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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 proportions in Euclid alternately.
The term homologous (oμóλoyos), 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. VII 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. vIII. It may be remarked that propositions are introduced in the Second and Third Books under the form of properties of rectangles, and are identical with propositions in the Sixth Book given in the form of proportion: as for instance, the first proportion in the Corollary to Euc. v1. 8, is identical to Euc. 11. 14. In the latter (see fig.) the square on EH is equal to the rectangle BE, EF: in the former (see fig.) AD is a mean proportional between BD and DC:
Prop. IX. The learner 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. v. def. 1. 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. XIII. The difference in the two propositions Euc. II. 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 ress.
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 by the learner, if he will bear in mind, that four magnitudes are reciprocally 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.
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 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, 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 is not restricted to triangles, but 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. v1. 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
fraction ; for expressing the ratio of two lines AB, BC: as well as that of a product AB × 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 Germetry. It must be admitted, however, that such notations in the language of
pure Geometry are liable to very serious objections, chiefly on the ground that pure Geometry does not admit the Arithmetical or Algebraical idea of a product or a quotient into its reasonings. On the other hand, it may be urged, that it is not the employment of symbols which renders a process of reasoning peculiarly Geometrical or Algebraical, but the ideas which are expressed by them. If symbols be employed in Geometrical reasonings, and be understood to express the magnitudes themselves and the conception of their Geometrical ratio, and not any measures, or numerical values of them, there would not appear to be any very great objections to their use, provided that the notations employed were such as are not likely to lead to misconception. It is however desirable, for the sake of avoiding confusion of ideas in reasoning on the properties of number and of magnitude, that the language and notations employed both in Geometry and Algebra should be rigidly defined and strictly adhered to, in all cases. At the commencement of his Geometrical studies, the student is recommended not to employ the symbols of Algebra in Geometrical demonstrations. How far it may be necessary or advisable to employ them when he fully understands the nature of the subject, is a question on which some difference of opinion exists.
Prop. xxv. There does not appear any sufficient reason why this proposition should be placed between Prop. XXIV. and Prop. xxvi.
Prop. xxvII. To understand this and the three following propositions more easily, it is to be observed:
1. "That a parallelogram is said to be applied to a straight line, when it is described upon it as one of its sides. Ex. gr. the parallelogram AC is said to be applied to the straight line AB.
2. But a parallelogram AE is said to be applied to a straight line AB, deficient by a parallelogram, when AD the base of AE is less than AB, and therefore AE is less than the parallelogram AC described upon AB in the same angle, and between the same parallels, by the parallelogram DC; and DC is therefore called the defect of AE.
3. And a parallelogram AG is said to be applied to a straight line AB, exceeding by a parallelogram, when AF the base of AG is greater than AB, and therefore AG exceeds AC the parallelogram described upon AB in the same angle, and between the same parallels, by the parallelogram BG."-Simson.
Props. XXVII-XXIX are not usually read: and Prop. xxx, is read as deduced from Euc. II. 10.
Prop. xxxx This proposition is the general case of Prop. 47, Book 1, for any similar rectilineal figure described on the sides of a right-angled triangle. The demonstration, however, here given is wholly independent of Euc. 1. 47.
Prop. XXXIII. In the demonstration of this important proposition, angles greater than two right angles are employed, in accordance with the criterion of proportionality laid down in Euc. v. def. 5.
This proposition forms the basis of the assumption of ares of circles for the measures of angles at their centers. One magnitude may be assumed as the measure of another magnitude of a different kind, when the two are so connected, that any variation in them takes place simultaneously, and in the same direct proportion. This being the case with angles at the center of a circle, and the arcs subtended by them; the arcs of circles can be assumed as the measures of the angles they subtend at the center of the circle.
Prop. B. The converse of this proposition does not hold good. when the triangle is isosceles.
QUESTIONS ON BOOK VI.
1. DISTINGUISH between similar figures and equal figures.
2. What is the distinction between homologous sides, and equal sides in Geometrical figures?
3. What is the number of conditions requisite to determine similarity of figures? Is the number of conditions in Euclid's definition of similar figures greater than what is necessary? Propose a definition of similar figures which includes no superfluous condition.
4. Explain how Euclid makes use of the definition of proportion in Euc. vI. 1. 5. Prove that triangles on the same base are to one another as their altitudes. 6. If two triangles of the same altitude have their bases unequal, and if one of them be divided into m equal parts, and if the other contain n of those parts; prove that the triangles have the same numerical relation as their bases, Why is this Proposition less general than Eue. vL 1?
7. Are triangles which have one angle of one equal to one angle of another, and the sides about two, other angles proportionals, necessarily similar?
