Euclids elements of geometry university of texas at austin. Proposition 14 equal straight lines in a circle are equally distant from the center, and those which are equally distant from the center equal one another. Byrnes treatment reflects this, since he modifies euclids treatment quite a bit. The proof is an interpolation at the end of book 10, i. If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another. Euclid s elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. Euclid is likely to have gained his mathematical training in athens, from pupils of plato. This historic book may have numerous typos and missing text.
Euclid says that the angle cbe equals the sum of the two angles cba and abe. But ab is to fe as db is to be, and bc is to fe as bg is to bf. The qualifying sentence, similarly we can prove that neither is any other straight line except bd, is meant to take care of the cases when e does not. If the sum of the angles between three straight lines sum up to 180 degrees, then the outer two lines form a single straight line.
This proof focuses more on the fact that straight lines are made up of 2. If two triangles have two sides respectively equal to. This proof focuses on the fact that vertical angles are equal to each other. Fundamentals of number theory definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. These does not that directly guarantee the existence of that point d you propose. 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. Mar 28, 2017 euclid s elements book 1 proposition 14 duration. The books cover plane and solid euclidean geometry. The above proposition is known by most brethren as the pythagorean proposition.
Book 1 proposition 14 at the end of a straight line, if two straight lines are attached to the first one and not lying on the same side, and the sum of the angles add to two right angles, then those two lines are straight with each other. For this reason we separate it from the traditional text. Book 1 outlines the fundamental propositions of plane geometry, includ. Although many of euclids results had been stated by earlier mathematicians, euclid was the first to show. 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. Definitions 1 4 axioms 1 3 proposition 1 proposition 2 proposition 3 proposition 1 proposition 2 proposition 3 definition 5 proposition 4. 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. For the hypotheses of this proposition, the algorithm stops when a remainder of 1 occurs.
I suspect that at this point all you can use in your proof is the postulates 15 and proposition 1. Project gutenbergs first six books of the elements of. Therefore the square on ab equals the square on lm. The verification that this construction works is also short with the help of proposition ii. These are sketches illustrating the initial propositions argued in book 1 of euclids elements. In the first proposition, proposition 1, book i, euclid shows that, using only the postulates and common notions, it is possible to construct an equilateral triangle on a given straight line. Proposition 20, side lengths in a triangle duration. Prop 3 is in turn used by many other propositions through the entire work.
Project euclid presents euclids elements, book 1, proposition 14 if with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles. That sum being mentioned is a straight angle, which is not to be considered as an angle according to euclid. Therefore the remainder, the pyramid with the polygonal. Then since the parallelogram ab equals the parallelogram bc, and fe is another parallelogram, therefore ab is to fe as bc is to fe v. To position at the given point a straightline equal to the given line. Cut off kl and km from the straight lines kl and km respectively equal to one of the straight lines ek, fk, gk, or hk, and join le, lf, lg, lh, me, mf, mg, and mh i. To construct a square equal to a given rectilinear figure. Mar 01, 2014 if the sum of the angles between three straight lines sum up to 180 degrees, then the outer two lines form a single straight line. Then, since ke equals kh, and the angle ekh is right, therefore the square on he is double the square on ek.
Byrnes treatment reflects this, since he modifies euclid s treatment quite a bit. Let ab and cd be equal straight lines in a circle abdc. The next stage repeatedly subtracts a 3 from a 2 leaving a remainder a 4 cg. On the given straight finite straightline to construct an equilateral triangle. 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. 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. Euclids elements, book i, proposition 14 proposition 14 if with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. This is the fourteenth proposition in euclids first book of the elements. The point d is in fact guaranteed by proposition 1 that says that given a line ab which is guaranteed by postulate 1 there is a equalateral triangle abd. The visual constructions of euclid book ii 91 to construct a square equal to a given rectilineal figure. In the hundred fifteenth proposition, proposition 16, book iv, he shows that it is possible to inscribe a regular 15gon in a circle. And the square on db equals the square on le, for eh was made equal to db.
