In the Third Book of the Elements are demonstrated the most elementary properties of the circle, assuming all the properties of figures demonstrated in the First and Second Books. It may be worthy of remark, that the word circle will be found sometimes taken to mean the surface included within the circumference, and sometimes the circumference itself. Euclid has employed the word (TEPLpipeia) periphery, both for the whole, and for a part of the circumference of a circle. If the word circumference were restricted to mean the whole circumference, and the word are to mean a part of it, ambiguity might be avoided when speaking of the circumference of a circle, where only a part of it is the subject under consideration. A circle is said to be given in position, when the position of its center is known, and in magnitude, when its radius is known. Def. 1. And it may be added, or of which the circumferences are equal. And conversely: if two circles be equal, their diameters and radii are equal; as also their circumferences. Def. 1. states the criterion of equal circles. Simson calls it a theorem ; and Euclid seems to have considered it as one of those theorems, or axioms, which might be admitted as a basis for reasoning on the equality of circles. Def. II. There seems to be tacitly assumed in this definition, that a straight line, when it meets a circle and does not touch it, must necessarily, when produced, cut the circle. A straight line which touches a circle, is called a tangent to the circle; and a straight line which cuts a circle is called a secant. Def. IV. The distance of a straight line from the center of a circle is the distance of a point from a straight line, which has been already explained in note to Prop. XI. page 53. Def. vI. X. An arc of a circle is any portion of the circumference; and a chord is the straight line joining the extremities of an arc. Every chord except a diameter divides a circle into two unequal segments, one greater than, and the other less than a semicircle. And in the same manner, two radii drawn from the center to the circumference, divide the circle into two unequal sectors which become equal when the two radii are in the same straight line. As Euclid, however, does not notice re-entering angles, a sector of the circle seems necessarily restricted to the figure which is less than a semicircle. A quadrant is a sector whose radii are perpendicular to one another, and which contains a fourth part of the circle. Def. vII. No use is made of this definition in the Elements. Def. XI. The definition of similar segments of circles as employed in the Third Book is restricted to such segments as are also equal. Props. XXIII. and xxiv. are the only two instances, in which reference is made to similar segments of circles. Prop. I. "Lines drawn in a circle," always mean in Euclid, such lines only as are terminated at their extremities by the circumference. If the point G be in the diameter CE, but not coinciding with the point F, the demonstration given in the text does not hold good. At the same time, it is obvious that G cannot be the centre of the circle, because GC is not equal to GE. Indirect demonstrations are more frequently employed in the Third Book than in the First Book of the Elements. Of the demonstrations of the forty-eight propositions of the First Book, nine are indirect: but of the thirty-seven of the Third Book, no less than fifteen are indirect demonstrations. The indirect is, in general, less readily appreciated by the learner, than the direct form of demonstration. The indirect form, however, is equally satisfactory, as it excludes every assumed hypothesis as false, except that which is made in the enunciation of the proposition. It may be here remarked that Euclid employs three methods of demonstrating converse propositions. First, by indirect demonstrations as in Euc. 1. 6: III. 1, &c. Secondly, by shewing that neither side of a possible alternative can be true, and thence inferring the truth of the proposition, as in Euc. 1. 19, 25. Thirdly, by means of a construction, thereby avoiding the indirect mode of demonstration, as in Euc. 1. 47. III. 37. Prop. II. In this proposition, the circumference of a circle is proved to be essentially different from a straight line, by shewing that every straight line joining any two points in the arc falls entirely within the circle, and can neither coincide with any part of the circumference, nor meet it except in the two assumed points. It excludes the idea of the circumference of a circle being flexible, or capable under any circumstances, of admitting the possibility of the line falling outside the circle. If the line could fall partly within and partly without the circle, the circumference of the circle would intersect the line at some point between its extremities, and any part without the circle has been shewn to be impossible, and the part within the circle is in accordance with the enunciation of the Proposition. If the line could fall upon the circumference and coincide with it, it would follow that a straight line coincides with a curved line. From this proposition follows the corollary, that "a straight line cannot cut the circumference of a circle in more points than two.' Commandine's direct demonstration of Prop. II. depends on the following axiom, "If a point be taken nearer to the center of a circle than the circumference, that point falls within the circle." Take any point E in AB, and join DA, DE, DB. (fig. Euc. 111. 