but the rectangle ED, DA is equal to the rectangle BD, DC; (111. 35.) therefore the rectangle BA, AC is equal to the rectangle BD, DC, together with the square of AD. Wherefore, if an angle, &c. Q.E.D. If from any angle of a triangle a straight line be drawn perpendicular to the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle. Let ABC be a triangle, and AD the perpendicular from the angle A to the base BC. The rectangle BA, AC shall be equal to the rectangle contained by AD and the diameter of the circle described about the triangle. B A E Describe the circle ACB about the triangle, (Iv. 5.) and draw its diameter AE, and join EC. because the right angle BDA is equal to the angle ECA in a semicircle, (III. 31.) and the angle ABD equal to the angle AEC in the same segment; (III. 21.) the triangles ABD, AEC are equiangular: therefore as BA to AD, so is EA to AC; (VI. 4.) and consequently the rectangle BA, AC is equal to the rectangle EA, AD. (vi. 16.) If therefore from an angle, &c. Q. E. D. PROPOSITION D. THEOREM. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle, is equal to both the rectangles contained by its opposite sides. Let ABCD be any quadrilateral figure inscribed in a circle, and join AC, BD. The rectangle contained by AC, BD shall be equal to the two rectangles contained by AB, CD, and by AD, BC. B E D Make the angle ABE equal to the angle DBC: (1.23.) and the angle BDA is equal to the angle BCE, because they are in the same segment: (III. 21.) therefore the triangle ABD is equiangular to the triangle BCE: wherefore, as BC is to CE, so is BD to DA; (vI. 4.) and consequently the rectangle BC, AD is equal to the rectangle BD, CE: (vi. 16.) again, because the angle ABE is equal to the angle DBC, and the angle BAE to the angle BDC, ˆ (111. 21.) the triangle ABE is equiangular to the triangle BCD: wherefore the rectangle BA, DC is equal to the rectangle BD, AE: but the rectangle BC, AD has been shewn equal to the rectangle BD, CE; therefore the whole rectangle AC, BD is equal to the rectangle AB, DC, together with the rectangle AD, BC. (11. 1.) Therefore the rectangle, &c. Q.E.D. This is a Lemma of Cl. Ptolemæus, in page 9 of his Meyaλn Zúvtaţis. NOTES TO BOOK VI. In this book, the theory of proportion exhibited in the fifth book, is applied to the comparison of the sides and areas of plane rectilineal figures, both to those which are similar, and those which are not similar. Def. I. In defining similar triangles, one condition is sufficient, namely, that similar triangles are those which have their three angles respectively equal; as in Prop. 4, Book VI, it is proved that the sides about the equal angles of equiangular triangles are proportionals. But in defining similar figures of more than three sides, both of the conditions stated in Def. 1, are requisite, as it is obvious, for instance, in the case of a square and a rectangle, which have their angles respectively equal, but have not their sides about their equal angles proportionals. The following definition has been proposed: "Similar rectilineal figures of more than three sides, are those which may be divided into the same number of similar triangles." This definition would, if adopted, require the omission of a part of Prop. 20, Book VI. Def. III. To this definition may be added the following: A straight line is said to be divided harmonically, when it is divided into three parts, such that the whole line is to one of the extreme segments, as the other extreme segment is to the middle part. Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion' is derived from the following fact in the Science of Acoustics, that three musical strings of the same material, thickness and tension, when divided in the manner stated in the definition, or numerically as 6, 4, and 3, produce a certain musical note, its fifth, and its octave. Def. IV. The term altitude, as applied to the same triangles and parallelograms, will be different according to the sides which may be assumed as the base. Prop. I. In the same manner may be proved, that triangles and parallelograms upon equal bases, are to one another as their altitudes. Prop. A. When the triangle ABC is isosceles, the line which bisects the exterior angle at the vertex is parallel to the base. In all other cases; If the line which bisects the angle BAC cut the base BC in the point G. Then the straight line BD is harmonically divided in the points G, C. For BG is to GC as BA is to AC; and BD is to