From the point a to the point b let the straight line ab be joined. In terms of the single variable x, this is equivalent to solving the quadratic equation, x b x c2. If two angles of a triangle are equal, then the sides opposite them will be equal. The purpose of this project is to facilitate research and study of the dense information of the books through intelligent navigation, robust taxonomies and comprehensive search. This is the sixth proposition in euclids second book of the elements. According to joyce commentary, proposition 2 is only used in proposition 3 of euclid s elements, book i. Thus, the remaining condition reduces to finding cd so that b 2 2 cd 2 c 2. This is the sixth proposition in euclid s second book of the elements.
Let p be the product of all the prime numbers in the list. With links to the complete edition of euclid with pictures in java by david joyce, and the well known. Geometor euclid restructuring euclids elements into a. If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally. To cut a given straight line so that the rectangle contained by the whole and one of the. 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. In terms of x alone, this is equivalent to solving the quadratic equation xx b c 2. Proposition 8 sidesideside if two triangles have two sides equal to two sides respectively, and if the bases are also equal, then the angles will be equal that are contained by the two equal sides. Euclids elements of geometry, book 12, proposition 2, joseph mallord william turner, c. 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. Classic edition, with extensive commentary, in 3 vols. If a straight line is bisected and some straightline is added to it on a straightone, the rectangle enclosed by the whole with the added line and the added line with the square from the half line is equal to the square from the line composed from the half and the added line. Now it is clear that the purpose of proposition 2 is to effect the construction in this proposition.
Thus it is required to place at the point a as an extremity a straight line equal to the given straight line bc. In right angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. Cut a line parallel to the base of a triangle, and the cut sides will be proportional. To place at a given point as an extremity a straight. If a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome. Given a triangle and a circle, create an equiangular triangle in the circle. For it was proved in the first theorem of the tenth book that, if two unequal magnitudes be set out, and if from the greater there be subtracted a magnitude greater than the half, and from that which is left a greater than the half, and if this be done continually, there will be left some magnitude which will be less than the lesser magnitude set out. Find two numbers x and y so that their difference x y is a known value b and their product is a known value c 2. Euclid proves this statement, which relies on propositions vi.
It appears that euclid devised this proof so that the proposition could be placed in book i. Book ii main euclid page book iv book iii byrnes edition page by page 71 7273 7475 7677 7879 8081 8283 8485 8687 8889 9091 9293 9495 9697 9899 100101 102103 104105 106107 108109 110111 1121 114115 116117 118119 120121 122 proposition by proposition with links to the complete edition of euclid with pictures in java by david joyce, and the well known comments. 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. Leon and theudius also wrote versions before euclid fl. Book iv main euclid page book vi book v byrnes edition page by page. Thus, the original rectangle equals the square eh 2. This is the second proposition in euclid s first book of the elements.
It will be shown that at least one additional prime number not in this list exists. I say that the sum of the rectangle ba by ac and the rectangle ab by bc equals the square on ab. 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. Using the postulates and common notions, euclid, with an ingenious construction in proposition 2, soon verifies the important sideangleside congruence relation proposition 4.
This proposition is set up to help in the solution of a quadratic problem of the following form. Note that gf equals gh, the hypotenuse of a right triangle ghe. 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. Euclid, who put together the elements, collecting many of eudoxus theorems, perfecting many of theaetetus, and also bringing to. Let abc be a rightangled triangle having the angle bac right.
Triangles and parallelograms which are under the same height are to one another as their bases. I say that the figure on bc is equal to the similar and similarly described figures on ba. Let the straight line ab be cut at random at the point c. Proposition four in book vi of the elements 1 states. If a straight line be drawn parallel to one of the sides of a triangle, it will cut the sides of the triangle proportionally. Euclid offered a proof published in his work elements book ix, proposition 20, which is paraphrased here. For it was proved in the first theorem of the tenth book that, if two unequal magnitudes be set out, and if from the greater there be subtracted a magnitude greater than the half, and from that which is left a greater than the half, and if this be. This proposition says that the product xy equals the square on bc which is b 2 minus the square on cd.
The goal of the proof is to show that the rectangle contained by the whole with the added portion and the added line segment add together with. Let ab be a rational straight line cut in extreme and mean ratio at c, and let ac be the greater segment. Consider any finite list of prime numbers p 1, p 2. Proposition 2 if a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally. This proposition is used in book i for the proofs of several propositions starting with i. Scholars believe that the elements is largely a compilation of propositions based on books by earlier greek mathematicians proclus 412485 ad, a greek mathematician who lived around seven centuries after euclid, wrote in his commentary on the elements. A reproduction of oliver byrnes celebrated work from 1847 plus interactive diagrams, cross references, and posters designed by nicholas rougeux. Book 1 definitions book 1 postulates book 1 common notions book 1 proposition 1.
There is something like motion used in proposition i. It is a collection of definitions, postulates, axioms, 467 propositions theorems and constructions, and mathematical proofs of the propositions. The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed. Let two spheres be conceived about the same centre a. The only basic constructions that euclid allows are those described in postulates 1, 2, and 3. Book i, propositions 9,10,15,16,27, and proposition 29 through pg. Given two spheres about the same centre, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface. If any number of magnitudes be equimultiples of as many others, each of each.
The incremental deductive chain of definitions, common notions, constructions. Euclid book v university of british columbia department. 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. Find two numbers x and y so that their sum is a known value b and their product is a known value c2. Only these two propositions directly use the definition of proportion in book v.
Let a be the given point, and bc the given straight line. If some straightline is drawn parallel to one of the sides of a triangle it will cut the sides of the triangle proportionally. Euclids elements, book vi, proposition 2 proposition 2 if a straight line is drawn parallel to one of the sides of a triangle, then it cuts the sides of the triangle proportionally. Begin by reading the statement of proposition 2, book iv, and the definition of segment of a circle given in book iii. Euclid presents a proof based on proportion and similarity in the lemma for proposition x. 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. Introduction main euclid page book ii book i byrnes edition page by page 1 2 3 45 6 7 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 elements is the oldest mathematical and geometric treatise consisting of books written by euclid in alexandria c. If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles will. Euclid prefers to prove a pair of converses in two stages, but in some propositions, as this one, the proofs in the two stages are almost inverses of each other, so both could be proved at once. The ratio of areas of two triangles of equal height is the same as the ratio of their bases. According to this proposition the rectangle ad by db, which is the product xy, is the difference of two squares, the large one being the square on the line cd, that is the square of x b 2, and the small one being the square on the line cb, that is, the square of b 2. To place at a given point as an extremity a straight line equal to a given straight line. If a straight line be drawn parallel to one of the.
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