2.) Then because DA is equal to DB in the triangle DAB; therefore the angle DAB is equal to the angle DBA; (1. 5.) but since the side AE of the triangle DAE is produced to B, therefore the exterior angle DEB is greater than the interior and opposite angle DAE; (1. 16.) but the angle DAE is equal to the angle DBE, therefore the angle DEB is greater than the angle DBE. And in every triangle, the greater side is subtended by the greater angle: therefore the side DB is greater than the side DE; but DB from the center meets the circumference of the circle, therefore DE does not meet it. Wherefore the point E falls within the circle: and E is any point in the straight line AB: therefore the straight line AB falls within the circle. Prop. VII. and Prop. VIII. exhibit the same property; in the former, the point is taken in the diameter, and in the latter, in the diameter produced. PROP. VIII. An arc of a circle is said to be convex or concave with respect to a point, according as the straight lines drawn from the point meet the outside or inside of the circular arc: and the two points found in the circumference of a circle by two straight lines drawn from a given point to touch the circle, divide the circumference into two portions, one of which is convex and the other concave, with respect to the given point. Prop. ix. This appears to follow as a Corollary from Euc. III. 7. Prop. XI. and Prop. xII. In the enunciation it is not asserted that the contact of two circles is confined to a single point. The meaning appears to be, that supposing two circles to touch each other in any point, the straight line which joins their centers being produced, shail pass through that point in which the circles touch each other. Prop. XIII. it is proved that a circle cannot touch another in more points than one, by assuming two points of contact, and proving that this is impossible. In Prop. XIII. The following is Euclid's demonstration of the case, in which one circle touches another on the inside. If possible, let the circle EBF touch the circle ABC on the inside, in more points than in one point, namely in the points B, D. (fig. Euc. III. 13.) Let P be the center of the circle ABC, and Q the center of EBF Join P, Q; then PQ produced shall pass through the points of contact B, D. For since P is the center of the circle ABC, PB is equal to PD, but PB is greater than QD, much more then is QB greater than QD. Again, since the point Q is the center of the circle EBF, QB is equal to QD; but QB has been shewn to be greater than QD, which is impossible. One circle therefore cannot touch anotheron the inside in more points than in one point. Prop. xvI. may be demonstrated directly by assuming the following axiom; "If a point be taken further from the center of a circle than the circumference, that point falls without the circle." If one circle touch another, either internally or externally, the two circles can have, at the point of contact, only one common tangent. Prop. XVII. When the given point is without the circumference of the given circle, it is obvious that two equal tangents may be drawn from the given point to touch the circle, as may be seen from the diagram to Prop. VIII. The best practical method of drawing a tangent to a circle from a given point without the circumference, is the following: join the given point and the center of the circle, upon this line describe a semicircle cutting the given circle, then the line drawn from the given point to the intersection will be the tangent required. Circles are called concentric circles when they have the same center. Prop. XVIII. appears to be nothing more than the converse to Prop. XVI., because a tangent to any point of a circumference of a circle is a straight line at right angles at the extremity of the diameter which meets the circumference in that point. Prop. xx. This proposition is proved by Euclid only in the case in which the angle at the circumference is less than a right angle, and the demonstration is free from objection. If, however, the angle at the circumference be a right angle, the angle at the center disappears, by the two straight lines from the center to the extremities of the arc becoming one straight line. And, if the angle at the circumference be an obtuse angle, the angle formed by the two lines from the center, does not stand on the same arc, but upon the arc which the assumed arc wants of the whole circumference. If Euclid's definition of an angle be strictly observed, Prop. xx. is geometrically true, only when the angle at the center is less than two right angles. If, however, the defect of an angle from four right angles may be regarded as an angle, the proposition is universally true, as may be proved by drawing a line from the angle in the circumference through the center, and thus forming two angles at the center, in Euclid's strict sense of the term. In the first case, it is assumed that, if there be four magnitudes, such that the first is double of the second, and the third double of the fourth, then the first and third together shall be double of the second and fourth together: also in the second case, that if one magnitude be double of another, and a part taken from the first be double of a part taken from the second, the remainder of the first shall be double the remainder of the second, which is, in fact, a particular case of Prop. v. Book v. Prop. xxI. Hence, the locus of the vertices of all triangles upon the same base, and which have the same vertical angle, is a circular arc. Prop. XXII. The converse of this Proposition, namely: