VARIABLES. LIMITS

Excerpts

• VARIABLES. LIMITS

  Now the greater the number of sides the more nearly does the apothem equal the radius in length.

• VARIABLES. LIMITS

  For *practical purposes* the lengths of such segments are approximated to any desired degree of accuracy, and their ratios are understood to be the ratios of these approximate numerical measures.

• VARIABLES. LIMITS

  Not every infinite sequence serves to single out a definite point in the manner shown above.

• VARIABLES. LIMITS

  That is, $1$ is the *smallest number* beyond which the sequence does not go.

• VARIABLES. LIMITS

  The Greeks dealt with the incommensurable cases in an interesting manner. Thus, to prove that two incommensurable ratios are equal they showed that neither can be less than the other.

• VARIABLES. LIMITS

  Let $a_1$, $a_2$, $a_3$, $\dotsc$ be a sequence of rational numbers whose limit is the side $a$.

• VARIABLES. LIMITS

  But the limit of this sequence is by definition the product of the limits of the three sequences $a_1$, $a_2$, $a_3$, $\dotsc$, $b_1$, $b_2$, $b_3$, $\dotsc$, $c_1$, $c_2$, $c_3$, $\dotsc$, or $abc$.

• VARIABLES. LIMITS

  That $U = V$ follows from the fact that the sum of the volumes of the circumscribed prisms exceeds that of the inscribed prisms by the volume of the lowest circumscribed prism, and this may be made as small as we please.

• VARIABLES. LIMITS

  The proof of this general theorem is more difficult than any thus far given, inasmuch as it involves sequences which *oscillate*; that is, which are neither constantly increasing nor constantly decreasing.

• VARIABLES. LIMITS

  Indeed, this theorem and also that of § [unit:437.]437 are special cases of what is known as **Cavalieri’s Theorem**.

• VARIABLES. LIMITS

  Hence, by § [unit:428.]428, $V = \tfrac{1}{3}rS$. But by § [unit:443.]443, $V = \tfrac{4}{3} \pi r^3$.

Equations

• VARIABLES. LIMITS

  a = bh

  The area of a rectangle equals its base times its altitude.

• VARIABLES. LIMITS

  d = \sqrt{2}

  The diagonal of a square whose side is unity has length the square root of 2.

• VARIABLES. LIMITS

  \dfrac{AD}{AB} = \dfrac{AE}{AC}

  A line DE parallel to side BC of triangle ABC, cutting sides AB and AC, divides them in the same ratio.

• VARIABLES. LIMITS

  \dfrac{k^2A}{A} = k^2 = \dfrac{r'^2}{r^2}

  For two circles whose radii are r and r' with r' = kr, the ratio of their areas equals the square of the ratio of their radii, k squared.

• VARIABLES. LIMITS

  V = \tfrac{4}{3} \pi r^3

  The volume of a sphere of radius r is four thirds of pi times r cubed.

• VARIABLES. LIMITS

  V = \tfrac{1}{3}rS

  The volume of a sphere equals one third of its radius times its surface area, by the limit of the volumes of the circumscribed polyhedrons.

• VARIABLES. LIMITS

  S = \tfrac{3}{r} \cdot \tfrac{4}{3} \pi r^3 = 4 \pi r^2

  The surface of a sphere of radius r equals four pi times r squared, obtained by substituting the volume formula into the relation V = r S / 3.

• VARIABLES. LIMITS

  U = V

  Defining the volume of a pyramid by circumscribed prisms gives the same limit as defining it by inscribed prisms.

• VARIABLES. LIMITS

  p = p'

  The perimeter of a convex closed curve is the same whether it is found as the limit of inscribed polygon perimeters or of circumscribed polygon perimeters.

• VARIABLES. LIMITS

  A = A'

  The area of a convex closed curve is the same whether it is found as the limit of inscribed polygon areas or of circumscribed polygon areas.

Problems

No exercises in this chapter.