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3. If two chords of a circle intersect, the product of the segments of either one is equal to the product of the segments of the other.

4. The perpendicular from any point on a circle to a diameter of the circle is the mean proportional between the segments of the diameter.

5. If from a point outside a circle a secant and a tangent are drawn, the tangent is the mean proportional between the secant and its external segment.

6. If from a point outside a circle two or more secants are drawn, the product of any secant and its external segment is equal to the product of any other secant and its external segment.

24. Similar Figures. 1. Similar polygons are polygons which have their corresponding angles equal and their corresponding sides proportional.

2. Two mutually equiangular triangles are similar.

3. If two angles of one triangle are equal respectively to two angles of another, the triangles are similar.

4. If two triangles have an angle of one equal to an angle of the other and the including sides proportional, the triangles are similar.

5. If two triangles have their sides respectively proportional, they are similar.

6. If two triangles have their sides respectively parallel or respectively perpendicular to one another, the triangles are similar.

7. The perpendicular from the vertex of the right angle of a right triangle to the hypotenuse divides the triangle into two triangles which are similar to the given triangle and to each other.

8. If two polygons are similar, they can be separated into the same number of triangles, similar each to each and similarly placed, and conversely.

9. The perimeters of two similar polygons have the same ratio as any two corresponding sides.

10. Two regular polygons of the same number of sides are similar.

25. Areas. 1. The area of a rectangle is the product of the base and the altitude.

2. The area of a parallelogram is the product of the base and the altitude.

3. The area of a triangle is half the product of the base and the altitude.

4. The area of a trapezoid is half the product of the altitude and the sum of the bases.

5. The area of a regular polygon is half the product of its apothem and its perimeter.

The apothem is the perpendicular from the center to a side.

6. The area of a circle is Tr2.

7. The area of a sector of a circle is half the product of the radius and the arc.

26. Ratios of Areas. 1. Rectangles with equal bases are to each other as their altitudes; rectangles with equal altitudes are to each other as their bases; any two rectangles are to each other as the products of their bases and altitudes; and similarly for parallelograms and triangles.

2. If an angle of one triangle is equal to an angle of another, the triangles are to each other as the products of the sides forming the equal angles.

3. The areas of two similar polygons are to each other as the squares of any two corresponding sides; and if the polygons are regular, to the squares of the radii of either the inscribed or the circumscribed circles.

4. The areas of two circles are to each other as the squares of the radii.

27. Regular Polygons and the Circle. 1. A circle can be inscribed in any regular polygon.

2. A circle can be circumscribed about any regular polygon.

28. Principles of Limits. 1. If a variable x approaches a finite limit l, and if c is a constant, then cx approaches the limit cl, and 2 approaches the limit.

с

C

When a variable (a quantity which may have different successive values) so approaches a constant (a quantity which has a fixed value) that the difference between the two may become and remain less than any assigned positive quantity, however small, the constant is the limit of the variable.

2. If, while approaching their respective limits, two variables are always equal, their limits are equal.

29. The Circle as a Limit. 1. The circumference of a circle is the limit of the perimeter of a regular inscribed or of a regular circumscribed polygon as the number of sides is indefinitely increased.

2. The area of a circle is the limit of the area of a regular inscribed or of a regular circumscribed polygon as the number of sides is indefinitely increased.

3. If the number of sides of a regular inscribed polygon is indefinitely increased, the apothem of the polygon approaches the radius of the circle as its limit.

II. LINES AND PLANES

30. Representation of Figures in Solid Geometry. A plane (§ 3, 9) has no thickness, and is understood to be indefinite in extent. Obviously we cannot draw a figure which shall represent such a surface; and hence when we wish to show a plane, we may represent it by a thin rectangular solid seen obliquely, as in any of the three ways shown below.

Since the student has been accustomed to seeing only plane figures, the drawing of a solid figure in the flat is confusing at first. The best way to meet this difficulty is to have a few pieces of cardboard, a few knitting needles filed to sharp points, a pine board about a foot square, and some small corks. The cardboard can be used to illustrate planes, and can be arranged to show parallel planes, perpendicular planes, or planes intersecting obliquely. The knitting needles may be stuek into the board to illustrate lines perpendicular or oblique to a plane. If two or more lines are to meet in a point, the needles may be held together by sticking them into one of the small corks.

The figures given in the text can also be illustrated in this manner. Such homely apparatus, which costs almost nothing and is put together in class, seems much more real and is usually more satisfactory than the models which are sold by dealers.

To have a model for each proposition, or even to have a photograph or a stereoscopic picture, is, however, a poor educational policy. The student must learn very early to visualize a solid from the flat figure in outline, just as a builder or a mechanic learns to read his working drawings. Teachers will find that the drawing of the different projections of a solid, as required in connection with the work on practical mensuration (§§ 259-266), is particularly helpful in this respect.

31. Postulates of Planes. Just as in plane geometry we assume certain postulates upon which to build the proofs of the propositions, we now assume certain postulates respecting planes. The following are the ones needed in the elementary part of solid geometry:

1. Two intersecting straight lines determine a plane.

Although the plane p may turn about one of its lines AB, as shown in the upper figure, and occupy any number of positions, as p', p', ..., it cannot turn if it must also pass through an intersecting line CD, as shown in the lower figure. In other words, the intersecting lines AB and CD determine the plane p.

This postulate may be taken to include the following statements:

B

p

B

p

A

p

A straight line and a point not on the line determine a plane.

For example, the line AB and the point C in the second figure are sufficient to determine the plane p.

Three points not in a straight line determine a plane.

If two of them are connected with the third, we have the case of the postulate as first stated.

Two parallel straight lines determine a plane.

By definition (§ 9, 1) they must lie in a plane, and one of the parallels and any point on the other determine the plane.

Any one of the four statements given above may be referred to as § 31, 1.

2. If two planes have one point in common, they have at least one other point in common.

It is evident that they must then coincide or else that they must intersect in a straight line,

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