If in any one calculation Ptolemy had had to invoke 83 epicycles all at once, while Copernicus never required more than one third this number, then ( in the sense obvious to Margenau ) Ptolemaic astronomy would be simpler than Copernican.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

If a nation wished to get a head start in physical fitness over all other nations, it would start its kindergarten students on a program of gymnastics the day they entered and thus eliminate a large number of the problems that plague American schools.

If no specific organization plan exists limiting the number of scientists at each salary level, the result is a department top-heavy with high-level, high-salaried personnel ''.

If we thus spent our very first day in the midst of a large number of your people honoring a new hero and a great national achievement, our last day, to us at least, was equally impressive and very moving, even though the crowds were absent and there was almost complete silence.

If a wedge-shaped coating of increasing thickness is removed from a substrate by an instrument like the Hesiometer with a knife of constant rake angle, a number of removal mechanisms are often observed which depend upon the thickness of the coating.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

If Rock shows up during the number -- or you hear anything -- give me the signal ''.

If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.

If so, it might be worth while to assign a future jazz show to a different department -- one with enough confidence in the musical material to cut down on the number of performers and give them a little room to display their talents.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If its minimal polynomial has degree, then the algebraic number is said to be of degree.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

If a ) testimonies conflict one another, b ) there are a small number of witnesses, c ) the speaker has no integrity, d ) the speaker is overly hesitant or bold, or e ) the speaker is known to have motives for lying, then the epistemologist has reason to be skeptical of the speaker's claims.

If we take in our hand any volume ; of divinity or school metaphysics, for instance ; let us ask, Does it contain any abstract reasoning concerning quantity or number?

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If the two carbon atoms at the double bond are linked to a different number of hydrogen atoms, the halogen is found preferentially at the carbon with fewer hydrogen substituents ( Markovnikov's rule ).

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If there is no number to its left, simply look at the column headed " 1 " in the previous row.

If the game has been scored correctly, the total number of plate appearances for a team should equal the total of that team's runs, men left on base, and men put out.

If the other player also knows to offer sacrifices, the expert also has to manipulate the number of available sacrifices through earlier play.

If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

If amber is heated under the right conditions, oil of amber is produced, and in past times this was combined carefully with nitric acid to create " artificial musk " – a resin with a peculiar musky odor.

If this particular quasar and our galaxy could be seen side by side at the same distance, the quasar would be 5 magnitudes ( or 100 times ) brighter than our galaxy.

If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.

Pasternak later said, " If, in a bad dream, we had seen all of the horrors in store for us after the war, we should have been sorry not to see Stalin go down together with Hitler: an end to the war in favour of our allies, civilized countries with democratic traditions, would have meant a hundred times less suffering for our people than that which Stalin again inflicted on it after his victory.

If a 4, 5, 6, 8, 9, or 10 is thrown on the come-out roll ( i. e., if a point is set ), most casinos allow pass line bettors to take odds by placing from one to five times ( and at some casinos, up to 100 times ) the pass line bet behind the pass line.

If he sometimes seems to approach the symbolic view, at other times he comes very close to a strong realistic doctrine.

If the late queen would have believed her men of war as she did her scribes, we had in her time beaten that great empire in pieces and made their kings of figs and oranges as in old times.

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

If anyone shall violate the truce of God he shall be admonished three times by the bishop to make satisfaction.

If one flips a coin 21 times in a row and obtains 21 heads, one might rationally conclude a high probability of bias towards heads, and hence conclude that future flips of this coin are also highly likely to be heads.

If you are listening carefully to everything I say, you just heard me say it three times.

* If a thief steals a cow, a sheep, a donkey, a pig, or a goat, he will pay ten times what it is worth.

If we neglect the possibility of overlapping states, which is valid if the temperature is high, then the number of times we count each state is approximately N < nowiki >!</ nowiki >.

If twice or three times that does not suffice, then not even the wealth of Croesus or of Persia will suffice.

If V is a formal language, then V < sub > i </ sub >, the i-th power of the set V, is a shorthand for the concatenation of set V with itself i times.

* If creatures similar to plesiosaurs lived in the waters of the Loch Ness, they would be seen very frequently as they would have to surface several times a day to breathe.

If the value being sought occurs k times in the list, and all orderings of the list are equally likely, the expected number of comparisons is

If the picture contains an area of the same color, it can be compressed without loss by saying " 200 red dots " instead of " red dot, red dot, ...( 197 more times )..., red dot.

If the brightness of the object exceeded the maximum limit of eight times the brightness of Canopus, the spacecraft would search for a new star.

If the force depends on time, the change in momentum ( or impulse ) between times and is

If the IMF has a southward component, this not only pushes the magnetopause boundary closer to Earth ( at times to about half its usual distance ), but it also produces an injection of plasma from the tail, much more vigorous than the one associated with substorms.

Saints All-Pro quarterback Drew Brees made a controversial tweet on June 20, 2012 stating, " If NFL fans were told there were " weapons of mass destruction " enough times, they'd believe it.

If the universe is homogeneous at a large scale, then there would be four times as many stars in a second shell between 2, 000, 000, 000 to 2, 000, 000, 001 light years away.