How To Use Euclid In A Sentence

The law of reflection was known as early as the time of Euclid, about 320 b.c., and to this geometrician was attributed, although probably erroneously, a "Treatise on Mirrors", in which the principles of catoptrics were correctly set forth. The Catholic Encyclopedia, Volume 12: Philip II-Reuss
His commentary to Euclid is of interest because of its discussion of unordered irrationals.
Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing.
Now the problem which had perplexed Bolyai most in his study of mathematics had been the independence of Euclid's Fifth postulate.
Euclid wished to discover whether there existed a simple geometrical proportionality between the apparent size of equal and parallel lines and their distances from the eye.

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Now, if Judge Douglas will demonstrate somehow that this is popular sovereignty, the right of one man to make a slave of another, without any right in that other, or anyone else to object, demonstrate it as Euclid demonstrated propositions, there is no objection. Speech of Hon. Abraham Lincoln
In Borken's messily vital urban context of supermarkets, small houses and traffic roundabouts, the bank's simple Euclidean form is a reassuringly calm, rooted presence.
For example, Spinoza's Ethics has the same format as Euclid's Elements, containing propositions and demonstrations.
Squared Euclidean distances were utilized in order to maximize the dissimilarity of unlike clusters.
His interest in mathematics was stimulated during his school years in Izmir by a teacher who encouraged him to solve problems in euclidean geometry.
Euclidean geometry, Fibonacci numbers, the digits of pi, the notion of algorithms, concepts of infinity, fractals, and other ideas furnished the mathematical underpinnings.
Euclid and Archimedes utilized two important techniques to prove theorems from their axioms: reductio ad absurdum arguments, and a method of exhaustion.
He worked on conjugate functions in multidimensional euclidean space and the theory of functions of a complex variable.
However he continued to work on topological ideas, in particular embedding complexes in Euclidean space.
His style shows how deeply he was influenced by Euclid's treatment of ration and proportions.
Tartaglia also wrote a popular arithmetic text and was the first Italian translator and publisher of Euclid's Elements in 1543.
The presentation took place at the mayor's parlour, in the Civic Offices, in Euclid Street.
Today we call these three geometries Euclidean, hyperbolic, and absolute.
He worked on conjugate functions in multidimensional euclidean space and the theory of functions of a complex variable.
The protest against Americans For Truth About Homosexuality's anti-gay "academy" will take place at 7: 30 PM sharp, Thursday, August 5 in front of "Christian Liberty Academy," 502 W. Euclid Avenue, Arlington Heights, IL. Andy Thayer: Why We Are Protesting Against "Americans For Truth About Homosexuality"
The second chapter presents a development of absolute and Euclidean geometry based on Hilbert's axioms.
Why do I refer to Euclidean geometry as a physical theory rather than a branch of mathematics?
Euclid [Euclidi Megarensi] "To Euclid of Megara, because he grasped the spaces of the [E] arth by means of lines and the compass, Federico gave this for a most precise invention. Architecture and Memory: The Renaissance Studioli of Federico da Montefeltro
Euclid's axioms form the foundation of his system of geometry.
She scans the trays of chicken feet and pork ears that beckon to her inside the Vinh Hung grocery near Euclid Avenue.
The work of both Aristaeus and Euclid on conics was, almost 200 years later, further developed by Apollonius.
Euclid's algorithm is here applied to 720 and 168: Just keep dividing and noting remainders so that the larger number 720 is 4 lots of the smaller number 168 with 48 left over.
(The toothache was the only malady to which Tom had ever been subject.) "Euclid, my lad; why, what's that?" said Mr. Tulliver. The Ontario Readers: Fourth Book
So when Euclid described his geometry, many believed it to be the one true geometry.
In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics.
Historically it has been convenient to explore them as an evolution from the works of Euclid.
Much of his career is spent working on physics and non-euclidean geometry.
Since Euclid's axiomatic formulation of geometry mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other four axioms.
The straight line must be one of the earliest curves studied, but Euclid in his Elementsalthough he devotes much study to the straight line, does not consider it a curve.
Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on.
In his Euclid a series of references is provided to the arithmetical treatises; several portions of his Archimedes are strictly related to his works on Apollonius and on conic sections, the latter referring in turn to researches in gnomonics.
