Euclids method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Let a be the given point, and bc the given straight line. Euclid may have been active around 300 bce, because there is a report that he lived at the time of the first ptolemy, and because a reference by archimedes to euclid indicates he lived before archimedes 287212 bce. Euclids predecessors employed a variety higher curves for this purpose. According to joyce commentary, proposition 2 is only used in proposition 3 of euclids elements, book i. If as many numbers as we please beginning from a unit are in continued proportion, and the number after the unit is square, then all the rest are also square. Book 9 contains various applications of results in the previous two books, and. The books on number theory, vii through ix, do not directly depend on book v since there is a different definition for ratios of numbers. What will be a sufficient condition for the angles that are contained by those sides to be equal, the angles a and d. The elements is a mathematical treatise consisting of books attributed to the ancient greek. If an equilateral pentagon is inscribed in a circle which has its diameter rational, then the side of the pentagon is the irrational straight line called minor. Although euclid is fairly careful to prove the results on ratios that he uses later, there are some that he didnt notice he used, for instance, the law of trichotomy for ratios. About logical converses, contrapositives, and inverses, although this is the first proposition about parallel lines, it does not require the parallel postulate post.
For the proof, see the wikipedia page linked above, or euclid s elements. Definitions from book v david joyces euclid heaths comments on definition 1 definition 2 definition 3 definition 4 definition 5 definition 6 definition 7 definition 8 definition 9 definition 10. Euclidean geometry is a mathematical system attributed to alexandrian greek mathematician euclid, which he described in his textbook on geometry. For one thing, the elements ends with constructions of the five regular solids in book xiii, so it is a nice aesthetic touch to begin with the construction of a regular triangle. For debugging it was handy to have a consistent not random pair of given lines, so i made a definite parameter start procedure, selected to look similar to. Book 2 proposition 9 if a straight line is cut into equal and unequal sections, then the sum of the sqares on the unequal sections is double the sum of the squares on one of the equal segments and the segment between the two cuts. In euclid s the elements, book 1, proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. The usual con struction really uses both, for prop. This proposition starts with a line that is randomly cut. Proposition 9 straight lines which are parallel to the same straight line but do not lie in the same plane with it are also parallel to each other. Euclid, elements of geometry, book i, proposition 9 edited by dionysius lardner, 1855 proposition ix. Euclids elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c.
Proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. Book xiii introduction select from book xiii book xiii intro xiii. Euclid, elements ii 9 translated by henry mendell cal. This is the ninth proposition in euclids first book of the elements. Euclid s construction problems i famous math problems 12 nj wildberger. A slight modification gives a factorization of the difference of two squares. Let each of the straight lines ab and cd be parallel to ef, but not in the same plane with it. If there be two straight lines, and one of them be cut into any number of segments. The incremental deductive chain of definitions, common notions, constructions. This proof is a construction that allows us to bisect angles. Inscribing and circumscribing circles and arbitrary triangles prop. Actually, the final sentence is not part of the lemma, probably because euclid moved that statement to the first book as i. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions.
This proposition is used in book x to prove a lemma for x. In euclids the elements, book 1, proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. Euclid is likely to have gained his mathematical training in athens, from pupils of plato. And, since the square on bk is five times the square on km, therefore the square on bk has to. Scholars believe that the elements is largely a compilation of propositions based on books by earlier greek mathematicians proclus 412485 ad, a greek mathematician who lived around seven centuries after euclid, wrote in his commentary on the elements. From a given straight line to cut off a prescribed part let ab be the given straight line. Use of this proposition this proposition is used in ii. Feb 26, 2014 49 videos play all euclid s elements, book 1 sandy bultena for the love of physics walter lewin may 16, 2011 duration. This is the ninth proposition in euclid s first book of the elements. This proposition admits of a number of different cases, depending on the relative.
Perseus provides credit for all accepted changes, storing new additions in a versioning system. However, euclids original proof of this proposition. For it was proved in the first theorem of the tenth book that, if two unequal magnitudes be set out, and if from the greater there be subtracted a magnitude greater than the half, and from that which is left a greater than the half, and if this be done continually, there will be left some magnitude which will be less. Prop 3 is in turn used by many other propositions through the entire work. I tried to make a generic program i could use for both the primary job of illustrating the theorem and for the purpose of being used by subsequent theorems, but it is simpler to separate those into two sub procedures. I do not see anywhere in the list of definitions, common notions, or postulates that allows for this assumption. Leon and theudius also wrote versions before euclid fl. It uses proposition 1 and is used by proposition 3.
