F 57 polygons are regular hexagons (fig. 57). Then we say, in the first place, they are equiangular with respect to each other, for each of the angles in both polygons is equal to one sixth of eight right angles (86). Again their homologous sides are proportional. For, by the definition of regular polygons (85) A B=B C=C D, &c. and G H=A I=I K, &c. Therefore, whatever be the ratio of A B to G H, the same must be the ratio of B C to H I, of C D to I K, &c., that is,

ABGH BC: HI:: CD: IK, and so on round the figure. Hence the two polygons are similar. The same reasoning would apply to any other number of sides.

90. THEOREM.-Every regular polygon may be inscribed in a circle, and circumscribed about a circle. A polygon is said to be inscribed, when all its vertices are in the circumference, and to be circumscribed, when all its sides are tangents. DEM. 1.-Let there be a F 59 regular polygon A B C D E F (fig. 59). Find the centre I of a circle (29) to pass through the three points B, C, D. We say the same will pass through all the other vertices. First it will pass through E. Draw the chords B D'and C E. Then, by the definition, the triangles B C D and C D E are equal (53), and if CD were placed upon B C, D E would fall upon C D. Accordingly the same circle which passes through B, C, D, will also pass through C, D, E. The same reasoning will apply to F and A, and to any number of vertices. 2.-Secondly, we say that this polygon may be circumscribed. Draw I H perpendicular to the middle of A B (28) and I G perpendicular to the middle of A F. Describe a circle with the radius I H, and A B will be a tangent (40). Now we say that A F will be a tangent to the same circle. Because the two right triangles A I H and A I G are equal (59) since they have the hypothenuse A I common, and A H half of A B A G half of A F. Therefore I H=I G, and A Fis a tangent. The same might he proved in like manner of all the other sides. Thus whenever a regular polygon is given, there may be a circle circumscribed about it, and a circle inscribed in it, or, in the words of the enunciation, the polygon may be inscribed and circumscribed.

91. PROBLEM.-To inscribe a square in a given circle. We cannot solve the general problem, having a given circle, to inscribe in it a regular polygon of any number of sides, since we have no means of dividing the circumference of a circle into any given number of equal parts. But there are certain particular cases in which the solution is possible. We begin with the square. SoL.-Let the given circle be A B C D (fig.60). F 60 Draw two diameters perpendicular to each other, and join their extremities by chords. A B C D is (83), because its sides are equal (53) and it

right angles (42).

are

are

reg

92. PROBLEM.-To inscribe in a given can ular hexagon and an equilateral triangle. SoL.-Take the radius AO (fig. 61) in the

compasses, and apply F 61

it round the circumference. We say that it will be contained exactly six times, or, in other words, that the side of an inscribed hexagon is equal to radius. Since A OB O, the angle O A B=0 B A. Then, supposing A B to be the side of a regular hexagon, the angle A O B must be equal to 60°, since the arc A B is a sixth part of the whole circumference. Then the

If

angles O A BO B A must be equal to 120° (46),
and since they are equal, each must be 60°. There-
fore the triangle A O B is equilateral (51), and A B,
the side of a hexagon, is equal to the radius A O.
now we would inscribe an equilateral triangle, it is only
necessary to join the alternate vertices A, C, E. In-
deed we may remark generally that when any polygon
of an even number of sides is already inscribed, we
may always inscribe one of half the number of sides,
by joining the alternate vertices. Also, by bisecting
the arcs, whether an even number or not, and drawing
chords to the half arcs, we may always inscribe one of
double the number of sides.

93. PROBLEM. To inscribe in a given circle a regular polygon of ten and one of fifteen sides. SOL.-First, to inscribe one of ten sides. Divide the radius OA (fig. 62) in extreme and mean ratio F 62 (82). Let O M be the greater part. Take the chord A BO M, and apply it round the circle. We say it will be contained exactly ten times, or in other words that the arc A B is a tenth part of the circumference.

To prove this we need only show that the angle
A O B=36°. We have by construction,

AMMO:: MO: AO,

or, drawing B M and substituting A B for M O, AM:AB::AB: A O.

Then the triangles A M B and A O B are similar (77) having the angle A common, and the sides including it proportional. But A O B is isosceles. Therefore

N

A M B is also isosceles, and B M—A B=O M. This
makes O M B isosceles, and the angle M O B M B O.
A, being an exterior angle (49) is equal to
thek
posite interior angles M O B+M B O=
twice B. Then B A M B M A=twice A O B,
and O B AM A B-twice A O B. Hence all the
angles of the triangle A O B or 180° five times
А О В. Then A O B=one fifth of 180°=36°, and
A B is the side of a regular decagon. 2.-If now,
secondly, we wish to inscribe a regular polygon of 15
sides, we have only to find one fifteenth of a circumfe-
rence. For this purpose, let AL be the side of a
hexagon and A B that of a decagon. Then B L will
be the arc required, for B L=÷— of a circumfe-
rence, that is, one fifteenth. Lastly, by joining the
alternate vertices of a decagon we should have a penta-
gon; and by bisecting the arcs which are one fifteenth
and drawing chords, we should have a polygon of 30
sides, and so on indefinitely.