8. What are the conditions, considered by Euclid, under which two triangles are similar to each other?
9. Apply Euc, VI. 2, to trisect the diagonal of a parallelogram.
10. When are three lines said to be in harmonical proportion? If both the interior and exterior angles at the vertex of a triangle (Euc. vi. 3, a.) be bisected by lines which meet the base, and the base produced in D, G; the segments BG GD, GC of the base shall be in harmonical proportion.
11. If the angles at the base of the triangle in the figure Euc. vi. A, be equal to each other, how is the proposition modified?
12. Under what circumstances will the bisecting line in the fig. Euc. vi. A, meet the base on the side of the angle bisected? Shew that there is an indeterminate case.
13. State some of the uses to which Euc. v. 4, may be applied.
14. Apply Euc. vi. 4, to prove that the rectangle contained by the segments of any chord passing through a given point within a circle is constant.
15. Point out clearly the difference in the proofs of the two latter cases in Euc. VI. 7.
16. From the corollary of Euc. vI. 8, deduce a proof of Euc. 1. 47.
17. Shew how the last two properties stated in Euc. vi. 8. Cor. may be deduced from Euc. 1. 47; 11. 2; VI. 17.
18. Given the nth part of a straight line, find by a Geometrical construction, the (n + 1)th part.
19. Define what is meant by a mean proportional between two given lines : and find a mean proportional between the lines whose lengths are 4 and 9 units respectively. Is the method you employ suggested by any propositions in any of the first Four Books?
20. Determine a third proportional to two lines of 5 and 7 units: and a fourth proportional to three lines of 5, 7, 9, units.
21. Find a straight line which shall have to a given straight line, the ratio of 1 to √5.
22. Define reciprocal figures. Enunciate the propositions proved respecting such figures in the Sixth Book.
23. Give the corollary, Euc. vi. 8, and prove thence that the Arithmetic mean is greater than the Geometric mean between the same extremes.
24. If two equal triangles have two angles together equal to two right angles, the sides about those angles are reciprocally proportional.
25. Give Algebraical proofs of Prop. 16 and 17 of Book vi.
26. Enunciate and prove the converse of Euc. vi. 15.
27. Explain what is meant by saying, that “similar triangles are in the duplicate ratio of their homologous sides."
28. What are the data which determine triangles both in species and magnitude? How are those data expressed in Geometry?
29. If the ratio of the homologous sides of two triangles be as 1 to 4, what is the ratio of the triangles? And if the ratio of the triangles be as 1 to 4, what is the ratio of the homologous sides?
30. Shew that one of the triangles in the figure, Euc. Iv. 10, is a mean prop、rtional between the other two.
31. What is the Algebraical interpretation of Euc. vi. 19?
32. From your definition of Proportion, prove that the diagonals of a square are in the same proportion as their sides.
33. What propositions does Euclid prove respecting similar polygons?
34. The parallelograms about the diameter of a parallelogram are similar to the whole and to one another. Shew when they are equal.
35. Prove Algebraically, that the areas of similar triangles and of similar parallelograms, are proportional to the squares on their homologous sides.
36. How is it shewn that equiangular parallelograms have to one another the ratio which is compounded of the ratios of their bases and altitudes ?
37. To find two lines which shall have to each other, the ratio compounded of the ratios of the lines A to B, and C to D.
38. State the force of the condition "similarly described;" and shew that, on a given straight line, there may be described as many polygons of different magnitudes, similar to a given polygon, as there are sides of different lengths in the polygon.
39. Describe a triangle similar to a given triangle, and having its area double that of the given triangle.
40. The three sides of a triangle are 7, 8, 9 units respectively; determine the length of the lines which meeting the base, and the base produced, bisect the interior angle opposite to the greatest side of the triangle, and the adjacent exterior angle.
41. The three sides of a triangle are 3, 4, 5 inches respectively; find the lengths of the internal and external segments of the sides determined by the lines which bisect the interior and exterior angles of the triangle.
42. What are the segments into which the hypotenuse of a right-angled triangle is divided by a perpendicular drawn from the right angle, if the sides containing it are a and 3a units respectively?
43. If the homologous sides of two triangles be as 3 to 4, and the area of one triangle be known to contain 100 square units; how many square units are contained in the area of the other triangle ?
44. Prove that if BD be taken in AB produced (fig. Euc. vi. 30.) equal to the greater segment AC, then AD is divided in extreme and mean ratio in the point B.
Shew also, that in the series 1, 1, 2, 3, 5, 8, &c. in which each term is the sum of the two preceding terms, the last two terms perpetually approach to the proportion of the segments of a line divided in extreme and mean ratio. Find a general expression (free from surds) for the nth term of this series.
45. The parts of a line divided in extreme and mean ratio are incommensurable with each other.