Prop. xx. Algebraically.

Let A1, A2, A, be three magnitudes, and a1, a2, as, other three, such that A: A2 :: α1: az

and A, A, Az: Az:

if A1> A,, then shall a1 > α3,

and if equal, equal; and if less, less.

and that A, > Az:

It follows that aj is > a3.

In the same way it may be shewn

that if A1 = Ag, then a1 = a; and if Abe < A3, then a < Azi

Prop. xxi. Algebraically.

Let A1, A2, A3, be three magnitudes,

and a1, a2, as three others,

such that A: A2 :: A2: A31
and A2 A3 A1: an.

If A > A, then shall a > a, and if equal, equal; and if less, less.

and A1: As :: α2: Az.

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

and similarly, if there were more than four magnitudes.

Prop. xx, Algebraically.

Let A1, A2, A, be three magnitudes,

and a1, a2, as other three,

such that A: A2 :: α2: ɑ3,

and A2 A3 :: A1: A2.

Then shall A1 : A3 :: α1⁄2 : αz.

For since A: A2 :: α2 : Az,

and since A2: A3 ¡¦ α1 : ɑg,

A1

=

A2

and similarly, if there be more than four magnitudes.

Prop. XXIV. Algebraically.

Let A a: Ag: α4,

and A ̧ : α2 :: A ̧ : α,

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

For since A, a2 :: A3 : as,

A1 A3

=

and.. A1 + As: a2 :: A3 + Ac: aş

COR. 1. Similarly may be shewn, that

A1- As: α1⁄2:: A3 – A6 : αs.

Prop. xxv. Algebraically.

Let A1a: A3: as,

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

Then shall A1 + αs > αz + Ag.

Since A: A3: α

but A1 > a, . A1 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 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