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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 prcperly, 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, :: = ;.
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, then, dividing these equals by bd, : a, or the ratio of the first to the second number, is equal to the ratio of the third to the fourth. Similarly may be shewn, that if = ; then ad=6. And conversely, if ad=6*; then
wa b .
= 9. 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. xxi. 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triplicate ratio, in the form in which it was propounded and practised by the ancient Geometers, has been almost wholly superseded.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 ap. plication 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 an; for expressing the ratio of two lines AB, BC: as well as that 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
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 is 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 AR, and therefore AG exceeds AC the parallelogram described upon AB in the same angle, and between the same parallels, by the parallelogram BG.”—Simson.
Both among Euclid's Theorems and Problems, cases occur in which the hypotheses of the one, and the data or quæsita of the other, are restricted within certain limits as to magnitude and position. The determination of these limits constitutes the doctrine of Maxima and Minima. Thus:- The theorem Euc. Vi, 27 is a case of the maximum value which a figure fulfilling the other conditions can have; and the succeeding proposition is a problem involving this fact among the conditions as a part of the data, in truth, perfectly analogous to Euc. I. 20, 22; wherein the limit of possible diminution of the sum of the two sides of a triangle described upon a given base, is the magnitude of the base itself: the limit of the side of a square which shall be equal to the rectangle of the two parts into which a given line may be divided, is half the line, as it appears from Euc. 11. 5:--the greatest line that can be drawn from a given point within a circle, to the circumference, Euc. III. 7, is the line which passes through the center of the circle ; and the least line which can be so drawn from the same point, is the part produced, of the greatest line between the given point and the circumference. Euc. I11. 8, also affords another instance of a maximum and a minimum when the given point is outside the given circle.
Prop. XXXI. 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. I. 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 arcs 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 Euc. vi. 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. vi. 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. I. 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 6 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 v5.
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 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 I 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 proportional 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 (1) of similar triangles and (2) of similar parallelograms are proportional to the squares of 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 al. titudes ?
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 external segments of the sides determined by the lines which bisect the exterior angles of the triangle.
42. What are the segments into which the hypotenuse of a rightangled 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 three sides of a triangle be 3, 4, 5 units respectively: what are the parts into which they are divided by the lines which bisect the angles opposite to them?
44. 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?
45. 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.
46. The parts of a line divided in extreme and mean ratio are incommensurable with each other.
47. Shew that in Euclid's figure (Euc. 11. 11.) four other lines, besides the given line, are divided in the required manner.
48. Enunciate Euc. vi. 31. What theorem of a previous book is included in this proposition ?
49. What is the superior limit, as to magnitude, of the angle at the circumference in Euc. VI, 33! Shew that the proof may be extended by withdrawing the usually supposed restriction as to angular magnitude; and then deduce, as a corollary, the proposition respecting the magnitudes of angles in segments greater than, equal to, or less than a semicircle.
50. The sides of a triangle inscribed in a circle are a, b, c, units respectively: find by Euc. vi. C, the radius of the circumscribing circle.
51. Enunciate the converse of Euc. VI. D.
52. Shew independently that Euc. VI. D, is true when the quadri. lateral figure is rectangular.
53. Shew that the rectangles contained by the opposite sides of a quadrilateral figure which does not admit of having a circle described about it, are together greater than the rectangle contained by the diagonals.
54. What different conditions may be stated as essential to the possibility of the inscription and circumscription of a circle in and about a quadrilateral figure ?
55. Point out those propositions in the Sixth Book in which Euclid's definition of proportion is directly applied.
56. Explain briefly the advantages gained by the application of analysis to the solution of Geometrical Problems.
57. In what cases are triangles proved to be equal in Euclid, and in what cases are they proved to be similar ?