Euclid s elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. I felt a bit lost when first approaching the elements, but this book is helping me to get started properly, for full digestion of the material. A lune also called a crescent is a region of nonoverlap of two intersecting circles. It uses proposition 1 and is used by proposition 3.
Euclid s 2nd proposition draws a line at point a equal in length to a line bc. Hippocrates quadrature of lunes proclus says that this proposition is euclids own, and the proof may be his, but the result, if not the proof, was known long before euclid, at least in the time of hippocrates a century before euclid. Triangles and parallelograms which are under the same height are to one another as their bases. A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. Now it is clear that the purpose of proposition 2 is to effect the construction in this proposition. Use of proposition 31 this construction is frequently used in the remainder of book i starting with the next proposition. Begin by reading the statement of proposition 2, book iv, and the definition of segment of a circle given in book iii. Euclid s elements is one of the most beautiful books in western thought. Definitions from book xi david joyces euclid heaths comments on definition 1 definition 2. Euclids elements, book vi, proposition 31 proposition 31 in rightangled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.
If two circles cut touch one another, they will not have the same center. The theory of the circle in book iii of euclids elements. If two straight lines are parallel, then a straight line that meets them makes the alternate angles equal, it makes the exterior angle equal to the opposite interior angle on the same side, and it makes the interior angles on the same side equal to two right angles. For euclid, an angle is formed by two rays which are not part of the same line see book i definition 8. Proposition 28 part 2, parallel lines 3 euclid s elements book 1.
He began book vii of his elements by defining a number as a multitude composed of units. Book x main euclid page book xii book xi with pictures in java by david joyce, and the well known comments from heaths edition at the perseus collection of greek classics. Let a be the given point, and bc the given straight line. The books cover plane and solid euclidean geometry. Book v main euclid page book vii book vi byrnes edition page by page 211 2122 214215 216217 218219 220221 222223 224225 226227 228229 230231 232233 234235 236237 238239 240241 242243 244245 246247 248249 250251 252253 254255 256257 258259 260261 262263 264265 266267 268 proposition by proposition with links to the complete edition of euclid with pictures. To place a straight line equal to a given straight line with one end at a given point. Therefore the straight line eaf has been drawn through the given point a parallel to the given straight line bc. Definitions heath, 1908 postulates heath, 1908 axioms heath, 1908 proposition 1 heath, 1908. The next two propositions depend on the fundamental theorems of parallel lines. This is the thirty first proposition in euclids first book of the elements. Apr 09, 2017 this is the thirty first proposition in euclid s first book of the elements. To construct an equilateral triangle on a given finite straight line. From a given point to draw a straight line equal to a given straight line. Proposition 31 is a generalization of the pythagorean theorem of book i.
A reproduction of oliver byrnes celebrated work from 1847 plus interactive diagrams, cross references, and posters designed by nicholas rougeux. Proposition 31, constructing parallel lines duration. For let ce be drawn through the point c parallel to the straight line ab. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions. On a given straight line to construct an equilateral triangle. In any triangle, the angle opposite the greater side is greater. Euclid presents a proof based on proportion and similarity in the lemma for proposition x. Feb 23, 2018 euclids 2nd proposition draws a line at point a equal in length to a line bc.
Since the straight line ad falling on the two straight lines bc and ef makes the alternate angles ead and adc equal to one another, therefore eaf is parallel to. In a triangle, if 2 lines drawn from the extremities of one side meet inside the triangle, the lines will be shorter but the angle will be bigger than any in the triangle. 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. 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 opposite angles of quadrilaterals in circles are equal to two right angles. For the proposition, scroll to the bottom of this post. Hippocrates quadrature of lunes proclus says that this proposition is euclid s own, and the proof may be his, but the result, if not the proof, was known long before euclid, at least in the time of hippocrates. This construction proof shows how to build a line through a given point. To cut off from the greater of two given unequal straight lines a straight line equal to the less. This has nice questions and tips not found anywhere else. Click anywhere in the line to jump to another position.
Any composite number is measured by some prime number. 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. It appears that euclid devised this proof so that the proposition could be placed in book i. Hide browse bar your current position in the text is marked in blue. Proclus says that this proposition is euclids own, and the proof may be his.
