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COR. 1. Similarly may be shewn, that A1- A ¦ α1⁄2 ï ̈¦ Д ̧ — Â¡¦ ɑs•
and let A, be the greatest, and consequently a, the least.
Then shall A1 + α1 > αg + A3•
Since A1 a2: Ag: α4, ..
Multiply these equals by
subtract 1 from each of these equals, ..
but Ã ̧ > α„, '.: A, is the greatest of the four magnitudes,
"The whole of the process in the Fifth Book is purely logical, that is, the whole of the results are virtually contained in the definitions, in the manner and sense in which metaphysicians (certain of them) imagine all the results of mathematics to be contained in their definitions and hypotheses. No assumption is made to determine the truth of any consequence of this definition, which takes for granted more about number or magnitude than is necessary to understand the definition itself. The latter being once understood, its results are deduced by inspection-of itself only, without the necessity of looking at any thing else. Hence, a great distinction between the fifth and the preceding books presents itself. The first four are a series of propositions, resting on different fundamental assumptions; that is, about different kinds of magnitudes. The fifth is a definition and its development; and if the analogy by which names have been given in the preceding Books had been attended to, the propositions of that Book would have been called corollaries of the definition."—Connexion of Number and Magnitude, by Professor De Morgan, p. 56.
The Fifth Book of the Elements as a portion of Euclid's System of Geometry ought to be retained, as the doctrine contains some of the most important characteristics of an effective instrument of intellectual Education. This opinion is favoured by Dr. Barrow in the following expressive terms: "There is nothing in the whole body of the Elements of a more subtile invention, nothing more solidly established, or more accurately handled than the doctrine of proportionals."
QUESTIONS ON BOOK V.
1. EXPLAIN and exemplify the meaning of the terms, multiple, submultiple, equimultiple.
2. What operations in Geometry and Arithmetic are analogous ?
3. What are the different meanings of the term measure in Geometry? When are Geometrical magnitudes said to have a common measure? What is meant by the greatest common measure and by the least common measure of two or more magnitudes ?
4. When are magnitudes said to have, and not to have, a ratio to one another? What restriction does this impose upon the magnitudes in regard to their species? 5. When are magnitudes said to be commensurable or incommensurable to each other? Do the definitions and theorems of Book v, include incommensurable quantities?
6. What is meant by the term Geometrical ratio? How is it represented? 7. Why does Euclid give no independent definition of the ratio of two Geometrical magnitudes ?
8. What sort of quantities are excluded from Euclid's idea of ratio, and how does his idea of ratio differ from the Algebraic definition ?
9. How is a ratio represented Algebraically? Is there any distinction between the terms, a ratio of equality, and equality of ratio?
10. In what manner are ratios, in Geometry, distinguished from each other as equal to, greater, or less than, one another? What objection is there to the use of an independent definition (properly so called) of ratio in a system of Geometry?
11. Point out the distinction between the geometrical and algebraical methods of treating the subject of proportion.
12. What is the geometrical definition of proportion? Whence arises the necessity of such a definition as this?
13. Shew the necessity of the qualification "any whatever" in Euclid's definition of proportion.
14. Must magnitudes that are proportional be all of the same kind?
15. To what objection has Euc. v. def. 5, been considered liable?
16. Point out the connexion between the more obvious definition of proportion and that given by Euclid, and illustrate clearly the nature of the advantage obtained by which he was induced to adopt it.
17. Why may not Euclid's definition of proportion be superseded in a system of Geometry by the following: "Four quantities are proportionals, when the first is the same multiple of the second, or the same part of it, that the third is of the fourth ?"
18. Apply Euclid's definition of proportion, to shew that if four quantities be proportional, and if the first and the third be divided into the same arbitrary number of equal parts, then the second and fourth will either be equimultiples of those parts, or will lie between the same two successive multiples of them.
19. The Geometrical definition of proportion is a consequence of the Algebraical definition; and conversely.
20. What Geometrical test has Euclid given to ascertain that four quantities are not proportionals? What is the Algebraical test?
Shew in the manner of Euclid, that the ratio of 15 to 17 is greater than that of 11 to13.
22. How far may the fifth definition of the fifth Book be regarded as an axiom? Is it convertible? If so, state the converse; if not, state why.
23. Def. 9, Book v. "Proportion consists of three terms at least." How is this to be understood?
24. Define duplicate ratio. How does it appear from Euclid that the duplicate ratio of two magnitudes is the same as that of their squares?
25. Shew that the ratio compounded of any ratio, and the reciprocal of that ratio, is a ratio of equality.
26. What is meant by "the ratio compounded of the ratio of A to B, and of C to D", when A, B, C, D are Geometrical magnitudes of the same species? How is the phrase translated into common Arithmetic when A, B, C, D are numbers?
27. By what process is a ratio found equal to the composition of two or more given ratios? Give an example, where straight lines are the magnitudes which express the given ratios.
28. What limitation is there to the alternation of a Geometrical proportion? 29. Explain the construction and sense of the phrases, ex æquali, and ex æquali in proportione perturbata, used in proportions.
