Let a be the given point, and bc the given straight line. 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. 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. 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.
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. Euclids elements of geometry university of texas at austin. Definitions 1 4 axioms 1 3 proposition 1 proposition 2 proposition 3 proposition 1 proposition 2 proposition 3 definition 5 proposition 4. T he logical theory of plane geometry consists of first principles followed by propositions, of which there are two kinds.
Then, since ke equals kh, and the angle ekh is right, therefore the square on he is double the square on ek. Euclid says that the angle cbe equals the sum of the two angles cba and abe. Euclid s elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. It is a collection of definitions, postulates, propositions theorems and constructions, and mathematical proofs of the propositions. This proof focuses more on the fact that straight lines are made up of 2 right angles. Definition 2 a number is a multitude composed of units. Definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. 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. The theorem that bears his name is about an equality of noncongruent areas. But the square on lm was also proved double the square on le. 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. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions. 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.
To construct a rectangle equal to a given rectilineal figure. The verification that this construction works is also short with the help of proposition ii. These does not that directly guarantee the existence of that point d you propose. These are sketches illustrating the initial propositions argued in book 1 of euclids elements. Therefore the parallelogram ab equals the parallelogram bc. An ambient plane is necessary to talk about the sides of the line ab. 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 above proposition is known by most brethren as the pythagorean proposition. Mar 28, 2017 euclid s elements book 1 proposition 14 duration. 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. If a, b, c, and d do not lie in a plane, then cbd cannot be a straight line. I suspect that at this point all you can use in your proof is the postulates 1 5 and proposition 1. Byrnes treatment reflects this, since he modifies euclid s treatment quite a bit. For this reason we separate it from the traditional text. Euclid also says that the sum of the angles cbe and ebd equals the sum of the three angles cba, abe, and ebd. 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 procedure, selected to. Book 1 outlines the fundamental propositions of plane geometry, includ. The proof is often thought to originate among the pythagoreans, though i dont know of any evidence for.
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. Therefore, in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional. The proof is an interpolation at the end of book 10, i. Learn this proposition with interactive stepbystep here. Is the proof of proposition 2 in book 1 of euclids.
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. Proposition 20, side lengths in a triangle duration. If bd is not in a straight line with bc, then produce be in a straight line with cb. Euclid is likely to have gained his mathematical training in athens, from pupils of plato. Euclidean geometry is a mathematical system attributed to alexandrian greek mathematician euclid, which he described in his textbook on geometry. Project gutenbergs first six books of the elements of. 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. The visual constructions of euclid book ii 91 to construct a square equal to a given rectilineal figure. Euclid, elements, book i, proposition 14 heath, 1908. But unfortunately the one he has chosen is the one that least needs proof. This proof focuses on the fact that vertical angles are equal to each other. But ab is to fe as db is to be, and bc is to fe as bg is to bf.
Euclids propositions are ordered in such a way that each proposition is only used by future propositions and never by any previous ones. Feb 24, 2018 proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. This proof focuses more on the fact that straight lines are made up of 2. 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. Let the number a be the least that is measured by the prime numbers b, c, and d. If two triangles have two sides respectively equal to. To position at the given point a straightline equal to the given line. Tex start of the project gutenberg ebook elements of euclid.
Definition 4 but parts when it does not measure it. We also know that it is clearly represented in our past masters jewel. 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. 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. Since the straight line ab stands on the straight line cbe. That sum being mentioned is a straight angle, which is not to be considered as an angle according to euclid. 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. 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. Therefore the remainder, the pyramid with the polygonal. According to joyce commentary, proposition 2 is only used in proposition 3 of euclid s elements, book i. Byrnes treatment reflects this, since he modifies euclids treatment quite a bit. Stoicheia is a mathematical treatise consisting of books attributed to the ancient greek mathematician euclid in alexandria, ptolemaic egypt c.
This is the fifteenth proposition in euclids first book of the elements. If two straightlines, not lying on the same side, make. Euclids elements is by far the most famous mathematical work of classical antiquity, and also has the distinction. Published on mar 28, 2017 this is the fourteenth proposition in euclids first book of the elements. Prop 3 is in turn used by many other propositions through the entire work. Fundamentals of number theory definitions definition 1 a unit is that by virtue of which each of the things that exist is called one. This is the fourteenth proposition in euclids first book of the elements. The visual constructions of euclid book i 63 through a given point to draw a straight line parallel to a given straight line. Let ab and cd be equal straight lines in a circle abdc. Therefore the square on ab equals the square on lm. To construct a square equal to a given rectilinear figure. In appendix a, there is a chart of all the propositions from book i that illustrates this. Euclids method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these. P ythagoras was a teacher and philosopher who lived some 250 years before euclid, in the 6th century b.
The books cover plane and solid euclidean geometry. 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. 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. Euclid, elements, book i, proposition 15 heath, 1908.
This historic book may have numerous typos and missing text. With two given, unequal straightlines to take away from the larger a straightline equal to the smaller. Proposition 3 looks simple, but it uses proposition 2 which uses proposition 1. On the given straight finite straightline to construct an equilateral triangle. 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.
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. For the hypotheses of this proposition, the algorithm stops when a remainder of 1 occurs. 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. Here euclid has contented himself, as he often does, with proving one case only. 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. I suspect that at this point all you can use in your proof is the postulates 15 and proposition 1. Therefore in the parallelograms ab and bc the sides about the equal angles are reciprocally proportional. The first six books of the elements of euclid subtitle. Euclids elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. Definition 3 a number is a part of a number, the less of the greater, when it measures the greater. Although euclid does not include a sidesideangle congruence theorem, he does have a sidesideangle similarity theorem, namely proposition vi.
Guide with this proposition, we begin to see what the arithmetic of magnitudes means to euclid, in particular, how to add angles. And the square on db equals the square on le, for eh was made equal to db. Although many of euclids results had been stated by earlier mathematicians, euclid was the first to show. 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. In the hundred fifteenth proposition, proposition 16, book iv, he shows that it is possible to inscribe a regular 15gon in a circle. 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. 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. Book v is one of the most difficult in all of the elements.
110 108 344 198 1493 998 1406 665 778 302 366 725 519 799 162 24 624 373 478 231 333 435 663 474 329 34 1319 1192 1250 111 639 486