This is the second proposition in euclid s second book of the elements. If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut line and each of the segments. 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. Definitions superpose to place something on or above something else, especially so that they coincide. Definition 2 a number is a multitude composed of units.
W e now begin the second part of euclid s first book. 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 than the lesser magnitude. Prop 3 is in turn used by many other propositions through the entire work. 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. These are sketches illustrating the initial propositions argued in book 1 of euclids elements. Euclids elements redux, volume 1, contains books iiii, based on john caseys translation. Euclid s plan and proposition 6 its interesting that although euclid delayed any explicit use of the 5th postulate until proposition 29, some of the earlier propositions tacitly rely on it. Oliver byrne mathematician published a colored version of elements in 1847. Euclid then builds new constructions such as the one in this proposition out of previously described constructions. Proposition 6 if a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up of the half and the added straight line.
Set out the circle efg of radius eh 23v2, and inscribe in that circle an equilateral triangle. If a straight line be drawn parallel to one of the. According to joyce commentary, proposition 2 is only used in proposition 3 of euclids elements, book i. In this proposition, there are just two of those lines and their sum equals the one line. Introduction euclids elements is by far the most famous mathematical work of classical antiquity, and also has the distinction. This proposition essentially looks at a different case of the distributive. To place at a given point as an extremity a straight line equal to a given straight line. On a given finite straight line to construct an equilateral triangle. I do not see anywhere in the list of definitions, common notions, or postulates that allows for this assumption.
Click anywhere in the line to jump to another position. Euclids elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally. Leon and theudius also wrote versions before euclid fl. If a straight line is bisected and a straight line is added to it in a straight line, then the rectangle contained by the whole with the added straight line and the added straight line together with the square on the half equals the square on the straight line made up. It is required to place a straight line equal to the given straight line bc with one end at the point a. Euclids elements of geometry, book 6, proposition 33. It was first proved by euclid in his work elements. If a straight line is bisected and some straightline is added to it on a straightone, the rectangle enclosed by the whole with the added line and the added line with the square from the half line is. We have accomplished the basic constructions, we have proved the basic relations between the sides and angles of a triangle, and in particular we have found conditions for triangles to be congruent.
Euclid prefers to prove a pair of converses in two stages, but in some propositions, as this one, the proofs in the two stages are almost inverses of each other, so both could be proved at once. Euclid s elements book 6 proposition 31 sandy bultena. Book 6 applies the theory of proportion to plane geometry, and contains theorems on similar. If a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome. According to this proposition the rectangle ad by db, which is the product xy, is the difference of two squares, the large one being the square on the line cd, that is the square of x b2, and the small one being the square on the line cb, that is, the square of b2. Lecture 6 euclid propositions 2 and 3 patrick maher. How to prove euclids proposition 6 from book i directly.
If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally. Cut a line parallel to the base of a triangle, and the cut sides will be proportional. It is required to inscribe a triangle equiangular with the triangle def in the circle abc. The books cover plane and solid euclidean geometry. Euclid here introduces the term irrational, which has a different meaning than the modern concept of irrational numbers. 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. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of. Return to vignettes of ancient mathematics return to elements ii, introduction go to prop. But the square on da is rational, for da is rational being half of ab which is rational, therefore the square on cd is also rational. Does euclids book i proposition 24 prove something that proposition 18 and 19 dont prove. David joyces introduction to book i heath on postulates heath on axioms and common notions. Euclids theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. Feb 23, 2018 euclids 2nd proposition draws a line at point a equal in length to a line bc. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions.
Euclids elements of geometry university of texas at austin. 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. There is a related pencil sketch in turners lecture notes. Make hk of length 43 and perpendicular to the plane of the triangle, and connect ke, kf, and kg. It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. One key reason for this view is the fact that euclids proofs make strong use of geometric diagrams. Then each side of the triangle will be 23v6, the same as ad. Definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. 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. For debugging it was handy to have a consistent not random pair of given lines, so i made a definite parameter start. W e now begin the second part of euclids first book. Shormann algebra 1, lessons 67, 98 rules euclids propositions 4 and 5 are your new rules for lesson 40, and will be discussed below. 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. He also gives a formula to produce pythagorean triples book 11 generalizes the results of book 6 to solid figures.
