THE Fourth Book of the Elements contains some particular cases of four general problems on the inscription and the circumscription of triangles and regular figures in and about circles. Euclid has not given any instance of the inscription or circumscription of rectilineal figures in and about other rectilineal figures. Any rectilineal figure, of five sides and angles, is called a pentagon; of seven sides and angles, a heptagon; of eight sides and angles, an octagon; of nine sides and angles, a nonagon; of ten sides and angles, a ḍecagon; of eleven sides and angles, an undecagon; of twelve sides and angles, a duodecagon; of fifteen sides and angles, a quindecagon, &c. These figures are included under the general name of polygons; and are called equilateral, when their sides are equal; and equiangular, when their angles are equal; also when both their sides and angles are equal, they are called regular polygons. Prop... An objection has been raised to the construction of this problem. It is said that in this and other instances of a similar kind, the lines which touch the circle at A, B, and C, should be proved to meet one another. This may be done by joining AB, and then since the angles KAM, KBM are equal to two right angles (III. 18.), therefore the angles BAM, ABM are less than two right angles, and consequently (ax. 12.), AM and BM must meet one another, when produced far enough. Similarly, it may be shewn that AL and CL, as also CN and BN meet one another. Prop. IV. This Problem is one case of the more general Problem :-To describe a circle which shall touch three straight lines. If the triangle be equilateral, the center of the inscribed circle is equidistant from the three angular points of the triangle. The centers of the circles inscribed in, and circumscribed about an equilateral triangle coincide, and the radius of one is double the radius of the other. Prop. v. is the same as "To describe a circle passing through three given points, provided that they are not in the same straight line." The corollary to this proposition appears to have been already demonstrated in Prop. 31. Book III. It is obvious that the square described about a circle is equal to double the square inscribed in the same circle. Also that the circumscribed square is equal to the square on the diameter, or four times the square on the radius of the circle. Prop. VII. It is manifest that a square is the only right-angled parallelogram which can be circumscribed about a circle, but that both a rectangle and a square may be inscribed in a circle. Prop. x. By means of this proposition, a right angle may be divided into five equal parts. Prop. XVI. The arc subtending a side of the quindecagon, may be found by placing in the circle from the same point, two lines respectively equal to the sides of the inscribed regular hexagon and pentagon. The centers of the inscribed and circumscribed circles of any regular polygon are coincident. Besides the circumscription and inscription of triangles and regular polygons about and in circles, some very important problems are solved in the constructions respecting the division of the circumferences of circles into equal parts. By inscribing an equilateral triangle, a square,. a pentagon, a hexagon, &c. in a circle, the circumference is divided into three, four, five, six, &c. equal parts. In Prop. 26, Book III., it has been shewn that equal angles at the centers of equal circles, and therefore at the center of the same circle, subtend equal arcs; by bisecting the angles at the center, the arcs which are subtended by them are also bisected, and hence, a sixth, eighth, tenth, twelfth, &c. part of the circumference of a circle may be found. If the right angle be considered as divided into 90 degrees, each degree into 60 minutes, and each minute into 60 seconds, and so on, according to the sexagesimal division of a degree; by the aid of the first corollary to Prop. 32, Book 1., may be found the numerical magnitude of an interior angle of any regular polygon whatever. Let ℗ denote the magnitude of one of the interior angles of a regular polygon of n sides, then no is the sum of all the interior angles. But all the interior angles of any rectilineal figure together with four right. angles, are equal to twice as many right angles as the figure has sides, that is, if π be assumed to designate two right angles, the magnitude of an interior angle of a regular polygon of n sides. By taking n = 3, 4, 5, 6, &c. may be found the magnitude in terms of two right angles, of an interior angle of any regular polygon whatever. Pythagoras was the first, as Proclus informs us in his commentary, who discovered that a multiple of the angles of three regular figures only, namely, the trigon, the square, and the hexagon, can fill up space round a point in a plane. It has been shewn that the interior angle of any regular polygon of n sides in terms of two right angles, is expressed by the equation 3 n Let 0, denote the magnitude of the interior angle of a regular figure of three sides, in