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DEMONSTRATION. For if the whole circumference of the given circle be divided into fifteen equal parts, the arc AC, because it is the third part of the whole circumference, contains five of these parts; in like manner the arc AB contains three of them, therefore the arc BC contains two, and therefore the arc BE is the fifteenth part of the whole circumference, and BE is the side of the required equilateral and equiangular quindecagon.

SCHOLIUM. The only regular polygons which the Greek Geometers could incribe geometrically in the circle were the trigon, or equilateral triangle, the tetragon, or square, the pentagon, the hexagon, and any others, such as the quindecagon, derived from them. M. Gause, however, in his Disquisitiones Arithmetica, has shown that a regular polygon of 2+1 sides is always capable of being inscribed geometrically in a circle, when 2 + 1 is a prime number.





1. A LESS magnitude is said to be a part 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.

SCHOLIUM. In ordinary use the word "part" means any portion whatever," but its geometrical sense in the above definition, and wherever subsequently employed, is that of an aliquot part or submultiple. It has already been explained in the scholium to the first proposition of the second book, that one magnitude is said to measure another when it is exactly contained in it any number of times without any remainder. The lesser magnitude is then said to be a part or submultiple of the greater, while the greater is said to be a multiple of the less.

In the four preceding books magnitudes have been compared simply as to their equality or inequality, but in the latter case no attempt has been made to determine how great or how small that inequality might be. The object, however, of the fifth book is to compare unequal magnitudes, and to determine with greater exactness their relative value. Now there are two ways in which two unequal magnitudes or quantities might be compared, namely,-1o, by subtracting the lesser from the greater, and so ascertaining how much one exceeded the other; thus if one line were represented by 50 and the other by 40, their difference thus estimated would be 10; this method, however, would fail to convey any idea of their relative values;2o, by ascertaining how often the greater contained the less, or, in other words, what multiple the greater was of the less; this latter method is the one employed by Euclid in the fifth book, and by it we are enabled to ascertain their relative value.

2. 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.

SCHOLIUM. It is necessary to observe the distinction between the expressions "measures" and "is contained in;" for example, 3 measures 15, being contained in it exactly 5 times without any remainder, but 3 does not measure 13, although it is contained in it 4 times, because there is a remainder of 1 over. It has already been explained, in the scholium to

II. 1, that when two magnitudes are multiples of the same magnitude, or, in other words, when they may both be measured by the same magnitude, they are said to be commensurable, but that when no magnitude could be found by which both the given magnitudes could be measured, they were said to be incommensurable, as in the case of the side and diagonal of a square.

3. Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity.

SCHOLIUM. This definition has been as severely criticised as perhaps any other portion of the Elements; but it should be borne in mind that no subsequent conclusions are deduced from, or made to depend upon it, but that Euclid doubtless introduced it as a mere explanation of the sense in which the word "ratio was to be afterwards employed. There is, however, a defect in the definition, inasmuch as it is not stated in what way the comparison of the two magnitudes is to be made, for we have already mentioned that two modes of comparison may be adopted, namely, either by finding the excess of one magnitude above the other, or by ascertaining what multiple one is of the other. In the following definition given by Wood in his Algebra, this objection is removed:-"Ratio is the relation which one quantity bears to another in respect of magnitude, the comparison being made by considering what multiple, part or parts, one is of the other."

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In order that two magnitudes may be capable of comparison so as to determine their ratio, it is essential that they should be of the same kind," that is to say, two lines, two angles, two surfaces, or two solids; or, as is expressed in the next definition, they must be such that "the less may be multiplied so as to exceed the greater."

It cannot be too strongly impressed on the learner that the ratio of two quantities is entirely irrespective of their actual magnitude, but is determined solely by their relative magnitude; so that if any ratio has been found to exist between any two quantities, that ratio will remain unaltered, although the original quantities may be both doubled or both halved, or, in fact, multiplied or divided by any other quantity, or submitted to any other operation.

The two quantities between which the ratio exists, are called the terms of the ratio; the first being named the antecedent and the second the consequent. Adopting the symbolism explained in the Scholium to II. 1, the two terms of a ratio may be represented by a, b, or any other two letters of the alphabet, and their ratio may be expressed by writing a : b (which is read) a is to b; or by which is read a divided by b; thus, if a represented 15




and b 5, then is the same as or as 15 divided by 5, namely 3,

which is the measure of the ratio of the two quantities represented by a and b.

4. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

5. The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third are taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of

the third is also less than that of the fourth; or if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth: or if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

SCHOLIUM. To render this definition as clear as possible, it may be symbolically expressed as follows:-Let A, B, C, and D represent four magnitudes, then, the first A is said to have the same ratio to the second B, which the third C has to the fourth D, when if A and C are multiplied by any number whatever as m, and B and D are multiplied by any other number, as n, it is found, that

If mA ben B, then m C is < nD,

if m An B, then m C = n D,

or if m A be > n B, then m C is > n D.

6. Magnitudes which have the same ratio are called proportionals.

SCHOLIUM. The arithmetical definition of proportion is as follows:-Four quantities are said to be proportional, or in proportion, when the quotient of the first divided by the second is equal to the quotient of the third divided by the fourth, whether these quotients be either integers or fractions.

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9 3

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= 3, and 3, therefore the numbers 15, 5, 9, and 3 are said

to be in proportion; and this is usually expressed by writing them thus, 15 593, which is read as 15 is to 5 so is 9 to 3.


Euclid's definition of proportion has been found fault with because it bears no resemblance to the common notions of the similitude of ratios employed in Arithmetic or Algebra; and with the view of removing this objection, Elrington has substituted the following, namely, " Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth, when any submultiple whatsoever of the first is contained in the second, as often as an equi-submultiple of the third is contained in the fourth." On the other hand, many of the most able geometers have maintained that the fifth book of Euclid is a masterpiece of skilful reasoning; and that none of the attempts which have been made to supersede it, have been successful in preserving the same unbroken chain of strict geometrical reasoning.

This objection, however, to Euclid's method of treating proportion, may be, to a great extent, removed by comparing his definition with the arithmétical one just given, and by showing that both lead to the same results. We have already explained that all species of geometrical magnitude may be expressed by letters and numbers, and we shall therefore proceed to illustrate and explain Euclid's definition by reasoning drawn from the properties of proportional numbers. We have just stated that four numbers are considered proportionals when the quotient arising from the division of the first by the second is equal to that arising from the division of the third by the fourth. Now in performing this division it may happen that the second term is not exactly contained in the first, but that a certain remainder is left; in such case we multiply this remainder by 10, and again divide by the second term, and if a fresh remainder arises, we again multiply it by 10 and repeat the division, and thus proceed either until no remainder is left, or until the remainder is too small to be of any consequence. And if instead of numbers we had two magnitudes (A and B) to deal with, we should proceed in a manner precisely similar, for, supposing B to be the

lesser, we should, by continual subtraction of B from A until a magnitude was left less than B, determine how often B was contained in A; the remaining magnitude we should then increase, say 10 times, and again subtract B until another remainder less than B was obtained, which should be again increased by 10, and the process continued until a sufficiently accurate result had been obtained. The series of products thus obtained should then be ranged in order, placing first the number of times that B was contained in A, then in the first remainder, then in the second, and so on through the whole series. And it is obvious that the process which we have described may be performed with any two magnitudes of the same kind, whether lines, surfaces, solids, or angles.

Now if in place of two magnitudes we have four, A, B, C, and D, and upon dividing A by B, and C by D, we in both cases obtain identical results, that is to say, that the two serieses of products, derived from the division of A by B, and of C by D, when arranged in similar order shall be identical, then the four magnitudes which A, B, C, and D represent will be in proportion.

Now if in place of multiplying any successive number of remainders by 10, the magnitude to be divided had, in the first instance, been multiplied by the product of that number of tens, and then divided by the second magnitude, the quotient obtained would be identical with that already derived by the first process. Thus, if instead of three successive remainders having been multiplied by 10, and the division subsequently performed upon them, the first magnitude had been multiplied by the product of 3 tens, or by 1000, and then the division performed, no difference would be found in the quotient obtained. Therefore our test for the proportionality of the four magnitudes may be thus expressed:-If the first, when multiplied any number of times by 10, and then divided by the second, gives the same quotient as the third multiplied the same number of times by 10, and divided by the fourth, the four magnitudes are proportional.

Again, it must be evident that any number might be substituted for 10, which has only been adopted in the foregoing explanation, because its use is familiar in arithmetic. And our test may therefore be generalized as follows:-If the first multiplied by any number, and divided by the second, gives the same quotient as the third multiplied by the same number, and divided by the fourth, the four magnitudes are proportional Or, to bring it still nearer to the language of Euclid's definition:-The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, the second is contained as often in the equimultiple of the first, as the fourth is contained in the equimultiple of the third.

Now let A, B, C, D, be four magnitudes determined to be in proportion by the test just given; let m be the number by which the first and third are to be multiplied, and n the quotient derived by the subsequent division by the other terms. Since, therefore, A, B, C, and D are proportional,

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mA m C



n, mA = n B, and, similarly, mC = n D,

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n, and therefore m A is < n B, and m C is < nD.

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