III. If two variables are proportional to two constants, their limits are proportional to the same constants.

case of Th. V, Bk. II, will be taken as an example of the method to be applied to all theorems having the two cases. Let the central angles AOB and COD in circle o have no common divisor. Suppose an exact divisor of AOB to be applied to COD as often as possible, leaving a remainder XOD, which is therefore less than the divisor used.

Then AOB and Cox have a common divisor, so ZAOB AB

=

LCOX CX

by the commensurable case.

But, if the divisor of AOB is taken smaller and smaller, that is, is made to approach the limit zero, the remainder XOD, being still smaller, will also approach the limit zero. Therefore Cox will approach the limit Z COD. Also its arc, cx, will approach the limit CD.

But, since the variables / Cox and CX are proportional to the constants / AOB and AB, their limits are also proZAOB АВ

portional to those constants; that is,

Z COD

=

CD

This method of proof will apply to Th. II, Bk. III, to Th. I, Bk. IV, and in fact to all proofs where the method of a common divisor is used.

349. Similar Figures.

Similar figures have already been defined (§ 282) as those which have their corresponding angles equal, and their corresponding sides proportional. This, of course, applies only to polygons. The following definition is sometimes given, and while it is not as convenient for use in Plane Geometry, it has the merit of applying to all kinds of figures, including those of Solid Geometry.

Two figures that may be placed in a pencil (or sheaf, in Solid Geometry) of lines, so that all pairs of corresponding points of the figures cut the respective lines of that pencil in the same ratio, are similar.

These two definitions are identical in result, as far as polygons are concerned, as may be shown by proving the two following statements:

1. If two polygons that are mutually equiangular and have their corresponding sides proportional are placed with one pair of corresponding sides parallel, the polygons lying on the same side of those lines, the lines joining their corresponding vertices will form a pencil which is cut proportionally by the vertices of the polygons.

2. If two polygons lie in a pencil of lines, and their vertices cut the lines proportionally, the polygons are mutually equiangular, and their corresponding sides are proportional.

350. The Evaluation of Pi. The ratio of the circumference of a circle to its diameter is obtained by the use of the ratios of the perimeters of regular inscribed and circumscribed polygons of a large number of sides to the diameter.

The circumference is longer than the

perimeter of any inscribed polygon, and shorter than the perimeter of any circumscribed polygon, so that if the lengths of the perimeters of two regular polygons of the same number of sides, one inscribed, the other circumscribed, can be obtained in terms of the diameter, the length of the circumference will lie between these values, and an approximate value can therefore be obtained.

The value, or π, cannot be obtained exactly, for it has been proved to be an incommensurable number. It is evident, however, that the value obtained will be more exact as the polygons used have a larger number of sides.

The approximation is started by finding the perimeters of some regular polygon, inscribed and circumscribed; squares or hexagons are the easiest to use. The perimeters of polygons of twice the number of sides are then worked out, and the doubling is continued until the value is found to the required degree of accuracy. The value has been carried to over 700 decimal places, but for ordinary purposes the value 3.14159+ is sufficiently accurate. 3.1416 is also much used, and 34 is a fair approximation for rough work.

The following theorems give the material for the numerical calculation:

I. The perimeter of a regular inscribed hexagon equals 3D; the perimeter of a regular circumscribed hexagon equals 2 DV3.

II. If the perimeters of regular inscribed and circumscribed polygons of any number of sides are known, the perimeter of the regular circumscribed polygon of double the number of sides equals twice their product divided by their sum.

III. If the perimeters of the regular inscribed poly

gon of any number of sides, and the regular circumscribed polygon of double that number of sides, are known, the perimeter of the regular inscribed polygon having double the first number of sides equals the square root of their product.

Applying the formula in II to the regular hexagons, calling the perimeter of a regular circumscribed dodecagon P:

Applying the formula in III to this value and the perimeter of the regular inscribed hexagon, and calling the perimeter of the regular inscribed dodecagon p:

p =√3 D × 12 D(2 – √3) = 6 D√2 −√3 = (3.10583−)d.

Applying these methods to the dodecagons, the perimeters of circumscribed and inscribed regular polygons of 24 sides are, respectively, (3.15966-)D and (3.13263 ̄)d.

It is already evident that the circumference of the circle, since its value is between those found, must be (3.1)D, and a continuance of the work will determine additional figures.