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.

He adds " If this highly speculative surmise be correct, it is also necessary to assume either that the existence of the door was forgotten or that the entrance was again blocked with facing stones " in order to explain why al-Ma ' mun could not find the entrance.

*" If stones could speak " featuring all cemeteries, monuments and relics related to the Gallipoli campaign.

The valuation of the Koh-i-Noor is given in the legend that one of Nader Shah's consorts supposedly said, " If a strong man should take five stones, and throw one north, one south, one east, and one west, and the last straight up into the air, and the space between filled with gold and gems, that would equal the value of the Koh-i-noor.

"… they are obliged to live in a separate part of town …; for they are considered as unclean creatures … Under the pretext of their being unclean, they are treated with the greatest severity and should they enter a street, inhabited by Mussulmans, they are pelted by the boys and mobs with stones and dirt … For the same reason, they are prohibited to go out when it rains ; for it is said the rain would wash dirt off them, which would sully the feet of the Mussulmans … If a Jew is recognized as such in the streets, he is subjected to the greatest insults.

If the opponent places a stone at B or D, the remaining hex can be filled to join the original two stones into a single group.

If the stone is positioned near a bone ( usually a rib in the case of kidney stones ), this treatment may be more uncomfortable because the shock waves can cause a mild resonance in the bone which can be felt by the patient.

If allowed to view the ultrasound or x-ray monitor, they may be able to see their stones change from a distinct bright point ( or dark spot depending on whether the fluoro unit is set up in native or bones white ) to a fuzzy cloud as the stone is disintegrated into a fine powder.

If traditional fixed placement of the handicap stones is used, nine stones is normally the maximum handicap.

If a stone moves between two enemy stones, it captures both stones.

If a kitsune offers a payment or reward that includes money or material wealth, part or all of the sum will consist of old paper, leaves, twigs, stones, or similar valueless items under a magical illusion.

If two matching Oxyd stones are activated they remain open, while if non-matching Oxyd stones are activated the first one closes.

If the Master disproves the guess, the Student may repeat the guessing process until he or she runs out of guessing stones.

If stone-faced, the stones are generally placed on edge, though around gateways the stones are often placed flat.

If the grykes are fairly straight and the clints are uniform in size, the resemblance to man-made paving stones is striking, but often they are less regular.

If the majority of the stones used proved with time very resistant, it was not in the same way for the called green stone Golfalina.

Under the pretext of their being unclean, they are treated with the greatest severity and should they enter a street, inhabited by Mussulmans, they are pelted by the boys and mobs with stones and dirt … For the same reason, they are prohibited to go out when it rains ; for it is said the rain would wash dirt off them, which would sully the feet of the Mussulmans … If a Jew is recognized as such in the streets, he is subjected to the greatest insults.

If pursued, they will not hesitate to take refuge in shallow water, going to the bottom and hiding under stones or debris.

If there are " running away " stones there must be " chasing " stones.