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 your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If not, actors frequently train in preparation for their roles, or the action director may rely more on stylized action or filmmaking tricks like camera angles, editing, doubles, undercranking, wire work, and computer-generated imagery.

If he takes 5 tricks, he scores 5.

If he takes 3 tricks, he scores-10.

For example, a player bids 4: If he takes exactly 4 tricks, he scores 14.

If he takes 3 or 5 tricks, he scores-14.

If he takes 2 or 6 tricks, he scores-24.

If a player fails to make the contract and takes overtrick ( s ), he / she scores the number of tricks taken.

If a player has overbid, he / she scores the difference between tricks bid and tricks taken times-10.

If the bidder makes more or fewer tricks than were bid, he / she loses 10 points for the first over or under trick + 15 points for each additional over or under trick.

If the high bid is " eight hearts ," then the team wins the hand if they take 8, 9, or all 10 tricks and are awarded points according to the table below.

If not, actors frequently train in preparation for their roles, or the action director may rely more on stylized action or filmmaking tricks.

If you had the knack, Puck might do minor housework for you, quick fine needlework or butter-churning, which could be undone in a moment by his knavish tricks if you fell out of favour with him.

* If the non-dealer asks for fewer cards than he has discarded, the dealer counts as tricks all cards which cannot be played to.

If the dealer play with fewer than five cards, the non-dealer counts as tricks all cards which cannot be played to.

If a player abandon his hand after he has made one or two tricks, his adversary is entitled to mark the point.

If only three or four of the tricks are taken while going alone, then only one point is scored.

If the bidder playing alone wins all five tricks in the hand, the team scores four points.

If West is holding the king ( it is " onside "), North-South will win two tricks, for a gain of one trick without losing a trick.

( If West actually plays the king on the first trick, of course, North-South win two tricks by covering with the ace.

If the position permits ( combined hands have sufficient number of intermediate cards, which can take the remaining tricks even if the three honors are played on the same trick ), it is often desirable to start by leading one of the high cards, in order to retain the lead in the same hand if the finesse is on, and spare the valuable entries.