Poplawski takes advantage of the Euclidean-based coordinate system called isotropic coordinates to describe the gravitational field of a black hole and to model the radial geodesic motion of a massive particle into a black hole. PhysOrg.com - latest science and technology news stories
The agglomerative, or grouping, schedule provided by Ward's method indicated a notable flattening of the curve of squared Euclidean distances after the five-cluster solution.
For example, in Euclidean geometry, the relevant invariants are embodied in quantities that are not altered by geometric transformations such as rotations, dilations, and reflections.
In this paper, empirical euclidean likelihood ratio statistics are constructed for parametric in a nonlinear model. And prove strong consistency and asymptotic normality of the estimation.
LiuWei shape . mental Euclid, Archimedes comparable.
Thus, for example, the notions of Euclidean geometry are invariant under similarity transformations, those of affine geometry under affine transformations, and those of topology under bicontinuous transformations. Logical Constants
The wormhole is invoked as a way of describing the concrete geographies of positionality and their non-Euclidean relationship to the Earth's surface.
However, this revelation did not bring about the destruction of Euclidean geometry, it simply added to it.
If Euclid's Postulate is denied, there are countless straight lines through Q, coplanar with a, that make acute angles with PQ but never meet a. Consider the set of real numbers which are the magnitudes of these acute angles. Nineteenth Century Geometry
Suppose them to be confronted with the problem of proving the entire equality, by the Euclidean method, of the triangles ABC and abc (Fig. 4), when it is granted that the side AB is equal to the side ab, AC is equal to ac, and the angle CAB is equal to the angle cab. The Fourth Dimension Simply Explained
The essential property and developmental course of Euclidean space, symplectic space and bogus space are studied.
It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied.
Moore wrote the sections on arithmetic, geometry, trigonometry and cosmography while the sections on algebra, Euclid and navigation were written by Perkins.
Clothes queue up in the wardrobe, an echo to the eye, or a jangle of Euclid.
For example, recall that in Euclidean geometry the sum of the angles of any triangle is always 1800.
Since Euclid's axiomatic formulation of geometry mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other four axioms.
The work of Bolyai and Lobachevsky are comparable in that sense, that they both challenge axiomatic assumptions, but their postulates are of Euclidean geometry.
References to Euclid's work on solid geometry clearly no longer looked intimidating.
In the classical differential geometry which deals with the theory of curves and surfaces of three dimensional Euclidean space, the most distinctive study is the Weingarten surface.
At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought.
non-Euclidean geometries discard or replace one or more of the Euclidean axioms
See this issue's mystery mix for a comparison of Euclidean, hyperbolic, and spherical geometry.
More recently, she completed the electrifying Euclid's Comet, a fresco secco in the Media Union of the University of Michigan, Ann Arbor.
Historically it has been convenient to explore them as an evolution from the works of Euclid.
Nouvelles pensées de Galilée sur les mécaniques" (Paris, 1639), both translations; "Cogitata physico-mathematica" (Paris, 1644); "Euclidis elementorum libri, Apollonii Pergæ conica, Sereni de sectione coni, etc. The Catholic Encyclopedia, Volume 10: Mass Music-Newman
Euclidean norm theory was used to select and optimize the contractors during the evaluation work of the construction projects.
The "ciphering" of the lower schools expands into elementary mathematics in the higher; into arithmetic, with a little algebra, a little Euclid. Science & Education
Perhaps Euclid's ghost is stalking the English countryside by night, leaving its distinctive mark wherever it happens to alight.
On the foundation of the conception of orthonormal basis in finite dimensional Euclidean space, this paper provides the theory of completely orthonormal system in infinite dimensional Euclidean space.
So far as experience goes, such a thing has no more real existence than a line without breadth; and hence an atomic theory based upon such an assumption may be as true ideally as any of the theorems of Euclid, but it can give only an approximatively true account of the actual universe. The Unseen World, and Other Essays
This theorem, also called the infinitude of primes theorem, was proved by Euclid in Proposition IX.20 of the Elements.
A standard axiomatisation for Euclidean plane geometry Finitism in Geometry
Kaluznin also worked in the area of geometrical algebra, particularly on arrangements of subspaces in Euclidean and unitary spaces.
He also realised that there were an infinite number of non-euclidean geometries and this, Taurinus claimed, was highly significant.
The following year he wrote on number theory, making a contribution to the theory of the Euclidean algorithm.
Up to this stage quantum theory was set up in Euclidean space and used Cartesian tensors of linear and angular momentum.
Supervised pattern classifiers with Euclidian Measurement are finally performed for recognition.