94. THEOREM.-The circle is a regular polygon of an infinite number of sides. DEM.-Inscribe in the F 63 circle (fig. 63) any one of the regular polygons before mentioned, for instance, a hexagon, as A B C D E F. Bisect the arcs B C, C D, &c., and join the half arcs by the chords B H, H C, C I, &c. Thus you have a regular polygon of 12 sides. Proceed in the same manner with this, and you have one of 24 sides, then one of 48 sides, and so on without limit. Now it is obvious that the polygon of 12 sides approaches nearer to a coincidence with the circle, than that of six sides. In the same manner the polygon of 24 sides, approaches nearer than that of 12, and the polygon of 48 sides approaches nearer than that of 24, and so on without a limit. But the difference between the first polygon and the circle is a finite or limited quantity, and we have seen that this difference constantly diminishes as we

increase the sides. Accordingly if the number of sides were increased to infinity, the difference would become nothing; for no one can doubt that the endless diminution of a limited quantity must bring it to nothing. Thus the polygon of an infinite number of sides would not differ from a circle. This idea of a circle agrees with the definition before given of a curved line (10) namely, that it is made up of infinitely small straight lines.

95. THEOREM.-The perimeters of regular polygons of the same number of sides are to each other as the radii of their circumscribed circles. By the perimeter of a polygon we mean the sum of its sides. Then we say that the perimeter A B C D E F (fig. 64), is to the perimeter G H I K L M as CN is to I 0. DEM.--Suppose the two polygons are hexagons. As they are similar (89) we have

BC HI:: C D ; IK,

For

Then (66)
6 times B C : 6 times HI:: CD: I K.
But 6 times B C is the perimeter of the first polygon,
and 6 times H I is the perimeter of the second. More-
over the triangles B N ̊C and H OI are similar.
the angle B N C=H O I since the arcs B C and H I
contain the same number of degrees, and BCN=
HIO (42) being inscribed in segments containing the
same number of degrees. Therefore

CD: IK :: CN: IO.

Accordingly by making the substitutions in the propor-
tion, 6 times BC: 6 times HI:: CD: IK,
we have the following;

perim. A B C D E F perim. G HIKLM::CN:10.
As the same reasoning might be employed for any other
number of sides than 6, the proposition is demonstrated.

96. THEOREM.-The circumferences of circles are to each other as their radii. DEM.-This follows directly from the two last propositions, for the circumfe rences of circles are the perimeters of regular polygons of an infinite, and therefore the same number of sides. Moreover the radii of the circumscribed circles become, in this case, the radii of the circles to be compared, the polygons being confounded with the circumscribed circles. COR. Similar arcs are to each other as their radii. By similar arcs we understand those

F 64

which contain the same number of degrees or measure equal angles at the centre. Now from the definition of a degree (15) such arcs are to each other as the circumferences of which they are a part. But these last are to each other as their radii. Therefore similar arcs

are to each other as their radii.

SECTION SECOND.

SURFACES.

97. DEF.-By the word Surface we understand, in the abstract, that magnitude which has length and breadth without thickness. But a more definite idea will be obtained if we introduce motion. Accordingly we may say-a surface is the space described by a line moving any other way than lengthwise. Thus we

have the origin of the two dimensions. For the line itself has one dimension, namely, length, and its motion makes another, namely, breadth. Speaking abstractly there is no thickness. But as you cannot make obvious to the senses, that which has absolutely no thickness, it is sufficiently near the truth to say-a surface has length and breadth with an infinitely small thickness. This is analogous to our definition of a line (2), for the infinitely small breadth and thickness of the moving line, would give an infinitely small thickness to the generated surface. Moreover as the boundaries of a line were points, so now, for a similar reason, the boundaries of a surface are lines.

98. DEF.-A plane surface is that with which a straight line will coincide in every direction. Thus, if we leave its thickness out of consideration, a sheet of paper perfectly smooth and even, may be taken to represent a plane surface, for in whatever direction we apply the straight edge of a rule to it, the rule will touch it in every point. Such a surface is usually designated by the word plane alone.-A polygonal surface is one which is composed of several planes. If a surface is neither plane nor composed of planes, it is a curved surface. But in order to give a definition which