Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

Given that one cannot be motivated by reason alone, requiring the input of the passions, Hume argued that reason cannot be behind morality.

Given the limited success of the Charity School, however, Wheelock intended his new College as one primarily for whites.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given good circumstances one might be able to discern the result of some human activity such as the changing of the Netherlands ' coast or the partial drying out of the Aral Sea, but even that would not be easy.

Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required.

Given our formula φ, we group strings of quantifiers of one kind together in blocks:

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given a general algorithm for integer factorization, one can factor any integer down to its constituent prime factors by repeated application of this algorithm.

Given a complete set of axioms ( see below for one such set ), modus ponens is sufficient to prove all other argument forms in propositional logic, and so we may think of them as derivative.

Given significant distance from the magnetic poles, one can figure which hand is which using a magnetic compass and the sun.

Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.

Given the same set of verifiable facts, some societies or individuals will have a fundamental disagreement about what one ought to do based on societal or individual norms, and one cannot adjudicate these using some independent standard of evaluation.

Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.

Given the above fact, one can ask:

Given a preorder on S one may define an equivalence relation ~ on S such that a ~ b if and only if a b and b a.

# Given any two distinct points, there is exactly one line incident with both of them.

# Given any two distinct lines, there is exactly one point incident with both of them.

# Given an array of values to be sorted, set up an auxiliary array of initially empty " pigeonholes ," one pigeonhole for each key through the range of the original array.

Given the unusual casting and production demands, Pacific Overtures remains one of the least-performed musicals by Stephen Sondheim.

Given the above expression, evidently the result of her ( local ) measurement is that the three-particle state would collapse to one of the following four states ( with equal probability of obtaining each ):

Given the great historical link between Persian and Jewish history, modern day Persian Jews are called " Esther's Children ".

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

" Given that there's a gulf of 100, 000, 000 light years, he checks his diary to find out what he had been doing that day, and finds out that it was something he wished to hide.

Given this is the feast day used by the Greek Orthodox Church, it has been argued that the discovery was probably made on 18 August, the feast day used by the Roman Catholic Church.

Given under my hand and seal this 12th day of January, 1847.

Given her familiarity with the woods and marshes of the region, it is likely that Tubman hid in these locales during the day.

Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week.

Given the maximum observed frequency ( MOF ) for a mode on each day of the month at a given hour, the MUF is the highest frequency for which an ionospheric communications path is predicted on 50 % of the days of the month.

Given a day mandated for seasonal rejoicing ( Shemini Atzeret ) with no specific obligations as to the focus of the rejoicing ( such as sukkah, lulav or etrog ), it is not surprising that the celebration of the conclusion of the annual reading was absorbed into the celebration of Shemini Atzeret.

Given the technical limitations of the day, transporting it was no easy feat — on the pedestal are drawn diagrams explaining the machinery that was used for the transportation.

Given that question time is the only time of day when all members of Parliament are in their respective chambers, the appearance of question time can be rowdy and boisterous compared to the normally sedate activity during the rest of the day.

Given under my hand this, the 1st day of March A. D. 1862.

Given Milan's notoriously damp and foggy climate, the Milanese consider it a fair-weather day when the Madonnina is visible from a distance, as it is so often covered by mist.

Given Vanier's physical state, Pearson was hesitant to advise the Queen to act along those lines, but his worry was short lived, as the following day, after hearing the Holy Sacrifice of the Mass and receiving Holy Communion in the chapel, the Governor General died.

Given the conventions of the day, within which journalism was seen as a disreputable profession for a gentleman, Burke was publicly reticent about his connection with The Annual Register.

Given the current human population and birth-rate, this equals five per day or 1800 per year.

Given that nearly all Jews of their day were poor or middle-class ( even the rich of that time were only rich relative to the poor ), they ruled that one should not give away more than a fifth of his income to charity, while yet being obligated to give away no less than 10 % of his income to charity.

Given at Our Court of Saint James's, the Third day of April 1906: in the Sixth Year of Our Reign.

Sun and moon, these minister for aye ; The laws of day and night cease nevermore: Given for signs to Jacob's seed that they Shall ever be a nation — till these be o ' er.

Given under Our Hand at Salisbury, this Eleventh day of November in the Year of Our Lord one thousand nine hundred and sixty-five.

An eyewitness account of the last day of Major André can be found in the book The American Revolution: From the Commencement to the Disbanding of the American Army Given in the Form of a Daily Journal, with the Exact Dates of all the Important Events ; Also, a Biographical Sketch of the Most Prominent Generals by James Thacher, M. D., a surgeon in the American Revolutionary Army:

Given the predominantly medium-haul route network of the airline, the choice of the Boeing 747SP continues to this day to raise interrogations.

Given the technical limitations of the day, transporting it was difficult — on the pedestal are diagrams explaining the machinery used for its transportation.