30. Explain the meaning of the word homologous as it is used in the Fifth Book of the Elements.
31. Why, in Euclid v. 11, is it necessary to prove that ratios which are the same with the same ratio, are the same with one another?
32. Apply the Geometrical criterion to ascertain, whether the four lines of 3, 5, 6, 10 units are proportionals.
33. Prove by taking equimultiples according to Euclid's definition, that the magnitudes 4, 5, 7, 9, are not proportionals.
34. Give the Algebraical proofs of Props. 17 and 18, of the Fifth Book.
35. What is necessary to constitute an exact definition? In the demonstration of Euc. v. 18, is it legitimate to assume the converse of the fifth definition of that Book? Does a mathematical definition admit of proof on the principles of the science to which it relates?
36. Explain why the properties proved in Book v, by means of straight lines, are true of any concrete magnitudes.
37. Enunciate Euc. v. 8, and illustrate it by numerical examples.
38. Prove Algebraically Euc. v. 25.
39. Shew that when four magnitudes are proportionals, they cannot, when equally increased or equally diminished by any other magnitude, continue to be proportionals.
40. What grounds are there for the opinion that Euclid intended to exclude the idea of numerical measures of ratios in his Fifth Book?
41. State the particular conclusions that may be arrived at by means of Euclid's definition of Proportion.
'42. Prove from a property of the circle, that if four quantities are proportionals, the sum of the greatest and least is greater than the sum of the other two. 43. "Number and magnitude do not correspond in all their relations." In what relations do they correspond? Does any intelligible relation exist between a point in Geometry and an unit in Arithmetic?
44. If four finite straight lines which are not proportionals be taken in order of magnitude: can any line be found Geometrically or Algebraically, which when added to, or taken from, each of the four lines, shall make the four lines so increased, or so diminished, to be proportionals?
45. If arcs of circles be admitted as measures of angles subtended by them
at the centers of circles; can an arc, and the angle it subtends at the centre of a circle, have a ratio to each other?
46. Is the following definition of proportion as comprehensive as Euclid's? If not, point out in what respects. "Four magnitudes are proportionals, if when the first and second are multiplied by two such numbers as make the products equal, the third and fourth being respectively multiplied by the same numbers, likewise make equal products." That is, if a, b, c, d be four magnitudes, and m and n any two numbers, such that ma = nb and mc = nd, then a, b, c, d are proportionals, or a is to b as c is to d.
47. What effect is produced on a ratio, if equal magnitudes be taken from both the terms of the ratio: first, when the antecedent is greater than the consequent; secondly, when it is less?
48. Shew Geometrically and Algebraically, that if four magnitudes be proportionals, and the first be the greatest; the first and fourth together are greater · than the second and third.
49. If four magnitudes of the same kind be proportionals, and if the first magnitude be the greatest, the fourth shall be the least; and if the first be the least, the fourth shall be the greatest.
50. If three magnitudes are proportionals, the sum of the extremes is greater than double the mean.
51. If a series of magnitudes be in continual proportion, and there be taken any series of equidistant terms; that series will also form a continual proportion.
52. If any number of magnitudes be continual proportionals, the successive sums of the first and second, second and third, &c. terms shall be proportionals: as also the successive differences.
53. In every series of magnitudes in continual proportion, the ratio of the first magnitude to the third is the same as the duplicate ratio of the first to the second: the ratio of the first to the fourth, the same as the triplicate ratio of the first to the second; the ratio of the first to the fifth, the same as the quadruplicate ratio of the first to the second; and so on, whatever may be the number of magnitudes in the series.
54. If four magnitudes be proportionals, and the first be the greatest, the difference between the first and third shall be greater than the difference between the second and fourth.
55. If the first magnitude of a ratio be greater than the second, the ratio is increased by adding equal magnitudes to both terms of the ratio: but if the first magnitude of a ratio be less than the second, the ratio is diminished by adding equal magnitudes to both terms of the ratio.
56. If the first of four magnitudes of the same kind has a greater ratio to the second than the third has to the fourth; the first shall have to the third a greater ratio than the second has to the fourth.
57. It has been objected to Euclid's definition of Proportion, that "it is cumbrous and difficult of comprehension to a learner." Examine these objections, and state how far Dr. Barrow's opinion is tenable, namely-"That there is nothing in the whole body of the Elements of a more subtile invention, nothing more solidly established, or more accurately handled, than the doctrine of proportionals."
58. What is the object of the Fifth Book of Euclid's Elements ?
SIMILAR rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.
Reciprocal figures, viz. triangles and parallelograms, are such as have their sides about two of their angles proportionals in such a manner, that a side of the first figure is to a side of the other, as the remaining side of the other is to the remaining side of the first."
A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less.
The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.
Triangles and parallelograms of the same altitude are one to the other as their bases.
Let the triangles ABC, ACD, and the parallelograms EC, CF, have the same altitude,
viz. the perpendicular drawn from the point A to BD or BD produced.
As the base BC is to the base CD, so shall the triangle ABC be to the triangle ACD,
and the parallelogram EC to the parallelogram CF.