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Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and has been solved many times in antiquity.
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Some Related Sentences
Without and requiring
On the statement about a decree requiring women to cover their heads, J. P. Kirsch comments in the Catholic Encyclopedia: " Without doubt this decree is apocryphal, and copied by the author of the Liber Pontificalis from the first Epistle of St. Paul to the Corinthians ( 11: 5 ) and arbitrarily attributed to the first successor of the Apostle in Rome.
Without requiring priests to face the people throughout the Mass, the Roman Missal now calls for the facing-the-people orientation to be made possible: " The altar should be built apart from the wall, in such a way that it is possible to walk around it easily and that Mass can be celebrated at it facing the people, which is desirable wherever possible.
Without requiring all right-holders ' consent, such an alteration necessarily entails the use or threat of force, whether legitimate or not.
Without a catalyst ( natural or artificial ), this reaction is very endothermic, requiring high temperatures ( at least 2500 K ).
Without and solution
Without stepping outside the focus of the PACS architecture, WADO becomes the solution to cross platform capability and can increase the distribution of images and reports to referring physicians and patients.
Without Julius Streicher and his Stürmer, the importance of a solution to the Jewish question would not be seen to be as critical as it actually is by many citizens.
Without the boundary conditions, the general solution to this equation is
Advocates on both sides of the issue, including the Arborway Committee and Better Transit Without Trolleys, present compelling arguments for improved service while the MBTA has not yet committed to a permanent transit solution.
In an open letter to Prime Minister Kevin Rudd, Reporters Without Borders states that the web filter is not the solution to combating child sex abuse, and the plan entails risks to freedom of expression.
Without paint pigments, the lime-sulfur solution bleaches wood to a bone-white color that takes time to weather and become natural-looking.
Without referring explicitly to Marx's solution of the problems of the labour theory of value of David Ricardo, the marginal utility theorist William Stanley Jevons clearly stated the main criticism of the concept of abstract labour in his 1871 treatise:
Without and by
Without really changing the general subject, I take this opportunity to confess that I am troubled by doubts, not only about pacifism, but also when asked to join in the protest against a law that most of those who consider themselves humane and liberal seem to regard as obviously barbarous ; ;
Without dissent, senators passed a bill by Sen. A. R. Schwartz of Galveston authorizing establishment in the future of a school for the mentally retarded in the Gulf Coast district.
`` Without any officious and improper interference on the subject, the price of labor or the wages of mechanics will be regulated by the demand for the manufactured article and the value of that which is paid for it ; ;
Without such a declaration, Paine concluded, “ he custom of all courts is against us, and will be so, until, by an independence, we take rank with other nations .”
Without RIMAs or the MAOI of MAO-A, DMT would be oxidised ( and thus rendered biologically inactive ) by monoamine oxidase enzymes in the digestive tract.
Without surviving male issue, Anton was succeeded as King by his nephew, Frederick Augustus II.
Chapter 17 of William Bates ' 1920 book Perfect Sight Without Glasses, in which the author argues that observation of the sun is beneficial to those with poor vision, includes a figure of somebody " Focussing the Rays of the Sun Upon the Eye of a Patient by Means of a Burning Glass.
Without using the term, 1920 saw the first rule that would be referred to today as defensive indifference, as stolen bases would not be credited, unless an effort was made to stop the runner by the defense.
Without firing a shot, by the mere display of military force, he has achieved a dominating position in Europe which Germany failed to win after four years of war.
Without protected memory, it is possible that a bug in one program will alter the memory used by another program.
Without being backed by gold or foreign currency, the money issued by Li and Yan quickly declined in value until it became virtually worthless.
Without valves, the player could only produce a harmonic series of notes like those played by the bugle and other " natural " brass instruments.
Guided by Arkyn, Corum, Rhalina and companion Jhary-a-Conel cross the planes and encounter the King Without A Country, the last of his people who in turn is seeking the City in the Pyramid.
Without experience of meditation, one's knowledge of the world is too limited to fully understand dukkha, as required by the first noble truth, and proceed to enlightenment.
The reunion led to other collaborations including a limited-edition single release version of Placebo's track " Without You I'm Nothing ", co-produced by Visconti, with Bowie's harmonised vocal added to the original recording.
Without data decoding, the carrier can be detected by much simpler ground antennas and receivers.
Without it, household behavior would be unaffected by uncertain employment and income prospects, financial and capital markets would reduce to exchange of a single instrument in each market period, and there would be no communications industry.
** Mountains and Rivers Without End by Gary Snyder ( composed 1965-1996 )
Without ambiguity, for all in, we denote by the unique inverse of.
Life Without Zoe was mostly panned by critics and was generally considered the segment that brought the film's overall quality down.
Without the availability of slave labour the plantations were soon taken over by the jungle, and the planters ruined.
Though used earlier in papers and discussions, the term 4GL was first used formally by James Martin in his 1982 book Applications Development Without Programmers to refer to non-procedural, high-level specification languages.
Without issue, after his death Frederick Augustus was succeeded by his younger brother, Johann.
Without Abkhazia and South Osetia, the population in the regions controlled by the central government of Georgia was 4, 321, 500 in 2005 and 4, 382, 100 in 2008 ( compare the 2008 figure with the CIA estimate of 4, 630, 841 for all of Georgia, including Abkhazia and South Osetia ).
Without and ruler
Without an authoritative ruler, the Balkan provinces fell rapidly into disorder.
Without and alone
Without the inclusion of clouds, water vapor alone contributes 36 % to 70 % of the greenhouse effect on Earth.
Without hesitation, alone and unaided, he repeatedly attacked this enemy formation, at close range in the face of intense combined machine gun and cannon fire.
" Without my husband, who alone makes me happy, I should have died a thousand deaths "
Without prior sedation, stunning or anesthetic, this method of slaughter causes a high degree of anxiety and should not be used alone.
In this state reading may cease, as well as deliberate thought ... When the feeling of prayer reaches the point where it becomes continuous, then spiritual prayer may be said to begin ... Without inner spiritual prayer there is no prayer at all, for this alone is real prayer, pleasing to God.
Without any formal marketing or advertising plans, news of their songs such as " That Was a Crazy Game of Poker ," " City on Down ," and " Night Shift " spread by word of mouth alone.
The central motif is ' Without you, I cannot be ..., With you, I am alone also ...'
After being left alone, Paige remembers her dream for Doctors Without Borders.
*" Without a family, man, alone in the world, trembles with the cold.
Without Bernadette knowing let alone caring about it, her apartment has been equipped with video cameras and microphones which can be used to compromise, and eventually blackmail, her customers.
Without Tanucci in government, the Queen alone ruled Naples and Sicily, assisted by her French-born, English favourite, John Acton, 6th Baronet, from 1778 onwards.
Without the support from the Allies, the Norwegian Army alone would not be able to defend its positions and a capitulation agreement for mainland Norway was signed.
In the song, Carey sings " Not more than three short years ago / I was abandoned and alone / Without a penny to my name / So very young and so afraid / No proper shoes upon my feet / Sometimes I couldn't even eat / I often cried myself to sleep.
Without Teachers, Dharma texts alone are of little benefit.
Without enough money for both of them to book passage on the same vessel, Loy took the trip on a regular ship and Cravan set out alone on a sailboat to Argentina.
Dr. Pope found a close affinity to the utterances of sincere devotion in such verses as ' Longing for devotion alone ', ' Without thy presence I pine ', ' Deadness of soul ', ' God all in all ', ' I am thine, save me ', ' His love demands my all '.
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