49. If the sides of a triangle be in arithmetical progression, and if R, r be the radii of the circumscribed and inscribed circles; then 6Rr is equal to the rectangle contained by the greatest and least sides.

50. Inscribe in a given circle a triangle having its three sides parallel to the three given lines.

51. If the sides AB, BC, &c., of a regular pentagon be bisected in the points A', B', C', D', E', and if the two pairs of alternate sides, BC, AE; AB, DE meet in the points A", E", respectively,

prove

▲A"AE" ▲ A'AE' = pentagon A'B'C'D'E'.

52. In a circle, prove that an equilateral inscribed polygon is regular, and also an equilateral circumscribed polygon, if the number of sides be odd.

53. Prove also that an equiangular circumscribed polygon is regular, and an equiangular inscribed polygon, if the number of

sides be odd.

54. The sum of the perpendiculars drawn to the sides of an equiangular polygon from any point inside the figure is con

stant.

55. Express the sides of a triangle in terms of the radii of its escribed circles.

BOOK V.

THEORY OF PROPORTION.

DEFINITIONS.

Introduction. Every proposition in the theories of ratio and proportion is true for all descriptions of magnitude. Hence it follows that the proper treatment is the Algebraic. It is, at all events, the easiest and the most satisfactory. Euclid's proofs of the propositions, in the Theory of Proportion, possess at present none but a historical interest, as no student reads them now. But although his demonstrations are abandoned, his propositions are quoted by every writer, and his nomenclature is universally adopted. For these reasons it appears to us that the best method is to state Euclid's definitions, explain them, or prove them when necessary, for some are theorems under the guise of definitions; and then supply simple algebraic proofs of his propositions.

1. A less magnitude is said to be a part or submultiple of a greater magnitude, when the less measures the greater that is, when the less is contained a certain number of times exactly in the greater.

II. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less-that is, when the greater contains the less a certain number of times exactly.

III. Ratio is the mutual relation of two magnitudes of the same kind with respect to quantity.

IV. Magnitudes are said to have a ratio to one another when the less can be multiplied so as to exceed the greater.

These definitions require explanation, especially Def. III., which has the fault of conveying no precise meaning, being, in fact, unintelligible.

The following annotations will make them explicit

:

1. If an integer be divided into any number of equal parts, one, or the sum of any number of these parts, is called a fraction. Thus, if the line AB represent the integer, and if it be divided into four equal parts in the points C, D, E, then AC is ;

A

C

D

E

B

+

+

+

AD, ; AE, 2. Thus, a fraction is denoted by two numbers parted by a horizontal line; the lower, called the denominator, denotes the number of equal parts into which the integer is divided; and the upper, called the numerator, denotes the number of these equal parts which are taken. Hence it follows, that if the numerator be less than the denominator, the fraction is less than unity. If the numerator be equal to the denominator, the fraction is equal unity; and if greater than the denominator, it is greater than unity. It is evident that a fraction is an abstract quantity—that is, that its value is independent of the nature of the integer which is divided.

=

2. If we divide each of the equal parts AC, CD, DE, EB into two equal parts, the whole AB will be divided into eight equal parts; and we see that AC ; AD ; AE = §; AB = §. Now, we saw in 1, that AE = 2 of the integer, and we have just shown that it is equal to . Hence = §; but would be got from by multiplying its terms (numerator and denominator) by 2. Hence we infer generally that multiplying the terms of any fraction by 2 does not alter its value. In like manner it may be shown that multiplying the terms of a fraction by any whole number does not alter its value. Hence it follows conversely, that dividing the terms of a fraction by a whole number does not alter the value. Hence we have the following important and fundamental theorem :-Two transformations can be made on any fraction without changing its value; namely, its terms can be either multiplied or divided by any whole number, and in either case the value of the new fraction is equal to the value of the original one.

3. If we take any number, such as 3, and multiply it by any whole number, the product is called a multiple of 3. Thus, 6, 9, 12, 15, &c., are multiples of 3; but 10, 13, 17, &c., are not, because the multiplication of 3 by any whole number will not produce them. Conversely, 3 is a submultiple, or measure, or part of 6, 9, 12, 15, &c., because it is contained in each of these without a remainder; but not of 10, 13, 17, &c., because in each case it leaves a remainder.

4. If we consider two magnitudes of the same kind, such as two lines AB, CD, and if we suppose that AB is equal to of CD, it is evident if AB be divided into 3 equal parts, and CD into 4 equal parts, one of the parts

into which AB is divided is A equal to one of the parts into

B

which CD is divided. And as there are 3 parts in AB, and 4 in CD, we express this relation by saying that AB has to

CD the ratio of 3 to 4; and we denote it thus, 3: 4. Hence the ratio 3 4 expresses the same idea as the fraction. In fact, both are different ways of expressing and writing the same thing. When written 3: 4 it is called a ratio, and when a fraction. In the same manner it can be shown that every ratio whose terms are commensurable can be converted into a fraction, and, conversely, every fraction can be turned into a ratio.

From this explanation we see that the ratio of any two commensurable magnitudes is the same as the ratio of the numerical quantities which denote these magnitudes. Thus, the ratio of two commensurable lines is the ratio of the numbers which express their lengths, measured with the same unit. And this may be extended to the case where the lines are incommensurable. Thus, if a be the side and b the diagonal of a square, the ratio of a: b is

When two quantities are incommensurable, such as the diagonal and the side of a square, although their ratio is not equal to that of any two commensurable numbers, yet a series of pairs of fractions can be found whose difference is continually diminishing, and which ultimately becomes indefinitely small; such that the ratio of the incommensurable quantities is greater than one, and less than the other fraction of each pair. These fractions are called convergents. By their means we can approximate as nearly as we please to the exact value of the ratio. In the case

of the diagonal and the side of a square, the following are the pairs of convergents :

14 15 141 142 1414 1415
10' 10' 100' 100
; 1000' 1000

&c.,

It is evident we may
Now if we denote the

and the ratio is intermediate to each pair. continue the series as far as we please. first of any of the foregoing pairs of fractions by the second

m + 1 will be

n

m

n

; and in general, in the case of two incommen

surable quantities, two fractions and

m
n

m+ 1

n

can always be found, where n can be made as large as we please, one of which is less and the other greater than the true value of the ratio. For let a and b be the incommensurable quantities; then, evidently, we cannot find two multiples na, mb, such that na = mb. In this case, take any multiple of a, such as na, then this quantity must lie between some two consecutive multiples of b, such as mb

small as n increases, we see that the difference between the ratio of two incommensurable quantities and that of two commensurable numbers m and n can be made as small as we please. Hence, ultimately, the ratio of incommensurable quantities may be regarded as the limit of the ratio of commensurable quantities.

5. The two terms of ratio are called the antecedent and the consequent. These correspond to the numerator and the denominator of a fraction. Hence we have the following definition :-"A ratio is the fraction got by making the antecedent the numerator and the consequent the denominator."

6. The reciprocal of a ratio is the ratio obtained by interchanging the antecedent and the consequent. Thus, 4: 3 is the reciprocal of the ratio of 3: 4. Hence we have the following theorem:-"The product of a ratio and its reciprocal is unity.'

7. If we multiply any two numbers, as 5 and 7, by any number such as 4, the products 20, 28 are called equimultiples of 5 and 7. In like manner 10 and 15 are equimultiples of 2 and 3, and 18 and 30 of 3 and 5, &c.