Perhaps two of the most easily recognized propositions from book xii by anyone that has taken high school geometry are propositions 2 and 18. If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the base greater than the base. Propositions dealing with ratios of lines are postponed until book vi, but any ratio concerning lines can be converted into a statement about areas of rectangles. Use of this proposition this proposition is used in ii. Book iv main euclid page book vi book v byrnes edition page by page. Euclids elements book one with questions for discussion. Book v is one of the most difficult in all of the elements. Euclid, elements of geometry, book i, proposition 1 edited by sir thomas l. To place a straight line equal to a given straight line with one end at a given point.
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. On a given straight line to construct an equilateral triangle. Euclids elements book 2 propositions flashcards quizlet. Prop 3 is in turn used by many other propositions through the entire work. Proposition 14, angles formed by a straight line converse. Euclid s elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. Choose an arbitrary point a and another arbitrary one d.
The only basic constructions that euclid allows are those described in postulates 1, 2, and 3. 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. The fragment contains the statement of the 5th proposition of book 2, which in the translation of t. If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions. 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. Euclidean geometry is a mathematical system attributed to alexandrian greek mathematician euclid, which he described in his textbook on geometry. Feb 24, 2018 proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. The fragment contains the statement of the 5th proposition of book 2. Introductory david joyces introduction to book i heath on postulates heath on axioms and common notions. He began book vii of his elements by defining a number as a multitude composed of units.
This is the second proposition in euclids second book of the elements. Stoicheia is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c. In obtuseangled triangles bac the square on the side opposite the obtuse angle bc is greater than the sum of the squares on the sides containing the obtuse angle ab and ac by twice the rectangle contained by one of the sides about the obtuse angle ac, namely that on which the perpendicular falls, and the stra. To place at a given point as an extremity a straight line equal to a given straight line. This is the second proposition in euclid s first book of the elements. Euclidis elements, by far his most famous and important work.
Second, euclid gave a version of what is known as the unique factorization theorem or the. Heath, 1908, on to place at a given point as an extremity a straight line equal to a given straight line. The original rectangle ah is the sum of the rectangles al and ch. 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. May 12, 2014 euclid s elements book 3 proposition 35 duration.
Is the proof of proposition 2 in book 1 of euclid s elements a bit redundant. Download it once and read it on your kindle device, pc, phones or tablets. I do not see anywhere in the list of definitions, common notions, or postulates that allows for this assumption. This proposition starts with a line that is randomly cut. The first, proposition 2 of book vii, is a procedure for finding the greatest common divisor of two whole numbers. It appears that euclid devised this proof so that the proposition could be placed in book i.
If a straight line is cut at random, then the sum of the rectangles contained by the whole and each of the segments equals the square on the whole. The national science foundation provided support for entering this text. The remaining four propositions are of a slightly different nature. A fter stating the first principles, we began with the construction of an equilateral triangle. The ideas of application of areas, quadrature, and proportion go back to the pythagoreans, but euclid does not present eudoxus theory of proportion until book v, and the geometry depending on it is not presented until book vi. This is the fourth proposition in euclid s second book of the elements. Start studying euclid s elements book 2 propositions. Although many of euclid s results had been stated by earlier mathematicians, euclid was the first to show.
The theorem that bears his name is about an equality of noncongruent areas. Euclid presents a proof based on proportion and similarity in the lemma for proposition x. 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 perseus collection of greek classics. Purchase a copy of this text not necessarily the same edition from.
There is something like motion used in proposition i. For it was proved in the first theorem of the tenth book that, if two unequal. This fundamental result is now called the euclidean algorithm in his honour. A slight modification gives a factorization of the difference of two squares. Up until this proposition, euclid has only used cutandpaste proofs, and such a proof can be made for this proposition as well. P ythagoras was a teacher and philosopher who lived some 250 years before euclid, in the 6th century b. Use features like bookmarks, note taking and highlighting while reading the thirteen books of the elements, vol. Euclid then builds new constructions such as the one in this proposition out of previously described constructions. Given two unequal straight lines, to cut off from the longer line. Byrnes treatment reflects this, since he modifies euclid s treatment quite a bit. The incremental deductive chain of definitions, common notions, constructions. 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.
According to joyce commentary, proposition 2 is only used in proposition 3 of euclid s elements, book i. Let a be the given point, and bc the given straight line. Circles are to one another as the squares on the diameters. 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. Euclid, elements, book i, proposition 2 heath, 1908. This proposition is used in book x to prove a lemma for x. Euclid s method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Leon and theudius also wrote versions before euclid fl. The works of archimedes dover books on mathematics archimedes. Euclid s 2nd proposition draws a line at point a equal in length to a line bc. It focuses on how to construct a line at a given point equal to a given line. If a straight line be cut at random, the rectangle contained by the whole and both of the segments is equal to the square on the whole for let the straight line ab be cut at random at the point c. Euclid, elements, book i, proposition 1 heath, 1908. In any triangle, the angle opposite the greater side is greater.
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. This is the first proposition in euclids second book of the elements. More recent scholarship suggests a date of 75125 ad. He later defined a prime as a number measured by a unit alone i. It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. If any number of magnitudes be equimultiples of as many others, each of each. How to construct a square, equal in area to a given polygon. The books cover plane and solid euclidean geometry. Proposition, angles formed by a straight line euclid s elements book 1. Begin by reading the statement of proposition 2, book iv, and the definition of segment of a circle given in book iii. If there be two straight lines, and one of them be cut into any number of segments whatever, the. Is the proof of proposition 2 in book 1 of euclids. Proposition 12, constructing a perpendicular line 2 euclid s elements book 1.
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. And, of course, the rectangles al and cm are equal. It uses proposition 1 and is used by proposition 3. I say that the rectangle contained by ab, bc together with the rectangle contained by ba, ac is equal to the square on ab. Feb 23, 2018 euclids 2nd proposition draws a line at point a equal in length to a line bc. In this proposition, there are just two of those lines and their sum equals the one line. By contrast, euclid presented number theory without the flourishes. Proposition 11, constructing a perpendicular line euclid s elements book 1.
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