Methods of Application

Excerpts

• Methods of Application

  The conclusion of no argument can be more certain than the assumptions from which it starts. All mathematical calculations about the course of nature must start from some assumed law of nature, such, for instance, as the assumed law of the cost of building stated above.

• Methods of Application

  The vital point in the application of mathematical formulæ is to have clear ideas and a correct estimate of their relevance to the phenomena under observation.

• Methods of Application

  there is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain.

• Methods of Application

  He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces.

• Methods of Application

  This ratio is the same for any lump of metal of the same material: it is now called the specific gravity of the material, and depends only on the intrinsic nature of the substance and not on its shape or quantity.

• Methods of Application

  Faraday was asked: “What is the use of this discovery?” He answered: “What is the use of a child---it grows to be a man.”

Equations

• Methods of Application

  20y = x

  If the cost of a house is y pounds and its volume x cubic feet, the assumed building-cost law makes x equal to 20 times y; the chapter uses this only to illustrate how a mathematical relation is applied to a fact.

• Methods of Application

  F = k\dfrac{mM}{d^{2}}

  The attraction between two bodies is a constant k times the product of their masses divided by the square of the distance between them, so the force F varies with the masses and inversely as the square of the distance.

• Methods of Application

  F = W - w

  The apparent weight of the crown in water, F, equals its weight in air W minus the weight w of the water it displaces, which is the upward force of the water.

• Methods of Application

  w = W - F

  The weight of the displaced water w equals the weight of the crown in air W minus its apparent weight F in water; this is the same relation as F = W - w, rearranged.

• Methods of Application

  \frac{W}{w} = \frac{W}{W - F}

  The ratio of the weight of the crown to the weight of an equal volume of water equals W divided by W minus F, so it can be computed from two weighings; this ratio is the specific gravity of the metal.

Problems

No exercises in this chapter.