It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions. Euclid, elements, book i, proposition 14 heath, 1908. 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. Euclid, elements, book i, proposition 15 heath, 1908. If bd is not in a straight line with bc, then produce be in a straight line with cb. Euclids elements is by far the most famous mathematical work of classical antiquity, and also has the distinction. This is the fifteenth proposition in euclids first book of the elements. Euclids propositions are ordered in such a way that each proposition is only used by future propositions and never by any previous ones. Therefore in the parallelograms ab and bc the sides about the equal angles are reciprocally proportional. Stoicheia is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c.
Definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. 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. 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. Since the straight line ab stands on the straight line cbe. If two straight lines cut one another, they make the vertical angles equal to one another. Definition 2 a number is a multitude composed of units. With two given, unequal straightlines to take away from the larger a straightline equal to the smaller. But ab is to bc as the square on ab is to the square on bd, therefore the square on ab is double the square on bd. Definition 3 a number is a part of a number, the less of the greater, when it measures the greater. If a, b, c, and d do not lie in a plane, then cbd cannot be a straight line. The first six books of the elements of euclid subtitle. Proposition 14 if with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. If two straightlines, not lying on the same side, make.
The proof is often thought to originate among the pythagoreans, though i dont know of any evidence for. The theorem that bears his name is about an equality of noncongruent areas. Euclids method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. Euclidean geometry is a mathematical system attributed to alexandrian greek mathematician euclid, which he described in his textbook on geometry. Although euclid does not include a sidesideangle congruence theorem, he does have a sidesideangle similarity theorem, namely proposition vi. 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. It is datable to the fourth century, though, as aristotle seems to make an allusion to the proof in prior analytics book 1, though i forget the reference.
An ambient plane is necessary to talk about the sides of the line ab. Definition 4 but parts when it does not measure it. In appendix a, there is a chart of all the propositions from book i that illustrates this. Let the number a be the least that is measured by the prime numbers b, c, and d.
T he logical theory of plane geometry consists of first principles followed by propositions, of which there are two kinds. 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. Guide with this proposition, we begin to see what the arithmetic of magnitudes means to euclid, in particular, how to add angles. Is the proof of proposition 2 in book 1 of euclids.
It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. But unfortunately the one he has chosen is the one that least needs proof. The visual constructions of euclid book i 63 through a given point to draw a straight line parallel to a given straight line. I suspect that at this point all you can use in your proof is the postulates 1 5 and proposition 1.
Learn this proposition with interactive stepbystep here. To construct a rectangle equal to a given rectilineal figure. Proposition 14 if a number is the least that is measured by prime numbers, then it is not measured by any other prime number except those originally measuring it. Published on mar 28, 2017 this is the fourteenth proposition in euclids first book of the elements. Feb 24, 2018 proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. P ythagoras was a teacher and philosopher who lived some 250 years before euclid, in the 6th century b. Tex start of the project gutenberg ebook elements of euclid. 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. So, one way a sum of angles occurs is when the two angles have a common vertex b in this case and a common side ba in this case, and the angles lie on opposite sides of their common side. Interpreting euclids axioms in the spirit of this more modern approach, axioms 1 4 are consistent with either infinite or finite space as in elliptic geometry, and all five axioms are consistent with a variety of topologies e. Here euclid has contented himself, as he often does, with proving one case only. Euclid also says that the sum of the angles cbe and ebd equals the sum of the three angles cba, abe, and ebd. Book v is one of the most difficult in all of the elements. We also know that it is clearly represented in our past masters jewel.
Let a be the given point, and bc the given straight line. But the square on lm was also proved double the square on le. Euclid does not include any form of a sidesideangle congruence theorem, but he does prove one special case, sidesideright angle, in the course of the proof of proposition iii. This proof focuses more on the fact that straight lines are made up of 2 right angles. Therefore, in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional. According to joyce commentary, proposition 2 is only used in proposition 3 of euclid s elements, book i.
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