Euclid, elements ii 6 translated by henry mendell cal. Jun 24, 2017 cut a line parallel to the base of a triangle, and the cut sides will be proportional. Book 9 contains various applications of results in the previous two books, and includes theorems. A fter stating the first principles, we began with the construction of an equilateral triangle. Lecture 6 euclid propositions 2 and 3 patrick maher scienti c thought i fall 2009. Proposition 6 if a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome. According to joyce commentary, proposition 2 is only used in proposition 3 of euclid s elements, book i.
Euclids elements redux, volume 2, contains books ivviii, based on john caseys translation. This proposition admits of a number of different cases, depending on the relative. Book 2 49 book 3 69 book 4 109 book 5 129 book 6 155 book 7 193 book 8 227 book 9 253 book 10 281 book 11 423 book 12 471 book 505 greekenglish lexicon 539. To inscribe a triangle equiangular with a given triangle in a given circle. With links to the complete edition of euclid with pictures in java by david joyce, and the well known. The elements is a mathematical treatise consisting of books attributed to the ancient greek. Use of proposition 2 the construction in this proposition is only used in proposition i.
Given two unequal straight lines, to cut off from the greater a straight line equal to the less. If in a triangle two angles be equal to one another, the sides which subtend the equal. Euclid s theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. 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. Jun 24, 2017 the ratio of areas of two triangles of equal height is the same as the ratio of their bases. Here euclid has contented himself, as he often does, with proving one case only. It uses proposition 1 and is used by proposition 3. Let abc be the given circle, and def the given triangle. If the ratio of the first of three magnitudes to the second be greater than the ratio of the first to the third, the second magnitude. Euclids 2nd proposition draws a line at point a equal in length to a line bc. Euclid s elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c.
Thus it is required to place at the point a as an extremity a straight line equal to the given straight line bc. Now it is clear that the purpose of proposition 2 is to effect the construction in this proposition. The ratio of areas of two triangles of equal height is the same as the ratio of their bases. Euclid is likely to have gained his mathematical training in athens, from pupils of plato. 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. Definitions from book i byrnes definitions are in his preface david joyces euclid heaths comments on the definitions. Definition 4 but parts when it does not measure it. The only basic constructions that euclid allows are those described in postulates 1, 2, and 3. But unfortunately the one he has chosen is the one that least needs proof. Let ab be a rational straight line cut in extreme and mean ratio at c, and let ac be the greater segment. Stoicheia is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c.
And, since the square on cd has not to the square on da the ratio which a square number has to a square number, therefore cd is incommensurable in length with da. If there are two straight lines, and one of them is cut into any number of segments whatever, then the rectangle contained by the two straight lines equals the sum of the rectangles contained by the uncut straight line and each of the segments. It may also be used in space, however, since proposition xi. Learn vocabulary, terms, and more with flashcards, games, and other study tools. This is the sixth proposition in euclids second book of the elements. Definition 3 a number is a part of a number, the less of the greater, when it measures the greater. 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. 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. Fundamentals of number theory definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. 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. Note that this constuction assumes that all the point a and the line bc lie in a plane. Let a be the given point, and bc the given straight line.
The proposition is the proposition that the square root of 2 is irrational. Proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. So if anybody is so inclined, where is the proposition in the english. Euclids plan and proposition 6 its interesting that although euclid delayed any explicit use of the 5th postulate until proposition 29, some of the earlier propositions tacitly rely on it. This proposition starts with a line that is bisected and then has some small. To place a straight line equal to a given straight line with one end at a given point. There is a free pdf file of book i to proposition 7.
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