A few good necessary and sufficient conditions are given by using length relation for symmetrical transformation on Euclidean space.
Other topics to interest Carslaw throughout his career, which we have not touched on above, included an interest in non-euclidean geometry, Green's functions and the history of Napier's logarithms.
Euclid also wrote Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set.
Moore wrote the sections on arithmetic, geometry, trigonometry and cosmography while the sections on algebra, Euclid and navigation were written by Perkins.
The position bears comparison with the development of geometrical reasoning by Euclid.
He goes in for rhomboids, quincunxes, and chess logic; even his ubiquitous infinities are of a linear, ‘Euclidean’ sort.
In fact there is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used such as that of an oblong, a rhombus, and a rhomboid.
Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved.
Something that exists nowhere and exists along the lines of Euclidean geometry, judging by what I understand of it, cannot exist.
This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent.
He was indeed a prodigious Scholar; he had learn'd the_ Alcoran, _and was well initiated into Human Learning before he was Ten years old; then he studied Logick and Arithmetick, and read over Euclid without any help, only his Master show'd him how to demonstrate the first five or six Propositions; Then he read_ Ptolemy's Almagest, The Improvement of Human Reason Exhibited in the Life of Hai Ebn Yokdhan
If you micropylar the entry level sales megahertz to a euclidian windsock, podicipediformes to that blessing in your progressive antique nitrochloroform and theropoda the predestinate overstrain. Rational Review
Some of his work on physical topics relates to his non-euclidean geometry for he examined how the gravitational potential as given by Newton would have to be modified in a space of negative curvature.
Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.
Our earliest glimpse of Euclidean material will be the most remarkable for a thousand years, six fragmentary ostraca containing text and a figure… found on Elephantine Island in 1906/07 and 1907 / 08…
The axiomatic form in Euclid is more complex, relying not just on first principles (communis animi conceptio), the only type of principle used by Boethius and Alan, but also on definitions, petitiones, theorems, etc. Literary Forms of Medieval Philosophy
However, this revelation did not bring about the destruction of Euclidean geometry, it simply added to it.
She believed that in order to get students excited about mathematics, it was essential to teach the revolutionary aspects of such fields as Galois theory of groups, non-Euclidean geometry, and modern logic. Lillian R. Lieber.
Why do I refer to Euclidean geometry as a physical theory rather than a branch of mathematics?
The concept of the cross product of vectors in Euclid space is introduced.
Apollonius did for conics what Euclid had done for elementary geometry: both his terminology and his methods became canonical and eliminated the work of his predecessors.
As well as constructions to divide a line in the golden ratio, Euclid gives applications such as the construction of a regular pentagon, an icosahedron and a dodecahedron.
It's extraordinary to think that whereas Albarn has been bringing himself up to speed with concepts of hermeticism, Euclidian geometry and Rosicrucianism, his erstwhile Britpop rival Liam Gallagher has formed Beady Eye. Dr Dee, Palace Theatre, Manchester | First night review
Based on Euclidean algorithm, this paper gives a new method that can judge similar zero and pole points of a system model and conception of similarly.
We take as given that the desired curve C is the unique Euclidean circle that is orthogonal to every geodesic through A.
He also studied birational contact transformations and non-euclidean and non-archimedean geometries.
A familiar Riemannian manifold is a Euclidean manifold (where one has to add a smoothly varying inner product on the tangent space of the standard Euclidean space), with the familiar Euclidean (distance) metric (our 3-space, for example). Bad Language: Metric vs Metric Tensor vs Matrix Form vs Line Element
Euclid was able to draw an equilateral triangle, square, pentagon and hexagon, but recall that the heptagon (which has seven sides) and the nonagon (nine) eluded him. HERE’S LOOKING AT EUCLID
It is vain to cite Euclidian postulates that "quantities which are equal to the same quantity are equal to each other.
Halley suggested to him that he might devote his considerable talents to the restoration of the work of the early Greek geometers, such as Euclid and Apollonius of Perga.
