If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.

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 there was realism in the theater, there would never be a third act.

* If the third baseman fields a ball and throws it to first, it is recorded as a 5 – 3 out.

If a Committee manages to adopt a joint text, it then has to be approved in a third reading by both the Council and Parliament or the proposal is abandoned.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

In particular, the third anathema reads: " If anyone divides in the one Christ the hypostases after the union, joining them only by a conjunction of dignity or authority or power, and not rather by a coming together in a union by nature, let him be anathema.

If a team fails to gain ten yards in two downs they usually punt the ball on third down or try to kick a field goal ( see below ), depending on their position on the field.

If objects are seen as moving within a rotating frame, this movement results in another fictitious force, the Coriolis force ; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced.

If the third layer is placed so that its atoms are directly above those of the first layer, the stacking will be ABA — this is the hcp structure, and it continues ABABABAB.

If the panegyric detailing the ceremony implied that the true center of the Empire was not Rome, but where the Emperor sat ("... the capital of the Empire appeared to be there, where the two emperors met "), it simply echoed what had already been stated by the historian Herodian in the early third century: " Rome is where the emperor is ".

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

If his son were guilty, he would still believe him innocent, on the basis of faith in his son ; this would violate the third subjunctive condition.

The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

If there is another functional group at a carbon, it may be named with the Greek letter, e. g., the gamma-amine in gamma-aminobutanoic acid is on the third carbon of the carbon chain attached to the carboxylic acid group.

If the team that batted second is substantially behind on first innings total, it may be required to bat again ( i. e., to immediately follow on from its first innings ) in the third innings of the match.

If anyone violates the truce of God and after the third admonition does not make satisfaction, he shall be anathematized.

If he disregards the third admonition the bishop, either with the advice of the metropolitan or with that of two or one of the neighboring bishops, shall pronounce the sentence of anathema against the violator and in writing denounce him to all the bishops.

If the Supreme Court grants certiorari and accepts the case, it will receive written briefs from each side ( and any amici curiae or friends of the court — usually interested third parties with some expertise to bear on the subject ) and schedule oral arguments.

If Language A borrowed a word from Language B, or both borrowed the word from a third language or inherited it from a common ancestor, and later the word shifted in meaning or acquired additional meanings in at least one of these languages, a native speaker of one language will face a false friend when learning the other.

If you listened very carefully, I have already stated to you what the third word is.

If you have read what I have told you, you will see that I have given you the third word.

If one carefully observes the third equation one will notice it also works for n

If the monopoly were permitted to charge individualised prices ( this is termed third degree price discrimination ), the quantity produced, and the price charged to the marginal customer, would be identical to that of a competitive company, thus eliminating the deadweight loss ; however, all gains from trade ( social welfare ) would accrue to the monopolist and none to the consumer.

If only the second and third conditions of the definition of measure above are met, and takes on at most one of the values, then is called a signed measure.

If a player is on 7's and it is wild for them and the other player is on a different number and player A ( wild 7 ) plays a 7 to make it a different suit, player B ( not on 7's ) may play a 7 to change the suit to whatever the card's suit is.

If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.

If she chose, and in that final decision discarded, what, above all, all of us value, life itself, must she not have risen to her fullest height, and transcending her murky self, felt at last the passion of a great moral decision??

* 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 you move them toward the beam, you count their value.

If you move away, you don't count their value.

If an alpha channel is used in an image, it is common to also multiply the color by the alpha value, to save on additional multiplications during compositing.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

* If the operation is commutative, ab = ba, then the value depends only on

If the hand value exceeds 21 points, it busts, and all bets on it are immediately forfeit.

* Split ( only available as first decision of a hand ): If the first two cards have the same value, the player can split them into two hands, by moving a second bet equal to the first into an area outside the betting box of the original bet.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

If a node has fewer than two children, some of the child pointers may be set to a special null value, or to a special sentinel node.

If the opponent drops the doubled stakes, he loses the game at the current value of the doubling cube.

If a house or car is above a certain value, the bankrupt can buy the interest back from the estate in order to keep the asset.

If a mutual is large, or is expected to grow if it can raise capital by a conversion, its managers derive more value from a conversion but do not suffer much loss of perks than if the bank were small.

If the shooter " hits " the point value again ( any value of the dice that sum to the point will do ; the shooter doesn't have to exactly repeat the value combination of the come-out roll ) before rolling a seven, the Pass line wins and a new round starts, usually with the dice staying with the current shooter.

If the roll is any other value, it establishes a point ; if that point is rolled again before a seven, the bet wins.

If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.

If its value is 127 and it is instructed to add one, the computer can not store the number 128 in that space.

If, i. e., it has a large norm with each value of s, and if, then Y ( s ) is approximately equal to R ( s ) and the output closely tracks the reference input.

If the rich man is to be saved, all he must do is to follow the two commandments, and while material wealth is of no value to God, it can be used to alleviate the suffering of our neighbor.