OF THE MENSURATION OF THE CIRCLE

Excerpts

OF THE MENSURATION OF THE CIRCLE
A magnitude or ratio, which is fixed in value by the conditions of the question, is called a Constant.
OF THE MENSURATION OF THE CIRCLE
If two variables are at every instant equal their limits are equal.
OF THE MENSURATION OF THE CIRCLE
Then if the number of sides of the polygons increase these two ratios vary but remain always equal to each other, therefore (Lemma) their limits are equal.
OF THE MENSURATION OF THE CIRCLE
The area of any circular sector is half the rectangle contained by its arc and the radius of the circle.
OF THE MENSURATION OF THE CIRCLE
Let a number of points be taken in a terminated curve line, and let straight lines be drawn from each point to the next, then if the number of points be conceived to increase and the distance between each two to diminish continually, the extremities remaining fixed, the limit of the sum of the straight lines is called the Length of the Curve.
OF THE MENSURATION OF THE CIRCLE
By the method of “continued fractions” it will be found that $\dfrac{22}{7}$ and $\dfrac{355}{113}$ are nearer approximations to the value of $\pi$ than any simpler fractions.
OF THE MENSURATION OF THE CIRCLE
Of these $\dfrac{22}{7}$ ($=3.14$) is the approximation discovered by Archimedes (killed, it is said, at the siege of Syracuse,

Equations

OF THE MENSURATION OF THE CIRCLE
\text{the circumference $÷$ the diameter $= \pi$,}
The ratio of the circumference of any circle to its diameter is one fixed number, denoted π.
OF THE MENSURATION OF THE CIRCLE
\dfrac{\pi}{2}
The ratio of the semi-circumference to the diameter, common to all circles, is denoted π/2 as is customary.
OF THE MENSURATION OF THE CIRCLE
\text{and the circumference $= 2\pi R$.}
The circumference of a circle equals 2π times its radius.
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AOC : AOB :: AC : AB
Angles at the centre of a circle are in the same ratio as the arcs they subtend (Euclid VI. 33).
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AOC = \frac{2}{\pi} × \text{a right angle},
The angle subtended at the centre by an arc equal to the radius is 2/π of a right angle, the same fixed fraction for all circles.
OF THE MENSURATION OF THE CIRCLE
\dfrac{1}{2}\, R × \text{circumference} = \pi R^2
The area of a circle is half the radius times the circumference, which equals π times the square of the radius.
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= \dfrac{\pi}{4}
The ratio of the area of a circle to the square on its diameter is π/4.
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r' = \frac{R + r}{2}
The inscribed radius of the second polygon (with twice the sides) is the arithmetic mean of the inscribed and circumscribed radii of the first polygon.
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R' = \sqrt{r' · R}
The circumscribed radius of the second polygon is the geometric mean of its inscribed radius and the circumscribed radius of the first polygon.
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EC: EF:: EF: EG
The circumscribed radius of the second polygon is a mean proportional between the circumscribed radius of the first polygon and the inscribed radius of the second.
OF THE MENSURATION OF THE CIRCLE
\dfrac{1}{3} (r + 2R)
When R and r nearly agree, (r + 2R)/3 is a very close approximation to the common limit of the two radii.
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\pi = \frac{20000000000}{6366197723}
π may be taken as 20000000000/6366197723, which equals 3.141592654 to the stated accuracy.
OF THE MENSURATION OF THE CIRCLE
\frac{2000000}{636621} < \pi < \frac{2000000}{636617}
Stopping at the 1024-sided polygon, π is bracketed between these two fractions (the transcription also carries a differing DPchg form, π < 2000000/636617 > 2000000/636621, which is flagged for review).

Problems

No exercises in this chapter.