and since it has been proved, that if the circumference BL be greater than EN, the sector BGL is greater than the sector EÉN; and if equal, equal; and if less, less : therefore, as the circumference BC is to the circumference EF, so is the sector BGC to the sector EHF. (v. def. 5.) Wherefore, in equal circles, &c. Q.E.D. PROPOSITION B. THEOREM. If an angle of a triangle be bisected by a straight line which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle. Let ABC be a triangle, and let the angle BAC be bisected by the straight line AD. The rectangle BA, AC shall be equal to the rectangle BD, DC, together with the square on AD. А B E Describe the circle ACB about the triangle, (IV. 5.) and produce AD to the circumference in Ē, and join ÉC. Then because the angle BAD is equal to the angle ČAE, (hyp.) and the angle ABD to the angle AEC, (11I. 21.) for they are in the same segment; the triangles ABD, AEC are equiangular to one another: (1. 32.) 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.) that is, to the rectangle ED, DA, together with the square on AD; (II. 3.) but the rectangle ED, DA is equal to the rectangle BD, DC; (III. 35.) therefore the rectangle BA, À C is equal to the rectangle BD, DC, together with the square on AD. PROPOSITION C. THEOREM. 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 de. scribed 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. a 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 A EC' in the same segment; (111. 21.) the triangles ABD, AEC are equiangular: therefore as BÀ 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 any 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. Make the angle ABE equal to the angle DBC: (I. 23.) then the angle ABD is equal to the angle EBC: and the angle BDĂ 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.) B E D 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, (III. 21.) therefore as BA to AE, so is BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE: but the rectangle BC, AD has been shewn to be equal to the rectangle BD, CE ; therefore the whole rectangle AC, B D is equal to the rectangle AB, DC, together with the rectangle AD, BC. (II. 1.) Therefore the rectangle, &c. Q. E. D. This is a Lemma of Cl. Ptolemæus, in page 9 of his Meyádn Eúvtažis. a NOTES TO BOOK VL 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 of those which are similar, and of those which are not similar. Def. 1. 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 vı, 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. 11. 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, unless they are equilateral. 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; (VI. 3.) therefore BD is to DC as BG is to GC, but BG BD – DG, and GC = GD - DC. Wherefore BD is to DC as BD – DG is to GD - DC. 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 pruportional, which is simply taking the proportion in Euclid alternately. The term homologous (õuólogos), 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 first case of equal triangles. The property of similar triangles, and that contained in Prop. 47, Book 1, are the most important theorems in Geometry. Prop. vil is the fourth case of similar triangles, and corresponds to the fourth case of equal triangles demonstrated in the note to Prop. 26, Book 1. Prop. ix. The learner here must not forget the different meanings of the word part, as employed in the Elements. The word here has the same meaning as in Euc. 1. ax. 9. It may be remarked, that this proposition is a more simple case of the next, namely, Prop. x. Prop. XI. This proposition is that particular case of Prop. XII, in which the second and third terms of the proportion are equal. These two problems exhibit the same results by a Geometrical construction, as are obtained by numerical multiplication and division. Prop. xul. The difference in the two propositions Euc. 11, 14, and Euc. vi. 13, is this : in the Second Book, the problem is, to make a rectangular figure or square equal in area to an irregular rectilinear figure, in which the idea of ratio is not introduced. In the Prop. in the Sixth Book, the problem relates to ratios only, and it requires to divide a line into two parts, so that the ratio of the whole line to the greater segment may be the same as the ratio of the greater segment to the less. The result in this proposition obtained by a Geometrical construction, is analogous to that which is obtained by the multiplication of two numbers, and the extraction of the square root of the product. 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 III, 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. This theorem will perhaps be more distinctly comprehended hy the learner, if he will bear in mind, that four magnitudes are reciprocally a a с be, a a a proportional, when the ratio compounded of these ratios is a ratio of equality. Prop. xvii is only a particular case of Prop. XVI, and more properly, might appear as a corollary: and both are cases of Prop. xiv. Algebraically, Let AB, CD, E, F, contain a, b, c, d units respectively Then, since a, b, c, d are proportionals, :.5 = d , : ā Multiply these equals by bd, .. ad = 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, , .65 or the ratio of the first to the second number, is equal to the ratio of the third to the fourth. b Similarly may be shewn, that if then ad=6*. b And conversely, if ad=; then őd Prop. XVIII. Similar figures are said to be similarly situated, when their homologous sides are parallel, as when the figures are situated on the same straight line, or on parallel lines : but when similar figures are situated on the sides of a triangle, the similar figures are said to be similarly situated when the homologous sides of each figure have the same relative position with respect to one another ; that is if the bases on which the similar figures stand, were placed parallel to one another, the remaining sides of the figures, if similarly situated, would also be parallel to one another. Prop. xx. It may easily be shewn, that the perimeters of similar polygons, are proportional to their homologous sides. Prop. xxi. This proposition must be so understood as to include all rectilineal figures whatsoever, which require for the conditions of similarity another condition than is required for the similarity of triangles. See note on Euc. vi. Def. 1. Prop. XXIII. The doctrine of compound ratio, including duplicate and triplicate ratio, in the form in which it was propounded and practised by the ancient Geometers, has been almost wholly superseded. However satisfactory for the purposes of exact reasoning the method of expressing the ratio of two surfaces, or of two solids by two straight lines, may be in itself, it has not been found to be the form best suited for the direct application of the results of Geometry. Almost all modern writers on Geometry and its applications to every branch of the Mathematical Sciences, have adopted the algebraical notation of a quotient AB : BC; or of a AB fraction ; for expressing the ratio of two lines AB, BC: as well as that BC of a product AB x BC, or AB.BC, for the expression of a rectangle The want of a concise and expressive method of notation to indicate the proportion of Geometrical Magnitudes in a form suited for the direct application of the results, has doubtless favoured the introduction of Algebraical symbols into the language of Geometry. It must be admitted, however, that such notations in the language of pure Geometry are liable |