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. "Proportion consists of three terms at least." 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.

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.

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

BOOK VI.

DEFINITIONS.

I.

SIMILAR rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.

II.

"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."

III.

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.

IV.

The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.

PROPOSITION I. THEOREM.

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.

Produce BD both ways to the points H, L,

and take any number of straight lines BG, GH, each equal to the base BC; (L. 3.)

and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, ÁL.

Then, because CB. BG, GH, are all equal,

the triangles AHG, AGB, ABC, are all equal: (1. 38.) therefore, whatever multiple the base HC is of the base BC, the same multiple is the triangle AHC of the triangle ABC: for the same reason whatever multiple the base LC is of the base CD, the same multiple is the triangle ALC of the triangle ADC: and if the base HC be equal to the base CL,

the triangle AHC is also equal to the triangle ALC: (1. 38.) and if the base HC be greater than the base CL, likewise the triangle AHC is greater than the triangle ALC; and if less, less;

therefore since there are four magnitudes,

viz. the two bases BC, CD, and the two triangles ABC, ACD; and of the base BC, and the triangle ABC, the first and third, any equimultiples whatever have been taken,

viz. the base HC and the triangle AHC;

and of the base CD and the triangle ACĎ, the second and fourth, have been taken any equimultiples whatever,

viz. the base CL and the triangle ALC;

and since it has been shewn, that, if the base HC be greater than the base CL,

the triangle AHC is greater than the triangle ALC;

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

therefore, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD. (v. def. 5.)

And because the parallelogram CE is double of the triangle ABC, (I. 41.)

and the parallelogram CF double of the triangle ACD,

and that magnitudes have the same ratio which their equimultiples have; (v. 15.)

as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF;

and because it has been shewn, that, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD;

and as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF;

therefore, as the base BC is to the base CD, so is the parallelogram EC to the parallelogram CF. (v. 11.)

Wherefore, triangles, &c. Q. E. D.

COR. From this it is plain, that triangles and parallelograms that have equal altitudes, are to one another as their bases.

Let the figures be placed so as to have their bases in the same straight line; and having drawn perpendiculars from the vertices of the triangles to the bases, the straight line which joins the vertices is parallel to that in which their bases are, (I. 33.) because the perpendiculars are both equal and parallel to one another. (1. 28.) Then, if the same construction be made as in the proposition, the demonstration will be the same.

PROPOSITION II. THEOREM.

If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or these produced, proportionally: and conversely, if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

Let DE be drawn parallel to BC, one of the sides of the triangle ABC. Then BD shall be to DA, as CE to EA.

Then the triangle BDE is equal to the triangle CDE, (1. 37.) because they are on the same base DE, and between the same parallels DE, BC;

but ADE is another triangle;

and equal magnitudes have the same ratio to the same magnitude; (v. 7.)

therefore, as the triangle BDE is to the triangle ADE, so is the triangle CDE to the triangle ADE:

but as the triangle BDE to the triangle ADE, so is BD to DA, (vI. 1.) because, having the same altitude, viz. the perpendicular drawn from the point E to AB, they are to one another as their bases; and for the same reason, as the triangle CDE to the triangle ADE, so is CE to EA:

therefore, as BD to DA, so is CE to EA. (v. 11.)

Next, let the sides AB, AC of the triangle ABC, or these sides produced, be cut proportionally in the points D, E, that is, so that BD may be to DA as CE to EA, and join DE.

Then DE shall be parallel to BC.

The same construction being made,

because as BD to DA, so is CE to EA;

and as BD to DA, so is the triangle BDE to the triangle ADE; (vI. 1.) and as CE to EA, so is the triangle CDE to the triangle ADE; therefore the triangle BDE is to the triangle ADE, as the triangle CDE to the triangle ADE; (v. 11.)