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 this triangulated graph is colorable using four colors or less, so is the original graph since the same coloring is valid if edges are removed.

If there is an edge from a vertex x to a vertex y, then the element is 1 ( or in general the number of xy edges ), otherwise it is 0.

If cutting tools were not used, individual stamps were torn off, as evidenced by the ragged edges of surviving such examples.

* If leather on the side of a shoe is scored by several parallel cuts, it was caused by someone who scraped around the edges of the sole in order to remove crusted mud.

If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron.

If the wipe proceeds from two opposite edges of the screen toward the center or vice versa, it is known as a barn door wipe ( named for its similarity to a pair of doors opening or closing ).

If a piece is placed that creates a connection between both players ' goal edges ( i. e. all edges are connected ), the winner is the player who placed the final piece.

If Keen narrowly misses a jump, he can grab onto the edges of most platforms, and climb up.

If the beam does not fill more than a third of the beam profiler ’ s sensor area, then there will be a significant number of pixels at the edges of the sensor that register a small baseline value ( the background value ).

If the baseline value is large or if it is not subtracted out of the image, then the computed D4σ value will be larger than the actual value because the baseline value near the edges of the sensor are weighted in the D4σ integral by x < sup > 2 </ sup >.

If there are n vertices in the graph, then each tree has n-1 edges.

If the mirrors have vertical edges then the left edge of the field of view is the plane through the right edge of the first mirror and the edge of the second mirror which is on the right when looked at directly, but on the left in the mirror image.

If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle, circuit, circle, or polygon ; see Cycle graph.

If the cut edges did not align correctly or the gluing was not done well, there might still be water leakage along the seam.

) If any of the edges ( r, s ), ( r, t ), ( s, t ) are also blue then we have an entirely blue triangle.

If not, then those three edges are all red and we have an entirely red triangle.

If this is done in such a way that, when glued together, neighboring edges are pointing in the opposite direction, then this determines an orientation of the surface.

If we have 2 nodes with an odd number of edges, the walk must begin at one such node and end at the other.

If U, V, W, u, v, w are lengths of edges of the tetrahedron ( first three form a triangle ; u opposite to U and so on ), then

However, many different DAGs may give rise to this same reachability relation: for example, the DAG with two edges a → b and b → c has the same reachability as the graph with three edges a → b, b → c, and a → c. If G is a DAG, its transitive reduction is the graph with the fewest edges that represents the same reachability as G, and its transitive closure is the graph with the most edges that represents the same reachability.

If a DAG G represents a partial order ≤, then the transitive reduction of G is a subgraph of G with an edge u → v for every pair in the covering relation of ≤; transitive reductions are useful in visualizing the partial orders they represent, because they have fewer edges than other graphs representing the same orders and therefore lead to simpler graph drawings.