to geometry | The story of how Euclid’s Parallel Postulate inspired all these developments is one of the most interesting in the history of mathematics. | parallel postulate, euclid |
identify the keywords in this sentence that relate to geometry | That story will unfold in the course of our study of geometry in this book. | geometry |
identify the keywords in this sentence that relate to geometry | It is only in the light of that story that the current organization of the foundations of geometry can be properly understood. | geometry |
identify the keywords in this sentence that relate to geometry | 4 A LOOK AT BOOK I OF THE ELEMENTS In order to give more substance to our discussion, we now take a direct look at parts of Book I of the Elements. | elements, parts |
identify the keywords in this sentence that relate to geometry | All Euclid’s postulates are stated below as well as selected definitions and propositions. | proposition, postulates, definitions, euclid |
identify the keywords in this sentence that relate to geometry | The excerpts included here are chosen to illustrate the points that will be made in the following section. | points |
identify the keywords in this sentence that relate to geometry | The translation into English is by Sir Thomas Little Heath (1861–1940). | |
identify the keywords in this sentence that relate to geometry | Some of Euclid’s Definitions Definition 1. | definitions, euclid |
identify the keywords in this sentence that relate to geometry | A point is that which has no part. | point, part |
identify the keywords in this sentence that relate to geometry | Definition 2. | definition |
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