If the opposite angles of a quadrilateral figure be equal to two right angles, a circle can be described about it, is not proved by Euclid. It is obvious from the demonstration of this proposition, that if any side of the inscribed figure be produced, the exterior angle is equal to the opposite angle of the figure. Prop. xx. It is obvious from this proposition that of two circular segments upon the same base, the larger is that which contains the smaller angle. Prop. xxv. The three cases of this proposition may be reduced to one, by drawing any two contiguous chords to the given arc, bisecting them, and from the points of bisection drawing perpendiculars. The point in which they meet will be the center of the circle. This problem is equivalent to that of finding a point equally distant from three given points. Props. XXVI-XXIX. The properties predicated in these four propositions with respect to equal circles, are also true when predicated of the same circle. Prop. XXXI. suggests a method of drawing a line at right angles to another when the given point is at the extremity of the given line. And that if the diameter of a circle be one of the equal sides of an isosceles triangle, the base is bisected by the circumference. Prop. xxxv. The most general case of this Proposition might have been first demonstrated, and the other more simple cases deduced from it. But this is not Euclid's method. He always commences with the more simple case and proceeds to the more difficult afterwards. The following process is the reverse of Euclid's method. Assuming the construction in the last fig. to Euc. 111. 35. Join FA, FD, and draw FK perpendicular to AC, and FL perpendicular to BD. Then (Euc. 11. 5.) the rectangle AE, EC with square on EK is equal to the square on AK: add to these equals the square on FK: therefore the rectangle AE, EC, with the squares on EK, FK, is equal to the squares on AK, FK. But the squares on EK, FK are equal to the square on EF, and the squares on AK, FK are equal to the square on AF. Hence the rectangle AE, EC, with the square on EF is equal to the square on AF. In a similar way may be shewn, that the rectangle BE, ED with the square on EF is equal to the square on FD. And the square on FD is equal to the square on AD. Wherefore the rectangle 4E, EC with the square on EF is equal to the rectangle BE. ED with the square on EF. Take from these equals the square on EF, and the rectangle AE, EC is equal to the rectangle BE, ED. The other more simple cases may easily be deduced from this general case. The converse is not proved by Euclid; namely,-If two straight lines intersect one another, so that the rectangle contained by the parts of one is equal to the rectangle contained by the parts of the other; then a circle may be described passing through the extremities of the two lines. Or, in other words:-If the diagonals of a quadrilateral figure intersect one another, so that the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other; then a circle may be described about the quadrilateral. Prop. xxxvI. The converse of the corollary to this proposition may be thus stated:-If there be two straight lines, such that, when produced to meet, the rectangle contained by one of the lines produced, and the part produced, be equal to the rectangle contained by the other line produced and the part produced; then a circle can be described passing through the extremities of the two straight lines. Or, If two opposite sides of a quadrilateral figure be produced to meet, and the rectangle contained by one of the sides produced and the part produced, be equal to the rectangle contained by the other side produced and the part produced; then a circle may be described about the quadrilateral figure. Prop. XXXVII. The demonstration of this theorem may be made shorter by a reference to the note on Euclid 111. Def. 2: for if DB meet the circle in B and do not touch it at that point, the line must, when produced, cut the circle in two points. It is a circumstance worthy of notice, that in this proposition, as well as in Prop. XLVIII. Book I. Euclid departs from the ordinary ex absurdo mode of proof of converse propositions. QUESTIONS ON BOOK III. 1. DEFINE accurately the terms radius, arc, circumference, chord, secant. 2. How does a sector differ in form from a segment of a circle? Are they in any case coincident? 3. What is Euclid's criterion of the equality of two circles? What is meant by a given circle? How many points are necessary to determine the magnitude and position of a circle? 4. When are segments of circles said to be similar? Enunciate the propositions of the Third Book of Euclid, in which this definition is employed. Is it employed in a restricted or general form? 5. In how many points can a circle be cut by a straight line and by another circle? 6. When are straight lines equally distant from the center of a circle? 7. Shew the necessity of an indirect demonstration in Euc. III. 1. 8. Find the centre of a given circle without bisecting any straight Line. 9. Shew that if the circumference of one of two equal circles pass through the center of the other, the portions of the two circles, each of which lies without the circumference of the other circle, are equal. 10. If a straight line passing through the center of a circle bisect a straight line in it, it shall cut it at right angles. Point out the exception; and shew that if a straight line bisect the arc and base of a segment of a circle, it will, when produced, pass through the center. |