No doubt, the geometrization of time also owed something to the revival of interest in Euclid in the TECHNOLOGY
It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Dubliners
Also, the precise mathematical form Poincaré chose had to make the Poincaré-line that joins any two points the shortest path between them called a geodesic, just as the usual line is the shortest path between points in Euclidean space. Euclid’s Window
One minute you're learning that Sir Issac Newton chuckled only once in his life (scoffing at Euclidean geometry) and that the term for such people who don't laugh is 'agelast'; the next that the apparently nonsensical elephant jokes that were popular in the Sixties are believed to be racist in origin; the next how Bertrand Russell put down a heckler during one of his lectures on logic. Chortle News RSS
He also thought in conic sections, squares and roots and ratios, and geometrized like Euclid. Pragmatism
Euclid was able to draw an equilateral triangle, square, pentagon and hexagon, but recall that the heptagon (which has seven sides) and the nonagon (nine) eluded him. HERE’S LOOKING AT EUCLID
This second commentary is on al-Samarqandi's famous short work of only 20 pages in which he discusses thirty-five of Euclid's propositions.
He worked on conjugate functions in multidimensional euclidean space and the theory of functions of a complex variable.
She scans the trays of chicken feet and pork ears that beckon to her inside the Vinh Hung grocery near Euclid Avenue.
For instance, it's also the square of the Euclidean norm on R2, the length of a two-dimensional vector, a part of the triangle inequality, and quite a bit more.
It is vain to cite Euclidian postulates that "quantities which are equal to the same quantity are equal to each other.
His work on non-euclidean geometries was used by Einstein in his general theory of relativity.
Although spatial intuition or observation remains the source of the axioms of Euclidean geometry, in Hilbert's writing the role of intuition and observation is explicitly limited to motivation and is heuristic.
Moreover, the geometries axiomatized in the book have consistent and decidable extensions, namely, Euclidean or hyperbolic geometry over real-closed fields.
Mathematically it corresponds to a linear transformation for a set of points in the Euclidean space.
Firstly, extracting face principal components by PCA, and computing the Euclid distances between testing sample and classes, then getting sub-decisions by a transform function.
See: on homotopy, where the space Y is in fact euclidean 3-space. Boing Boing: July 25, 2004 - July 31, 2004 Archives
Each innovation destined to dwarf the one extensive accomplishment of the Greeks - Euclidean geometry.
Speculative geometry contains elementary geometry which is not all based on Euclid.
In 1903 he published Geometrie der Dynamen which considered euclidean kinematics and the mechanics of rigid bodies.
O'Connell knew well the use of sound in the vituperation, and having to deal with an ignorant scold, determined to overcome her in volubility, by using all the _sesquipedalia verba_ which occur in Euclid. Irish Wit and Humor Anecdote Biography of Swift, Curran, O'Leary and O'Connell
This appendix proved to be one of the foundations of non-Euclidean geometry; it was a mathematical landmark worth far more than anything else in the book.
The conformal coordinate z is to be identified as exp (tau + i sigma), where tau is euclidean time, and and 0 infinity, and we typically do not want to impose boundary conditions in the infinite future. String Theory is Losing the Public Debate
Euclidian geometry
It follows from Kant's view that we know a priori that non-Euclidean geometry cannot be applied in physics.
It is also mathematics, of course, but Euclidean geometry is by no means the only conceivable mathematical geometry.
One of the first papers which he published after arriving in the United States was on the Euclidean algorithm in principal ideal domains.
At the time when these papers were written he had received no instruction in mathematics beyond a few books of Euclid and the merest elements of algebra.
Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied.
Euclid was trying to convey his idea of a geometrical point.
Euclid's Elements is full of algorithms for geometry, including one to find the greatest common divisor of two numbers.
I try to think of a descriptive metaphor as we head west on Euclid Avenue, passing abandoned houses, new and gated condominiums, and apartment buildings with boards nailed to all the ground-level windows.
The Lorentz transformation of orthogonal bases is derived by means of matrix Euclidean four dimension space.
That is, the pre-Euclidean, Classical Greek geometers, typified by the Pythagoreans, and the School of Plato.
In Euclidean geometry, light travels on straight lines.
This is far from an end to the arguments about Euclid the mathematician.
A very important thing to remember: The dark star calculations are done for a three-dimensional Euclidean space, while the black hole calculations are done for a four-dimensional Lorentzian spacetime (there is a big difference between the two). Objections to Kaku and Liu’s “How to Time Travel?”
It had always sounded strangely in my ears, like the word gnomon in the Euclid and the word simony in the Catechism. Dubliners
Theaetetus was the first to study the octahedron and the icosahedron and it is believed that Book XIII of Euclid's Elements is based on his work.
Axiomatized theories - such as Euclid's presentation of plane geometry - offer a compelling example of foundationalism in action.

How To Use Euclid In A Sentence

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