90. Definition. The surfaces described by the arcs A B, BC, CD, &c. are called zones.

91. The area of a zone is equal to the product of its altitude by the circumference of a great circle.

92. Zones on the same or equal spheres are as their altitudes.

93. The surface of a sphere is four times the surface of one of its great circles. (62; III. 32.)

94. Definition. A polyedron is circumscribed about a sphere when its faces are each tangents to the sphere. In this case the sphere is inscribed in the polyedron.

95. The surface of a sphere is equal to the convex surface of the circumscribed cylinder. (62; 15.)

96. Definition. A Spherical Sector is the solid described by any sector of a semicircle as the semicircle revolves about its diameter.

97. The volume of a spherical sector is equal to the product of the surface of the zone forming its base by one third of the radius of the sphere of which it is a part.

98. A Spherical Segment is a part of a sphere included by two parallel planes cutting or touching the sphere. When one plane touches and one cuts the sphere, the spherical segment is called a spherical segment of one base; when both cut, a spherical segment of two bases.

99. How can the volume of a spherical segment of one base be found? A spherical segment of two bases?

100. A sphere is two thirds of the circumscribed cylinder.

101. A cone, hemisphere, and cylinder having equal bases and the same altitude are as the numbers 1, 2, 3.

BOOK VI.

PROBLEMS OF CONSTRUCTION.

IN the preceding demonstrations we have assumed that our figures were already constructed. The Problems of Construction given in this Book depend for their solution upon the principles of the preceding Books. In some of the problems the construction and demonstration are given in full; in others the construction is given and the propositions necessary to prove the construction referred to in the order in which they are to be used, and the pupil must complete the demonstration. In a few instances references are made to the Exercises appended to the previous Books. In such cases either the propositions to which reference is made can be demonstrated or the problem omitted.

PROBLEM I.

1. To bisect a given straight line.

From

Let A B be the given straight line. A and B as centres with a radius greater than half of A B, describe arcs cutting one another at C and D; join C and D cutting A B at E, and the line A B is bisected at E. For C and D being each equally distant from A and B, the line CD must be perpendicular to AB at its middle point (converse of I. 53).

PROBLEM II.

2. From a given point without a straight line to draw a perpendicular to that line.

Let C be the point and A B the line.
From C as a centre describe an arc

cutting A B in two points E and F; with

E and F as centres, with a radius greater Athan half E F, describe arcs intersecting

at D.

Draw CD, and it is the perpen

dicular required (converse of I. 53).

PROBLEM III.

3. From a given point in a straight line to erect a perpendicu lar to that line.

Let C be the given point and AB the given line.

With C as a centre describe an arc cutting A B in D and E; with D and E

F

X

B

C

E

as centres, with a radius greater than A DC, describe arcs intersecting at F. Draw CF, and it is the perpendicular required (converse of

I. 53).

G

Second Method. With C as a centre describe an arc DEF; take the distances DE and EF equal to CD, and from E and Fas centres, with a radius greater than half the distance from E to F, describe arcs intersecting at G. Draw CG, and it is the perpendicular required (III. 33; III. 16; III. 15).

Α

D

E

F

-B

C

ap.

Third Method. With any point, D, without the line A B, with a radius equal to the distance from D to C, describe an arc cutting A B at E; draw the diameter EDF. Draw CF, and it is the perpendicular required (III. 23).

PROBLEM IV.

4. To bisect a given arc, or angle.

1st. Let AB be the given arc. Draw the chord AB and bisect it with a perpendicular (1; III. 16).

2d. Let C be the given angle.

With C as a centre describe an arc cutting the sides of the angle in A and B; bisect the arc A B with the line CD, and it will also bisect the angle C (III. 11).

PROBLEM V.

C

B

5. At a given point in a straight line to make an angle equal to a given angle.

Let A be the given point in the line A B, and C the given angle. With C as a centre describe an arc DE cutting the sides of the angle C; with A as a centre, with the same radius, describe an arc; with F as a centre, with a radius equal to the distance from D to E, describe an arc cutting the arc FG. Draw A G.

The angle A

G

F

E

D

= C (III. 12; III. 11).

PROBLEM VI.

6. Through a given point to draw a line parallel to a given straight line.

7. Two angles of a triangle given, to find the third.

8. The three sides of a triangle given, to construct the triangle.

C

Take A B equal to one of the given sides; with A as a centre, with a radius equal to another of the given sides, describe an arc, and with B as a centre, with a radius equal to the remaining side, describe an arc intersecting the first arc at C. Draw AC and CB, and A CB is evidently the triangle required.

A

B