Thanks for contributing an answer to mathematics stack exchange. 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 statement of this proposition includes three parts, one the converse of i. 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. Definition 2 a number is a multitude composed of units.
Book 1 proposition 28 if a straight line falls though two straight lines, making the exterior and interioropposite angles equal, or making the sum of the interior angles on. Definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. 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. When teaching my students this, i do teach them congruent angle construction with straight edge and. Euclids elements of geometry, book 11, propositions 1 and 3. Euclids definitions, postulates, and the first 30 propositions of book i. I dont understand a statement in euclids proof of prop. Next, since the sum of the angles bgh and ghd equals two right angles. It uses proposition 1 and is used by proposition 3. This proof focuses more on the properties of parallel. Use of proposition 28 this proposition is used in iv. Hide browse bar your current position in the text is marked in blue.
If a straight line crosses two other straight lines, and the exterior to opposite angles are equal, or the sum of the interior angles equals 180. Index introduction definitions axioms and postulates propositions other. On a given finite straight line to construct an equilateral triangle. A plane angle is the inclination to one another of two. But the angle abe was proved equal to the angle bah. This is the first part of the twenty eighth proposition in euclids first book of the elements. If a straight line falling on two straight lines makes the exterior angle equal to the interior and. Describe ebfg similar and similarly situated to d on eb, and complete the parallelogram ag i. But avoid asking for help, clarification, or responding to other answers. 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.
And they are alternate, therefore ab is parallel to cd therefore if a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight. This proposition is also used in the next one and in i. Euclids elements, book iii clay mathematics institute. Prop 3 is in turn used by many other propositions through the entire work. Euclids method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Click anywhere in the line to jump to another position. Actually, the final sentence is not part of the lemma, probably because euclid moved that statement to the first book as i. One part of a right line cannot be in a plane superficies, and another part above it, while the right corresponds to proposition 3.
Therefore the remaining angle agh equals the remaining angle ghd. And they are alternate, therefore ab is parallel to cd. Project gutenbergs first six books of the elements of. 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. Book 1 proposition 28 if a straight line falls though two straight lines, making the exterior and interioropposite angles equal, or making the sum of the interior angles on the same side two right angles, they are parallel. A straight line is a line which lies evenly with the points on itself.
If then ag equals c, that which was proposed is done, for the parallelogram ag equal to the given rectilinear figure c has been applied to the given straight line ab but falling short by a parallelogram gb similar to d. For if we start at angle 1 and go around, those angles alternate. The visual constructions of euclid book i 63 through a given point to draw a straight line parallel to a given straight line. On the given straight finite straightline to construct an equilateral triangle.
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. Euclid s theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. Feb 24, 2018 proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. We now begin the second part of euclids first book. See introduction, royal academy perspective lectures. 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 28 29 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.
Note that euclid does not consider two other possible ways that the two lines could meet. The books cover plane and solid euclidean geometry. After having read the first book of the elements, the student will find no difficulty in proving that the triangles c f e and c d f are equilateral. If two straight lines are parallel, then a straight line that meets them makes the alternate angles equal, it makes the exterior angle. For example, proposition 16 says in any triangle, if one of the sides be extended, the exterior angle is greater than either of the interior and opposite. The proposition states that if two numbers are relatively prime, then their powers are also relatively prime. 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. Euclids theorem is a special case of dirichlets theorem for a d 1. This proposition states two useful minor variants of the previous proposition. Given two unequal straight lines, to cut off from the greater a straight line equal to the less. With two given, unequal straightlines to take away from the larger a straightline equal to the smaller. T he next two propositions depend on the fundamental theorems of parallel lines. Sketchbook, diagrams and related material circa 180928. Euclids elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c.
And, since the straight line ba equals ae, therefore the angle abe also equals the angle aeb. Every case of dirichlets theorem yields euclids theorem. Although many of euclids results had been stated by earlier mathematicians, euclid was the first to show. 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. This is the second part of the twenty eighth proposition in euclids first book of the elements.
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. If a straight line falling on two straight lines makes the exterior angle equal to the interior and opposite angle on the same side, or the sum of the interior angles on the same side equal to two right angles, then the straight lines are parallel to one another. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions. If then ag equals c, that which was proposed is done, for the parallelogram ag equal to the given rectilinear figure c has been applied to the given straight line ab but falling short by a parallelogram gb similar to d but, if not, let he be greater than c. Hence the straight line he also equals ea, that is, ab. If a straight line falls on two straight lines, then if the alternate angles are not equal, then the straight lines meet on a certain side of the line. Stoicheia is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c. Definition 3 a number is a part of a number, the less of the greater, when it measures the greater. Euclid s elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c.
W e now begin the second part of euclid s first book. If two triangles have two sides respectively equal to. I do not see anywhere in the list of definitions, common notions, or postulates that allows for this assumption. In any triangle, if one of the sides be produced, the exterior angle is greater than either of the. To place at a given point as an extremity a straight line equal to a given straight line. For the next 27 proposition, we do not need the 5th axiom of euclid, nor any continuity axioms, except for proposition 22, which needs circlecircle intersection axiom. Explicitly, it only says that their squares are relatively prime, and their cubes are relatively prime, but the way it is used in viii. Euclid, book i, proposition 27 heaths edition if a straight line falling on two straight lines make the alternate angles. Although this is the first proposition about parallel lines, it does not require the parallel postulate post. Let a be the given point, and bc the given straight line. About logical converses, contrapositives, and inverses, although this is the first proposition about parallel lines, it does not require the parallel postulate post. But the angle abe was proved equal to the angle bah, therefore the angle bea also equals the angle bah. In appendix a, there is a chart of all the propositions from book i that illustrates this.
In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of. Euclidean geometry is a mathematical system attributed to alexandrian greek mathematician euclid, which he described in his textbook on geometry. To position at the given point a straightline equal to the given line. It was first proved by euclid in his work elements. Like those propositions, this one assumes an ambient plane containing all the three lines. 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. Euclid s method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Euclids propositions are ordered in such a way that each proposition is only used by future propositions and never by any previous ones. This is the first proposition which depends on the parallel postulate. Feb 23, 2018 euclids 2nd proposition draws a line at point a equal in length to a line bc. Euclids elements all thirteen books complete in one volume, based on heaths translation, green lion press isbn 1888009187. Thus, bisecting the circumferences which are left, joining straight lines, setting up on each of the triangles pyramids of equal height with the cone, and doing this repeatedly, we shall leave some segments of the cone which are less than the solid x let such be left, and let them be the segments on hp, pe, eq, qf, fr, rg, gs, and sh. If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.
The three statements differ only in their hypotheses which are easily seen to be equivalent with the help of proposition i. Therefore the remainder, the pyramid with the polygonal. It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. Therefore the angle dfg is greater than the angle egf. The diagrams in the present section are based on plates in samuel cunns euclids elements of geometry london 1759. The theory of parallels in book i of euclids elements of geometry.