Sidebilder
PDF
ePub

299. If through the angles of a regular inscribed polygon tangents be drawn to the circle, they form a regular circumscribed polygon of the same number of sides.

For the line drawn through the point of contact of any tangent and the centre is a symmetrical axis of the figure (fig. 287), and therefore all the sides and all the angles of a circumscribed polygon are equal.

If tangents be drawn parallel to the sides of a regular inscribed polygon, they form a regular circumscribed polygon of the same number of sides. For a reason similar to that in the preceding proposition (fig. 290).

Hence, whenever we know how to inscribe a regular polygon in a circle, we are able also to describe about the same circle a regular polygon of the same number of sides.

To inscribe in and describe about a given circle a regular polygon of a given number of sides.

300. 1. The equilateral triangle. Describe a circle and draw a diameter AB (fig. 291). With the same

[blocks in formation]

radius and A as centre, mark off the points C and D, and join them with the point B. BCD is the inscribed equilateral triangle; for DC is the chord of an arc of 120°, CB and D B are therefore also each arcs of 120°.

2. The hexagon. Having described a circle, and drawn a diameter AB (fig. 292), with A and B as centres, and the same radius, describe two circles intersecting the first in C, D, E, F; these are the remaining angular points of the hexagon. Each of the sides A C, AD, BF, BE, subtends an arc of 60°, and therefore also C E and D F are arcs of 60°.

3. The dodecagon. Draw a diameter perpendicular to AB (fig. 292); it will bisect the arcs EC and F D. With the extremities of this diameter as centres, and the radius of the circle, describe two arcs; they will bisect the remaining arcs. Join these points, and the regular dodecagon will be inscribed; for it will readily be seen that each side subtends an arc of 30°. The construction therefore consists simply of circles described from the extremities of two diameters at right-angles, with radii equal to that of the given circle (fig. 293).

Fig. 293.

4. The square. Draw two diameters at right-angles (fig. 294). Join their extremities, and a square will be inscribed; for each side will subtend an arc of 90°.

5. The octagon. Draw through the centre lines parallel to the sides of the square; the extremities of these diameters with the vertices of the square will be

Fig. 294.

Fig. 295.

the eight angular points of a regular octagon, for each side subtends an arc of 45°. The regular circumscribed octagon is most easily obtained by describing four arcs from the angles of a circumscribed square, passing through the centre of the inscribed circle (fig. 295).

6. The decagon.

Divide the radius in extreme

and mean ratio, and take upon the circumference ten times consecutively, the greater segment A B (fig. 296): the tenth point will coincide with A, and the lines joining the points of division will form a regular decagon, for each side will subtend an arc of 36°.

7. The pentagon. The same construction gives AC, the side of the regular inscribed pentagon.

C

Fig. 296.

B

This figure may be directly constructed in the following manner :— -Draw two diameters A B, O C, perpendicular to each other (fig. 297). Take the middle M of OA, and from M as centre, describe the arc CD; O D will be the side of the decagon, C D that of the pentagon. For it may be demonstrated that a right-angled triangle having for the sides containing the right-angle the radius

A

E

B

M

Fig. 297.

and the side of the decagon, has the side of the pentagon for hypothenuse.

8. The quindecagon. The angle at the centre of a quindecagon is 24°; that is, the difference between the angle 60° at the centre of a hexa

gon, and the angle 36° at the centre of a decagon. Having described the circle (fig. 298), mark off the chord AB, equal to the radius, and from A mark off AC the side of the decagon; the chord B C subtending the difference

Fig. 298.

of the two arcs is the side of the quindecagon.

P

To inscribe by means of the auxiliary curve a regular polygon which is not readily inscribed geometrically.

301. Upon a straight line take equidistant points (fig. 299); number them in order, and at each point erect a perpendicular to this straight line. Upon perpen

[merged small][merged small][ocr errors][merged small]

dicular 3, take the side of an inscribed equilateral triangle; upon 4, that of an inscribed square; upon 5, that of the pentagon; and so on with the divisions, 8, 10, 12, 15, 16, etc.

Pass through these perpendiculars a curved line; the part intercepted upon the perpendicular 7 will be the side of the polygon of seven sides inscribed in the same circle; the part intercepted upon No. II would be the side of a polygon of eleven sides, and so on.

302. Having inscribed a polygon of a certain number of sides, we may easily proceed to the polygon with twice that number of sides, so that we have found methods of inscribing or describing about a circle,— Ist. A polygon of 3, 4, 5, 15 sides.

2nd. A polygon of 6, 8, 10, 30, or of 12, 16, 20, 60 sides, etc.

3rd. A polygon of any number of sides not included amongst the preceding.

The first two are solved by a direct geometrical process; the third by means of an auxiliary curve drawn by hand.

Application to the graduation of circles and verniers, the drawing of toothed wheels, pinions, etc. 303. All instruments used for measuring angles have a circular vernier, or limb, generally made of metal,

and divided according to its size into degrees and parts of a degree. Such is the graphometer (fig. 300), used in land surveying for.measuring angles. LL is the limb, A B a fixed cross-staff, CD a movable one. The graduated limb, or scale, gives the magnitude of the angle D O B.

[graphic][subsumed][merged small]

For astronomical observations, where great precision is necessary, the limbs are very large, and are sub

« ForrigeFortsett »