Feb 23, 2018 euclids 2nd proposition draws a line at point a equal in length to a line bc. Definition 2 a number is a multitude composed of units. Euclids elements, book ii, proposition 9 proposition 9 if a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half and the square on the straight line between the points of section. A fter stating the first principles, we began with the construction of an equilateral triangle. Here i bisect the angle fcg instead of the line fg. The expression here and in the two following propositions is.
Definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. Archimedes, after euclid, created two constructions. An exterior angle of a triangle is greater than either of the interior angles not adjacent to it. Stoicheia is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c.
It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. The visual constructions of euclid book i 63 through a given point to draw a straight line parallel to a given straight line. The books cover plane and solid euclidean geometry. Note that euclid does not consider two other possible ways that the two lines could meet, namely, in the directions a and d or toward b and c. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate. This is the fourth proposition in euclids second book of the elements. Feb 24, 2018 proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. Book i, propositions 9,10,15,16,27, and proposition 29 through pg. About logical converses, contrapositives, and inverses, although this is the first proposition about parallel lines, it does not. The square created by the whole line is equal to the sum of the.
The goal of euclid s first book is to prove the remarkable theorem of pythagoras about the squares that are constructed of the sides of a right triangle. This is the same as proposition 20 in book iii of euclids elements although euclid didnt prove it this way, and seems not to have considered the application to angles greater than from this we immediately have the. Euclid s method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions theorems from these. Many other equimultiples might be selected, which would tend to. Fundamentals of number theory definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. Although many of euclids results had been stated by earlier mathematicians, euclid was the first to show. If a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments. S uppose that two sides of one triangle are equal respectively to. If a straight line is cut into equal and unequal segments, then the sum of the squares on the unequal segments of the whole is double the sum. This work is licensed under a creative commons attributionsharealike 3.
The goal of euclids first book is to prove the remarkable theorem of pythagoras about the squares that are constructed of the sides of a right triangle. According to joyce commentary, proposition 2 is only used in proposition 3 of euclid s elements, book i. As euclid states himself i3, the length of the shorter line is measured as the radius of a circle directly on the longer line by letting the center of the circle reside on an extremity of the longer line. If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double of the square on the half. When teaching my students this, i do teach them congruent angle construction with straight edge and.
Logical structure of book ii the proofs of the propositions in book ii heavily rely on the propositions in book i involving right angles and parallel lines, but few others. By pappus time it was believed that angle trisection was not possible using. Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers and the construction of all even perfect numbers. Definition 3 a number is a part of a number, the less of the greater, when it measures the greater. This sequence demonstrates the developmental nature of mathematics. Definition 4 but parts when it does not measure it. Introduction main euclid page book ii book i byrnes edition page by page 1 23 45 67 89 1011 12 1415 1617 1819 2021 2223 2425 2627 2829 3031 3233 3435 3637 3839 4041 4243 4445 4647 4849 50 proposition by proposition with links to the complete edition of euclid with pictures in java by david joyce, and the well known comments from heaths edition at the. An xml version of this text is available for download, with the additional restriction that you offer perseus any modifications you make. Because, if those angles are equal, then the triangles will be congruent, sideangleside.
In this proposition, there are just two of those lines and their sum equals the one line. Return to vignettes of ancient mathematics return to elements ii, introduction go to prop. Euclid, who put together the elements, collecting many of eudoxus theorems, perfecting many of theaetetus, and also bringing to. Euclid s elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. Now it is clear that the purpose of proposition 2 is to effect the construction in this proposition. Let a straight line ac be drawn through from a containing with ab any angle. If a straight line is cut into equals and unequals, the squares from the unequal segments of the whole are double that from the half and the square from the line between the cuts. Using the postulates and common notions, euclid, with an ingenious construction in proposition 2, soon verifies the important sideangleside congruence relation proposition 4.
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