Next if there be four magnitudes, and other four such, that

and similarly, if there were more than four magnitudes.

Prop. XXII, Algebraically.

Let A1, A2, A, be three magnitudes,

and a1, a2, as other three,

such that A1: A2 :: α2 : α3,

and A2 A3 :: ɑ1 : а2.

Then shall A1: Az :: α1 : αz.

A2

and similarly, if there be more than four magnitudes.

Prop. xxiv. Algebraically.

Let A1: a2 :: Ag: α4,

and A, : a2 :: A。 : α,

Then shall A1 + As: a2 :: A3 + A ̧: αş.

For since A, az :: A3 : as,

and since A,: α1⁄2:: A ̧: as,

A3

..

=

a2 as

and.. A1 + A5 : 02 :: A3 + A ̧: α ̧·

COR. 1. Similarly may be shewn, that

Prop. xxv. Algebraically.

Let A a: Ag: 49

and let A, be the greatest, and consequently a the least.

Then shall A1 + αs > αz + Az.

Since A 2 :: A3: αs

but A1> a, A, is the greatest of the four magnitudes,

.. also A1- A3 > α2 — α4

add Ag + a to each of these equals,

.. A1 + as > a2 + A3.

"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 developement; 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?

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 ratio?

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, 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. Point out the defect of the following definition: "Four magnitudes are proportional when equimultiples may be taken of the first and the third, and also of the second and fourth, such that the multiples of the first and second are equal, and also those of the third and fourth."

19. 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.

20. The Geometrical definition of proportion is a consequence of the Algebraical definition; and conversely.

21. What Geometrical test has Euclid given to ascertain that four quantities are not proportionals? What is the Algebraical test?

22. Shew in the manner of Euclid, that the ratio of 15 to 17 is greater than that of 11 to 13.

23.

How far may the fifth definition of the fifth Book be regarded as an axiom? Is it convertible? 24. Def. 9, Book v. 66 How is this to be understood? 25. Define duplicate ratio. How does it appear from Euclid that the duplicate ratio of two magnitudes is the same as that of their squares? 26. When is a ratio compounded of any number of ratios? What is the ratio which is compounded of the ratios of 2 to 5, 3 to 4, and 5 to 6? 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.

Proportion consists of three terms at least."

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. Exemplify 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. What is the object of the Fifth Book of Euclid's Elements ?