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. XXIII. 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. 11. 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 AE, 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 11. 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 1. 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. 11. If any point be taken within a circle, and a right line be drawn from it to the circumference, how many lines can generally be drawn equal to it? Draw them. 12. Find the shortest distance between a circle and a given straight line without it. 13. Shew that a circle can only have one center, stating the axioms upon which your proof depends. 14. Why would not the demonstration of Euc. 1. 9, hold good, if there were only two such equal straight lines? 15. Two parallel chords in a circle are respectively six and eight inches in length, and one inch apart; how many inches is the diameter in length? 16. Which is the greater chord in a circle whose diameter is 10 inches; that whose length is 5 inches, or that whose distance from the center is 4 inches? 17. What is the locus of the middle points of all equal straight lines in a circle? 18. The radius of a circle BCDGF, (fig. Euc. III. 15.) whose center is E, is equal to five inches. The distance of the line FG from the center is four inches, and the distance of the line BC from the center is three inches, required the lengths of the lines FG, BC. 19. If the chord of an arc be twelve inches long, and be divided into two segments of eight and four inches by another chord: what is the length of the latter chord, if one of its segments be two inches? 20. What is the radius of that circle of which the chords of an arc and of double the arc are five and eight inches respectively? 21. If the chord of an arc of a circle whose diameter is 8 inches, be five inches, what is the length of the chord of double the arc of the same circle? 22. State when a straight line is said to touch a circle, and shew from your definition that a straight line cannot be drawn to touch a circle from a point within it. 23. Can more circles than one touch a straight line in the same point? 24. Shew from the construction, Euc. III. 17, that two equal straight lines, and only two, can be drawn touching a given circle from a given point without it: and one, and only one, from a point in the circumference. 25. What is the locus of the centers of all the circles which touch a straight line in a given point? 26. How may a tangent be drawn at a given point in the circumference of a circle, without knowing the center? 27. In a circle place two chords of given length at right angles to each other. 28. From Euc. III. 19, shew how many circles equal to a given circle may be drawn to touch a straight line in the same point. 29. Enunciate Euc. III. 20. Is this true, when the base is greater than a semicircle? If so, why has Euclid omitted this case? 30. The angle at the center of a circle is double of that at the circumference. How will it appear hence that the angle in a semicircle is a right angle? 31. What conditions are essential to the possibility of the inscription and circumscription of a circle in and about a quadrilateral figure? 32. What conditions are requisite in order that a parallelogram may be inscribed in a circle? Are there any analogous conditions requisite that a parallelogram may be described about a circle? 33. Define the angle in a segment of a circle, and the angle on a seg ment; and shew that in the same circle, they are together equal to two right angles. 34. State and prove the converse of Euc. II. 22. 35. All circles which pass through two given points have their centers in a certain straight line. 36. Describe the circle of which a given segment is a part. Give Euclid's more simple method of solving the same problem independently of the magnitude of the given segment. 37. In the same circle equal straight lines cut off equal circumferences. If these straight lines have any point common to one another, it must not be in the circumference. Is the enunciation given complete? 38. Enunciate Euc. 111. 31, and deduce the proof of it from Euc. III. 20. 39. What is the locus of the vertices of all right-angled triangles which can be described upon the same hypotenuse? 40. How may a perpendicular be drawn to a given straight line from one of its extremities without producing the line? 41. If the angle in a semicircle be a right angle; what is the angle in a quadrant? 42. The sum of the squares of any two lines drawn from any point in a semicircle to the extremity of the diameter is constant. Express that constant in terms of the radius. 43. In the demonstration of Euc. 11. 30, it is stated that "equal straight lines cut off equal circumferences, the greater equal to the greater, and the less to the less:" explain by reference to the diagram the meaning of this statement. 44. How many circles may be described so as to pass through one, two, and three given points? In what case is it impossible for a circle to pass through three given points? 45. Compare the circumference of the segment (Euc. III. 33.) with the whole circumference when the angle contained in it is a right angle and a half. 46. Include the four cases of Euc. 111. 35, in one general proof. 47. Enunciate the propositions which are converse to Props. 32, 35 of Book III. 48. If the position of the center of a circle be known with respect to a given point outside a circle, and the distance of the circumference to the point be ten inches: what is the length of the diameter of the circle, if a tangent drawn from the given point be fifteen inches? 49. If two straight lines be drawn from a point without a circle, and be both terminated by the concave part of the circumference, and if one of the lines pass through the center, and a portion of the other line intercepted by the circle, be equal to the radius: find the diameter of the circle, if the two lines meet the convex part of the circumference, a, b, units respectively from the given point. 50. Upon what propositions depends the demonstration of Euc. IIL 35? Is any extension made of this proposition in the Third Book? 51. What conditions must be fulfilled that a circle may pass through four given points? 52. Why is it considered necessary to demonstrate all the separate cases of Euc. III. 35, 36, geometrically, which are comprehended in one formula, when expressed by Algebraic symbols? 53. Enunciate the converse propositions of the Third Book of Euclid which are not demonstrated ex absurdo: and state the three methods which Euclid employs in the demonstration of converse propositions in the First and Third Books of the Elements. GEOMETRICAL EXERCISES ON BOOK III. PROPOSITION I. THEOREM. If AB, CD be chords of a circle at right angles to each other, prove that the sum of the arcs AC, BD is equal to the sum of the arcs AD, BC. Draw the diameter FGH parallel to AB, and cutting CD in H. Then the arcs FDG and FCG are each half the circumference. Also since CD is bisected in the point H, the arc. FD is equal to the arc FC, and the arc FD is equal to the arcs FA, AD, of which, AF is equal to BG, therefore the arcs AD, BG are equal to the arc FC; add to each CG, therefore the arcs AD, BC are equal to the arcs FC, CG, which make up the half circumference. Hence also the arcs AC, DB are equal to half the circumference. Wherefore the arcs AD, BC are equal to the arcs AC, DB. PROPOSITION II. PROBLEM. The diameter of a circle having been produced to a given point, it is required to find in the part produced a point, from which if a tangent be drawn to the circle, it shall be equal to the segment of the part produced, that is, between the given point and the point found. Analysis. Let AEB be a circle whose center is C, and whose diameter AB is produced to the given point D. Suppose that G is the point required, such that the segment GD is equal to the tangent GE drawn from G to touch the circle in E. F E B G Join DE and produce it to meet the circumference again in F; join also CE and CF. Then in the triangle GDE, because GD is equal to GE, |