which case, n = 3. that is, six angles, each equal to the interior angle of an equilateral triangle, are equal to four right angles, and therefore six equilateral triangles may be placed so as completely to fill up the space round the point at which they meet in a plane. In a similar way, it may be shewn that four squares and three hexagons may be placed so as completely to fill up the space round a point in a plane. Also it will appear from the results deduced, that no other regular figures besides these three, can be made to fill up the space round a point; for any multiple of the interior angles of any other regular polygon, will be found to be in excess above, or in defect from four right angles. The equilateral triangle or trigon, the square or tetragon, the pentagon, and the hexagon, were the only regular polygons known to the Greeks, capable of being inscribed in circles, besides those which may be derived from them. M. Gauss in his Disquisitiones Arithmeticæ, has extended the number by shewing that in general, a regular polygon of 2" + 1 sides is capable of being inscribed in a circle by means of straight lines and circles, in those cases in which 2" + 1 is a prime number. The case in which n = 4, in 2" + 1, was proposed by Mr. Lowry of the Royal Military College, to be answered in the seventeenth number of Leybourn's Mathematical Repository, in the following form: Required a geometrical demonstration of the following method of constructing a regular polygon of seventeen sides in a circle : Draw the radius CO at right angles to the diameter AB; on OC and OB, take OQ equal to the half, and OD equal to the eighth part of the radius; make DE and DF each equal to DQ, and. EG. and FH respectively equal to EQ and FQ; take OK a mean proportional between OH and OQ, and through K, draw KM parallel to AB, meeting the semicircle described on OG in M, draw MN parallel to OC cutting the given circle in N, the arc AN is the seventeenth part of the whole circumference. A demonstration of the truth of this construction has been given by Mr. Lowry himself, and will be found in the fourth volume of Leybourn's Repository. The demonstration including the two lemmas occupies more than eight pages, and is by no means of an elementary character. QUESTIONS ON BOOK IV. 1. WHAT is the general object of the Fourth Book of Euclid? 2. What consideration renders necessary the first proposition of the Fourth Book of Euclid? 3. When is a circle said to be inscribed within, and circumscribed about a rectilineal figure ? 4. Shew that a circle cannot be described about a rhombus. Can a circle be inscribed in a rhombus ? 5. When is one rectilineal figure said to be inscribed in, and circumscribed about another rectilineal figure? 6. Modify the construction of Euc. IV. 4, so that the circle may touch one side of the triangle and the other two sides produced. 7. The sides of a triangle are 5, 6, 7 units-respectively; find the radii of the inscribed and circumscribed circles. 8. Give the constructions by which the centers of circles described about, and inscribed in triangles are found. In what triangles will they coincide?. 9. How is it shewn that the radius of the circle inscribed in an equilateral triangle, is half the radius described about the same triangle?. 10. The equilateral triangle inscribed in a circle is one-fourth of the equilateral triangle circumscribed about the same circle. 11. If a circle be inscribed in an equilateral triangle, the triangle formed by joining the points of contact will also be equilateral. 12. What relation subsists between the square inscribed in, and the square circumscribed about the same circle? 13. Enunciate Euc. III. 22: and extend this property to any inscribed polygon having an even number of sides. 14. If any triangle be inscribed in a circle, the sum of the three angles in the segments cut off by the sides, is equal to four right angles.. 15. Trisect a quadrantal arc of a circle, and show that every arc which th part of a quadrantal arc, may be trisected geometrically: m and n being whole numbers. 16. If one side of a quadrilateral figure inscribed in a circle be produced, the exterior angle is equal to the interior and opposite angle of the figure. Is this property true of any inscribed polygon having an even number of sides? 17. In what parallelograms can circles be inscribed ? 18. What Geometrical Problem must be solved as a condition precedent to the construction of a regular pentagon? 19. Shew that in the figure Euc. Iv. 10, there are two triangles possessing the required property. 20. How is it made to appear that the line BD is the side of a regular decagon inscribed in the larger circle, and the side of a regular pentagon inscribed in the smaller circle? fig. Euc. IV. 10. 21. In the construction of Euc. IV. 3, Euclid has omitted to shew that the tangents drawn through the points A and B will meet in some point M. How may this be shewn? 