Euclid s assumptions about the geometry of the plane are remarkably weak from our modern point of view. Since the straight line ad falling on the two straight lines bc and ef makes the alternate angles ead and adc equal to one another, therefore eaf is parallel to bc. Euclidean geometry is a mathematical system attributed to alexandrian greek mathematician euclid, which he described in his textbook on geometry. Proposition 30, relationship between parallel lines euclid s elements book 1. This construction proof shows how to build a line through a given point that is parallel to a given line. To place a straightline equal to a given straightline.
Proposition 28 part 1, parallel lines 2 euclid s elements book 1. To draw a straight line through a given point parallel to a given straight line. Euclid, book iii, proposition 2 proposition 2 of book iii of euclid s elements shows that any straight line joining two points on the circumference of a circle falls within the circle. The elements book iii euclid begins with the basics. It is also frequently used in books ii, iv, vi, xi, xii, and xiii. Euclids elements book one with questions for discussion. Each proposition falls out of the last in perfect logical progression. Stoicheia is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c. If on the circumference of a circle two points be take at random, the straight line joining the points will fall within the circle.
Proposition 29, parallel lines converse euclid s elements book 1. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. By contrast, euclid presented number theory without the flourishes. In the first proposition, proposition 1, book i, euclid shows that, using only the postulates. He later defined a prime as a number measured by a unit alone i. Proposition 22 to construct a triangle given by three unequal lines. If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally. By appealing to the impossibility of an infinite regress of natural numbers, his demonstration takes the form of a reductio ad absurdum. Use of proposition 32 although this proposition isnt used in the rest of book i, it is frequently used in the rest of the books on geometry, namely books ii, iii, iv, vi, xi, xii, and xiii. Feb 26, 2017 euclid s elements book 1 mathematicsonline. The theory of the circle in book iii of euclids elements of. Euclid s method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions theorems from these.
This is a very useful guide for getting started with euclid s elements. Euclid s compass could not do this or was not assumed to be able to do this. Oct 06, 2015 in book vii of his elements euclid sets forth the following. The part of this proposition which says that an angle inscribed in a semicircle is a right angle is often called thales theorem. Describe the square adeb on ab, and draw cf through c parallel to.
It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. It is required to place a straight line equal to the given straight line bc with one end at the point a. Introduction main euclid page book ii book i byrnes edition page by page 1 2 3 45 67 89 1011 12 1415 1617 1819 2021 2223 2425 2627 2829 30 31 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. Introductory david joyces introduction to book iii. T he next two propositions depend on the fundamental theorems of parallel lines. Let the straight line ab be cut at random at the point c. The national science foundation provided support for entering this text.
Definitions from book iii byrnes edition definitions 1, 2. Leon and theudius also wrote versions before euclid fl. The statements and proofs of this proposition in heaths edition and caseys edition differ, though the proofs are related. The sum of the exterior angles of any convex rectilinear figure together equal four right angles. Given two unequal straight lines, to cut off from the longer line. Now ae is the square on ab af is the rectangle contained by ba, ac, for it is contained by da, ac, and ad is equal to ab. Purchase a copy of this text not necessarily the same edition from. The proof youve just read shows that it was safe to pretend that the compass could do this, because you could imitate it via this proof any time you needed to. So, to euclid, a straight angle is not an angle at all, and so proposition 31 is not a special case of proposition 20 since proposition 20 only applies when you have an angle at the center.
Let abc be a rightangled triangle having the angle bac right. Introduction to the works of euclid melissa joan hart. 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. In this proposition, there are just two of those lines and their sum equals the one line. To a given straight line to apply a parallelogram equal to a given rectilineal figure and deficient by a parallelogrammic figure similar to a given one. Is the proof of proposition 2 in book 1 of euclids. The parallel line ef constructed in this proposition is the only one passing through the point a. The visual constructions of euclid book i 63 through a given point to draw a straight line parallel to a given straight line. Book 2 proposition in an acute angled traingle, the square on the length opposite of the acute angle is less than the sum of the squares of the other two lengths by the rectangle made by one of the lengths and the cut segment making it right. 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. If there were another, then the interior angles on one side or the other of ad it makes with bc would be less than two right angles, and therefore by the parallel postulate post. It is required to draw a straight line through the point a parallel to the straight line bc.
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