DC as BA is to AC, (vI. 3.) (vI. a.) Hence BD, DG, DC, are in harmonical proportion. Prop. IV is the first case of similar triangles, and corresponds to the third case of equal triangles, Prop. 26, Book 1. Sometimes the sides opposite to the equal angles in two equiangular triangles, are called the corresponding sides, and these are said to be proportional, which is simply taking the proportion in Euclid alternately. The term homologous (¿μóλoyos), has reference to the places the sides of the triangles have in the ratios, and in one sense, homologous sides may be considered as corresponding sides. The homologous sides of any two similar rectilineal figures will be found to be those which are adjacent to two equal angles in each figure. Prop. v, the converse of Prop. IV, is the second case of similar triangles, and corresponds to Prop. 8, Book 1, the second case of equal triangles. Prop. vi is the third case of similar triangles, and corresponds to Prop. 4, Book 1. The property of similar triangles, and that contained in Prop. 47, Book 1, are the most important theorems in Geometry. Prop. VII is the fourth case of similar triangles, and corresponds to the fourth case of equal triangles pointed out in the note to Prop. 26, Book 1, p. 49. Prop. XIII may be compared with Prop. XVI, Book 11. It may be observed, that half the sum of AB and BC is called the Arithmetic mean between these lines; also that BD is called the Geometric mean between the same lines. To find two mean proportionals between two given lines is impossible by the straight line and circle. Pappus has given several solutions of this problem in Book 111, of his Mathematical Collections; and Eutocius has given, in his Commentary on the Sphere and Cylinder of Archimedes, ten different methods of solving this problem. Prop. XIV, depends on the same principle as Prop. xv, and both may easily be demonstrated from one diagram. Join DF, FE, EG in the fig. to Prop. XIV, and the figure to Prop. xv is formed. We may add, that there does not appear any reason why the properties of the triangle and parallelogram should be here separated and not in the first proposition of the sixth book. Prop. xv, holds good when one angle of one triangle is equal to the defect from what the corresponding angle in the other wants of two right angles. Prop. XVII is only a particular case of Prop. XVI, and more properly, might appear as a corollary. Algebraically, Let AB, CD, E, F, contain a, b, c, d units respectively. Then, since a, b, c, d are proportionals, .. a с Multiply these equals by bd, .. ad = bc, = or, the product of the extremes is equal to the product of the means. And conversely, If the product of the extremes be equal to the product of the means, or ad = bc, or the ratio of the first to the second number, is equal to the ratio of the third to the fourth. Prop. XVIII. Similar figures are said to be similarly situated when their homologous sides are parallel. Prop. xx. It may easily be shewn, that the perimeters of similar polygons are proportional to their homologous sides. Prop. XXXI. This proposition is an extension of Prop. 47, Book 1, to similar rectilineal figures, and may be deduced from Prop. 22, Book v1, and Prop. 47, Book 1. Prop. B. The converse of this proposition does not hold good when the triangle is isosceles. The seventh, eighth, ninth and tenth books of the Elements treat of numbers, and employ the Greek numerical notation. BOOK XI. DEFINITIONS. I. A SOLID is that which hath length, breadth, and thickness. II. That which bounds a solid is a superficies. III. A straight line is perpendicular, or at right angles, to a plane, when it makes right angles with every straight line meeting it in that plane. IV. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes, are perpendicular to the other plane. V. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane. VI. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one upon one plane, and the other upon the other plane. VII. Two planes are said to have the same, or a like inclination to one another, which two other planes have, when the said angles of inclination are equal to one another. VIII. Parallel planes are such as do not meet one another though produced. IX. A solid angle is that which is made by the meeting, in one point, of more than two plane angles, which are not in the same plane. X. Equal and similar solid figures are such as are contained by similar planes equal in number and magnitude. |