For Euclid ’ s method to succeed, the starting lengths must satisfy two requirements: ( i ) the lengths must not be 0, AND ( ii ) the subtraction must be “ proper ”, a test must guarantee that the smaller of the two numbers is subtracted from the larger ( alternately, the two can be equal so their subtraction yields 0 ).

Euclid ( c. 325-265 BC ), of Alexandria, probably a student of one of Plato ’ s students, wrote a treatise in 13 books ( chapters ), titled The Elements of Geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry.

We know from other references that Euclid ’ s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost.

His work surrounding Euclid ’ s Elements, for example, was of great help in providing training that would help future scholars understand the relationships between demonstrative and geometrical proofs.

According to Allen Weiss, in Mirrors of Infinity, this optical effect is a result of the use of the tenth theorem of Euclid ’ s Optics which asserts that “ the most distant parts of planes situated below the eye appear to be the most elevated .” In Fouquet ’ s time, interested parties could cross the canal in a boat, but walking around the canal provides a view of the woods that mark what is no longer the garden and shows the distortion of the grottos previously seen as sculptural.

For over 2, 000 years, Euclid ’ s Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century.

* Dotty O ’ Neill: Frank's somewhat eccentric 4th class teacher who loves Euclid.

His political philosophy can be seen from a postcard he sent in the 1930s, on which he wrote what can be seen as an epitaph for Mussolini ’ s Italy: Empires die, but Euclid ’ s theorems keep their youth forever.

* Bjørn Felsager ( 2004 ), Through the Looking Glass-A glimpse of Euclid ’ s twin geometry, the Minkowski geometry, ICME-10 Copenhagen ; pages 6 & 7.

U. S. Secretary of Commerce Donald L. Evans visits Lincoln Electric ’ s Euclid headquarters, citing the Company as a prime example of America ’ s manufacturing strength.

* President George W. Bush visits Lincoln Electric ’ s world headquarters in Euclid, Ohio, tours the manufacturing campus and praises the Company ’ s employees for their global competitiveness.

Since most of the later proofs ( presented by Euclid and others ) are geometrical in nature, the Sulba Sutra ’ s numerical proof was unfortunately ignored.

The only evidence comes from traditions recorded in works such as Proclus ’ commentary on Euclid written centuries later.

Thanks to Margaret Harris, who waived her right to a life-rent of her brother ’ s estate, a girls ’ school was built across Euclid Crescent in two stages between 1886 and 1890.

They found a home in a farmhouse donated by Cleveland ’ s industrialist Francis Drury located at East 85th and Euclid Avenue which ultimately became the site of a long-lasting home of CPH.

After one pair of Bézout coefficients ( x, y ) has been computed ( using extended Euclid or some other algorithm ), all pairs may be found using the formula

* 300 Euclid, Greek mathematician, publishes Elements, treating both geometry and number theory ( see also Euclidean algorithm ).

Knowledge of the golden mean goes back at least as far as 300BC, when Euclid described the method of geometric construction in Book 6, Proposition 30 of his book the Elements.

Euclid never used the definition of rhomboid again and introduced the word parallelogram in Proposition 31 of Book I ; " In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas.

Some notable mathematicians include Archimedes of Syracuse, Leonhard Euler, Carl Gauss, Johann Bernoulli, Jacob Bernoulli, Aryabhata, Brahmagupta, Bhaskara II, Nilakantha Somayaji, Omar Khayyám, Muhammad ibn Mūsā al-Khwārizmī, Bernhard Riemann, Gottfried Leibniz, Andrey Kolmogorov, Euclid of Alexandria, Jules Henri Poincaré, Srinivasa Ramanujan, Alexander Grothendieck, David Hilbert, Alan Turing, von Neumann, Kurt Gödel, Joseph-Louis Lagrange, Georg Cantor, William Rowan Hamilton, Carl Jacobi, Évariste Galois, Nikolay Lobachevsky, Rene Descartes, Joseph Fourier, Pierre-Simon Laplace, Alonzo Church, Nikolay Bogolyubov and Pierre de Fermat.

A gnomon as in Euclid book II

As the university continued to grow, particularly after World War II, students moved into many of the houses along Euclid Avenue and its side streets.

Proclus ( 410-485 ), author of Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry.

For two centuries Euclid had been taught from two Latin translations taken from an Arabic source ; these contained errors in Book V, the Eudoxian theory of proportion, which rendered it unusable.

* Proclus ' Commentary on Euclid, Book I. PDF scans of Friedlein's Greek edition, now in the public domain ( Classical Greek )

Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with.

The Saccheri quadrilateral was first considered by Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.

For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.

It has been argued that, given some differences between the two models, it is more likely that Copernicus could have taken the ideas found in the Tusi couple from Proclus's Commentary on the First Book of Euclid.

# a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals ( see extracts from Pappus ' comm.

Euclid introduced the term in his Elements in Book I, Definition 22,

See Aristotle, Metaphysics, 1020a14 and Euclid, Elements, Book VII, Definition 1 and 2.

: In the Second Book of Euclid.

Euclid showed in his second proposition ( Book I of the Elements ) that such a collapsing compass could be used to transfer a distance, proving that a collapsing compass could do anything a real compass can do.

Similarly, the name Dulcarnon is given to the 47th proposition of Book I of Euclid, better known as the Pythagorean theorem, after the Arabic Dhū ' l qarnain ذ ُ و ٱل ْ ق َ ر ْ ن َ ي ْ ن, meaning " the owner of the two horns ", because diagrams of the theorem showed two smaller squares like horns at the top of the figure.

It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements.

Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic thereto ( Books VII to IX of Euclid's Elements ).

* Barrow, William C., The Euclid Heights Allotment: A Palimpsest of the Nineteenth Century Search for Real Estate Value in Cleveland's East End, Chapter VII, Cleveland State University ( Master's thesis )( 1997 ),

Euclid IX 21 — 34 is very probably Pythagorean ; it is very simple material (" odd times even is even ", " if an odd number measures divides an even number, then it also measures divides half of it "), but it is all that is needed to prove that

In Orléans, one of the pre-eminent centres of classical studies, he read ancient Roman literature ( known simply as " the Authors ") with Hilary of Orléans, and learned mathematics (" especially Euclid ") with William of Soissons.

Frankie Yankovic, a Slovenian from Cleveland Ohio traveled the world playing his version (" Cleveland Style " as per Polka Hall of Fame, Euclid Ohio ) of the Viennese Waltzes.

The word " diagonal " derives from the ancient Greek διαγώνιος diagonios, " from angle to angle " ( from διά-dia -, " through ", " across " and γωνία gonia, " angle ", related to gony " knee "); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus (" slanting line ").

Subsequently in 2004, the U. S. Food and Drug Administration ( FDA ) approved a design from Euclid Systems Corporation (" Emerald ").

The motion picture Les Amants (" The Lovers ") was first shown locally in 1959 at the Heights Art Theatre, then located at the intersection of Coventry Road and Euclid Heights Boulevard.