22. Shew that if the points of intersection of the circles in Euclid's figure, Book IV. Prop. 10, be joined with the vertex of the triangle and with each other, another triangle will be formed 'equiangular and equal to the former. 23. Divide a right angle into five equal parts.. How may an isosceles triangle be described upon a given base, having each angle at the base onethird of the angle at the vertex ? 24. What regular figures may be inscribed in a circle by the help of Euc. Iv. 10? 25. The difference of the squares described on the straight lines joining the extremities of the base of the constructed triangle in the figure of Euc. 1v. 10, with the other point of intersection of the circles, is equal to the square on the side of the triangle. 26. What is Euclid's definition of a regular pentagon? Would the stellated figure, which is formed by joining the alternate angles of a regular pentagon, as described in the Fourth Book, satisfy this definition ? 27. Shew that each of the interior angles of a regular pentagon inscribed in a circle, is equal to three-fifths of two right angles. 28. If two sides not adjacent, of a regular pentagon, be produced to meet; what is the magnitude of the angle contained at the point where they meet? 29. Is there any method more direct than Euclid's for inscribing a regular pentagon in a circle? 30. State briefly the mode of describing an equilateral and equiangular hexagon about a given circle. 31. In what sense is a regular hexagon also a parallelogram? Would the same observation apply to all regular figures with an even number of sides? 32. Why has Euclid not shewn how to inscribe an equilateral triangle in a circle, before he requires the use of it in Prop. 16, Book Iv.? 33. An equilateral triangle is inscribed in a circle by joining the first, third, and fifth angles of the inscribed hexagon. 34. If the sides of a hexagon be produced to meet, the angles formed by these lines will be equal to four right angles. 35. Shew that the area of an equilateral triangle inscribed in a circle, is onehalf of a regular hexagon inscribed in the same circle. 36. If a side of an equilateral triangle be six inches: what is the radius of the inscribed circle? 37. Find the area of a regular hexagon inscribed in a circle whose diameter is twelve inches. What is the difference between the inscribed and the circumscribed hexagon? 38. Which is the greater, the difference between the side of the square and the side of the regular hexagon inscribed in a circle whose radius is unity; or the difference between the side of the equilateral triangle and the side of the regular pentagon inscribed in the same circle? 39. The regular hexagon inscribed in a circle, is three-fourths of the regular circumscribed hexagon. 40. Assuming the construction of Euc. Iv. 6; how may a regular octagon be inscribed in a circle? 41. The sum of the squares on the three diagonals of a regular hexagon that terminate in one angular point is equal to ten times the square on one side. 42. All the interior angles of an octagon are equal to twelve right angles. 43. What figure is formed by the production of the alternate sides of a regular octagon ? ་་ 44. How many square inches are in the area of a regular octagon whose side is eight inches? 45. If an irregular octagon be capable of having a circle described about it, shew that the sums of the angles taken alternately are equal. 46. Find an algebraical formula for the number of degrees contained by an interior angle of a regular polygon of n sides. 47. What are the three regular figures which can be used in paving a plane area? Shew that no other regular figures but these will fill up the space round a point in a plane. 48. Into what number of equal parts may a right angle be divided geometrically? What connection has the solution of this problem with the possibility of inscribing regular figures in círcles? 49. Assuming the demonstrations in Euc. IV, shew that any equilateral figure of 3.2", 4.2′′, 5.2", or 15.2" sides may be inscribed in a circle, when n is any of the numbers, 0, 1, 2, 3, &c. 50. With a pair of compasses only, shew how to divide the circumference of a given circle into twenty-four equal parts. 51. Shew that if any polygon inscribed in a circle be equilateral, it must also be equiangular. Is the converse true? 52. The area of a regular polygon of n sides is equal to n times the area of the triangle whose base is a side of the polygon and altitude equal to the radius of the inscribed circle. 53. Shew that if the circumference of a circle pass through three angular points of a regular polygon, it will pass through all of them. 54. Similar polygons are always equiangular: is the converse of this proposition true? 55. What are the limits to the Geometrical inscription of regular figures in circles? What does Geometrical mean when used in this way? 56. What is the difficulty of inscribing geometrically an equilateral and equiangular undecagon in a circle? Why is the solution of this problem said to be beyond the limits of plane geometry? Why is it so difficult to prove that the geometrical solution of such problems is impossible? |