and the angle FED stands upon the circumference FABCD, and the angle AFE upon EDCBA; therefore the angle AFE is equal to FED: (III. 27.) in the same manner it may be demonstrated, that the other angles of the hexagon ABCDEF are each of them equal to the angle AFE or FED: therefore the hexagon is equiangular; and it is equilateral, as was shewn; and it is inscribed in the given circle ABCDEF. Q.E.F. COR.-From this it is manifest, that the side of the hexagon is equal to the straight line from the center, that is, to the semi-diameter of the circle. And if through the points A, B, C, D, E, F there be drawn straight lines touching the circle, an equilateral and equiangular hexagon will be described about it, which may be demonstrated from what has been said of the pentagon: and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon. PROPOSITION XVI. PROBLEM. To inscribe an equilateral and equiangular quindecagon in a given circle. Let ABCD be the given circle. It is required to inscribe an equilateral and equiangular quindecagon in the circle ABCD. A B E F D Let A Cbe the side of an equilateral triangle inscribed in the circle, (Iv.2.) and AB the side of an equilateral and equiangular pentagon inscribed in the same; (IV. 11.) therefore, of such equal parts as the whole circumference ABCDF contains fifteen, the circumference ABC, being the third part of the whole, contains five; and the circumference AB, which is the fifth part of the whole, contains three; therefore BC, their difference, contains two of the same parts: bisect BC in E; (III. 30.) therefore BE, EC are, each of them, the fifteenth part of the whole circumference ABCD: therefore if the straight lines BE, EC be drawn, and straight lines equal to them be placed round in the whole circle, (IV. 1.) an equilateral and equiangular quindecagon will be inscribed in it. Q. E. F. And in the same manner as was done in the pentagon, if through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon will be described about it: and likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it. 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 decagon; 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. III. 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 KĀM, KBM are equal to two right angles (11. 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. 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. Reference has already been made to the distinction between analysis and synthesis, and that all Euclid's direct demonstrations are synthetic, properly so called. There is however a single exception in Prop. 16. Book IV, where the analysis only is given of the Problem. The two methods are so connected in all processes of reasoning, that it is very difficult to separate one from the other, and to assert that this process is really synthetic, and that is really analytic. In every operation performed in the construction of a problem, there must be in the mind a knowledge of some properties of the figure which suggest the steps to be taken in the construction of it. Let any Problem be selected from Euclid, and at each step of the operation, let the question be asked, “Why that step is taken?" It will be found that it is because of some known property of the required figure. As an example will make the subject more clear to the learner, the Analysis of Euc. Iv. 10, is taken from the "Analysis of Problems" in the larger edition of the Euclid, and to which the learner is referred for more complete information. In Euc. Iv. 10, there are five operations specified in the construction: (1) Take any straight line AB. (2) Divide the line AB in C, so that the rectangle AB, BC, may be equal to the square on AC. (3) Describe the circle BDE with center A and radius AB. (4) Place the line BD in that circle, equal to the line AC. (5) Join the points A, D. Why should either of these operations be performed rather than any others? And what will enable us to foresee that the result of them will be such a triangle as was required? The demonstration affixed to it by Euclid does undoubtedly prove that these operations must, in conjunction, produce such a triangle: but we are furnished in the Elements with no obvious reason for the adoption of these steps, unless we suppose them accidental. To suppose that all the constructions, even the simpler ones, are the result of accident only, would be supposing more than could be shewn to be admissible. No construction of the problem could have been devised without a previous knowledge of some of the properties of the figure. In fact, in directing the figure to be constructed, we assume the possibility of its existence; and we study the properties of such a figure on the hypothesis of its actual existence. It is this study of the properties of the figure that constitutes the Analysis of the problem. Let then the existence of a triangle BAD be admitted, which has each of the angles ABD, ADB double of the angle BAD, in order to ascertain any properties it may possess which would assist in the construction of such a triangle. Then, since the angle ADB is double of BAD, if we draw a line DC to bisect ADB and meet AB in C, the angle ADC will be equal to CAD; and hence (Euc. 1. 6.) the sides AC, CD are equal to one another. Again, since we have three points A, C, D, not in the same straight line, let us examine the effect of deseribing a circle through them: that is, describe the circle ACD about the triangle ACD. (Euc. Iv. 5.) Then, since the angle ADB has been bisected by DC, and since ADB is double of DAB, the angle CDB is equal to the angle DAC in the alternate segment of the circle; the line BD therefore coincides with a tangent to the circle at D. (Converse of Euc. 111. 32.) Whence it follows, that the reetangle contained by AB, BC, is equal to the square on BD. (Euc. II. 36.) But the angle BCD is equal to the two interior opposite angles CAD, CDA; or since these are equal to each another, BCD is the double of CAD, that is, of BAD. And since ABD is also double of BAD, by the conditions of the triangle, the angles BCD, CBD are equal, and BD is equal to DC, that is, to AC. It has been proved that the rectangle AB, BC, is equal to the square on BD; and hence the point C in AB, found by the intersection of the bisecting line DC, is such, that the rectangle AB, BC is equal to the square on AC. (Eue. 11. 11.) Finally, since the triangle ABD is isosceles, having each of the angles ABD, ADB double of the same angle, the sides AB, AD are equal, and hence the points B, D, are in the circumference of the circle described about A with the radius AB. And since the magnitude of the triangle is not specified, the line AB may be of any length whatever. From this "Analysis of the problem," which obviously is nothing more than an examination of the properties of such a figure supposed to exist already, it will be at once apparent, why those steps which are prescribed by Euclid for its construction, were adopted.. The line AB is taken of any length, because the problem does not prescribe any specific magnitude to any of the sides of the triangle. The circle BDE is described about A with the distance AB, because the triangle is to be isosceles, having AB for one side, and therefore the other extremity of the base is in the circumference of that circle. The line AB is divided in C, so that the rectangle AB, BC shall be equal to the square on AC, because the base of the triangle must be equal to the segment AC. And the line AD is drawn, because it completes the triangle, two of whose sides, AB, BD are already drawn. Whenever we have reduced the construction to depend upon problems which have been already constructed, our analysis may be terminated; as was the case where, in the preceding example, we arrived at the division of the line AB in C; this problem having been already constructed as the eleventh of the second book. 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 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 111, 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 0 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 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 Let es 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. 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? K |