any such fractions may be reduced to other equivalent fractions having one common denominator, and that fraction whose denominator is the common denominator, and whose numerator is unity, will measure any one of the fractions. Two magnitudes having a common measure can be represented by two numbers which express the number of times the common measure is contained in both the magnitudes: but two incommensurable magnitudes cannot be exactly represented by any two whole numbers or fractions whatever.

Incommensurability arises out of the attempt to express numerically the relation of one magnitude to another of the same kind, either by taking one of them as a standard unit to measure the other, or by taking some third magnitude as a standard unit by which to measure both. The possibility of finding a standard unit, however small, which will exactly measure any given magnitudes, is not a likely occurrence: for instance, the side and the diagonal of a square are two lines, and the square on the diagonal is double the square on the side universally and constantly: but what is the relation between the side and diagonal themselves? The answer generally given is, if the side of the square be unity, the diagonal is the square root of 2 or √2: but what is √2? Simply no number at all in the original sense of the word as the ancients understood the term. It cannot be expressed in the decimal scale, nor in any other conceivable scale. An approximation to its value may be made to any assigned degree of accuracy. Numbers may be assigned at any stage of the approximation, between which its value lies, but its true and exact value can never be expressed. For, it may be shewn, numerically, that if the side of a square contain one unit of length, the diagonal contains more than one, but less than two units of length. If the side be divided into 10 units, the diagonal contains more than 14, but less than 15 such units. Also if the side contain 100 units, the diagonal contains more than 141, but less than 142 such units. And again, if the side contain 1000 units, the diagonal contains more than 1414, but less than 1415 such units. It is also obvious, that as the side is successively divided into a greater number of equal parts, the error in the magnitude of the diagonal will be diminished continually, but never can be entirely exhausted; and therefore into whatever number of equal parts the side of a square be divided, the diagonal will never contain an exact number of such parts. Thus the diagonal and side of a square having no common measure, cannot be exactly represented by any two numbers.

The term equimultiple in Geometry is to be understood of magnitudes of the same kind, or of different kinds, taken an equal number of times, and implies only a division of the magnitudes into the same number of equal parts. Thus, if two given lines are trebled, the trebles of the lines are equimultiples of the two lines: and if a given line and a given triangle be trebled, the trebles of the line and triangle are equimultiples of the line and triangle: as (v1. 1. fig.) the straight line HC and the triangle AHC are equimultiples of the line BC and the triangle ABC: ́ and in the same manner, (v1, 33. fig.) the arc EN and the angle EHN are equimultiples of the arc EF and the angle EHF.

Def. III. Λόγος ἐστὶ δύο μεγεθών ὁμογενῶν ἡ κατὰ πηλικότητα πρὸς ἄλληλα Toià σxécis. By this definition of ratio is to be understood the conception of the mutual relation of two magnitudes of the same kind, as two straight lines, two angles, two surfaces, or two solids. To prevent any misconception, Def. iv. lays down the criterion, whereby it may be known what kinds of magnitudes can have. a ratio to one another; namely, Λόγον ἔχειν πρὸς ἄλληλα μεγέθη λέγεται, ἃ δύναται πολλαπλασιαζόμενα ἀλλήλων ὑπερέχειν. Magnitudes are said to have a ratio to one another, which, when they are multiplied, can exceed one another;" in other words, the magnitudes which are capable of mutual comparison must be of the

same kind. The former of the two terms is called the antecedent; and the latter, the consequent of the ratio. If the antecedent and consequent are equal, the ratio is called a ratio of equality; but if the antecedent be greater or less than the consequent, the ratio is called a ratio of greater or of less inequality. Care must be taken not to confound the expressions "ratio of equality", and "equality of ratio:" the former is applied to the terms of a ratio when they, the antecedent and consequent, are equal to one another, but the latter, to two or more ratios, when they are equal.

The ratios which form the subject of the propositions of the Fifth Book are the ratios of finite magnitudes, which are not supposed to admit of any variation. No property is founded on a single ratio, nor on the magnitudes which constitute a ratio; but all the reasonings on ratios in this Book are restricted to two or more ratios, as being equal to, greater than, or less than one another. The questionwhat do ratios become when one of the terms, or both of the terms of the ratios are increased or diminished indefinitely, does not form any part of the subject of the Fifth Book of the Elements.

The simple idea of ratio itself, absolutely considered, could not, in any way, lead to any conclusion respecting the properties of figures, any more than the mere idea of magnitude. It is by the comparison of two or more magnitudes subjected to some specified conditions in the first Four Books of the Elements, that all the propositions have been demonstrated: and it is by the comparison of the ratios of two or more pairs of correlative magnitudes, subject to specified conditions, that the properties of figures depending on ratio are to be established. As each ratio involves the idea of two magnitudes, the least number of magnitudes between which a comparison of ratios is possible, is four, two for each of the ratios: and when these ratios are equal, the sixth definition gives the name of proportionals to the four magnitudes which constitute the two ratios.

Arithmetical ratio has been defined to be the relation which one number bears to another with respect to quotity; the comparison being made by considering what multiple, part or parts, one number is of the other.

An arithmetical ratio, therefore, is represented by the quotient which arises from dividing the antecedent by the consequent of the ratio; or by the fraction which has the antecedent for its numerator and the consequent for its denominator, Hence it will at once be obvious that the properties of arithmetical ratios will be made to depend on the properties of fractions.

It must ever be borne in mind that the subject of Geometry is not number, but the magnitude of lines, angles, surfaces, and solids; and its object is to demonstrate their properties by a comparison of their absolute and relative magnitudes.

Also, in Geometry, multiplication is only a repeated addition of the same magnitude; and division is only a repeated subtraction, or the taking of a less magnitude successively from a greater, until there be either no remainder, or a remainder less than the magnitude which is successively subtracted.

The Geometrical ratio of any two given magnitudes of the same kind will obviously be represented by the magnitudes themselves; thus, the ratio of two lines is represented by the lengths of the lines themselves; and, in the samė manner, the ratio of two angles, two surfaces, or two solids, will be properly represented by the magnitudes themselves.

In the definition of ratio as given by Euclid, all reference to a third magnitude of the same geometrical species, by means of which, to compare the two, whose ratio is the subject of conception, has been carefully avoided. The ratio of the two magnitudes is their relation one to the other, without the intervention of any standard unit whatever, and all the propositions demonstrated in the Fifth Book respecting

the equality or inequality of two or more ratios, are demonstrated independently of any knowledge of the exact numerical measures of the ratios; and their generality includes all ratios, whatever distinctions may be made, as to the terms of them being commensurable or incommensurable.

In measuring any magnitude, it is obvious that a magnitude of the same kind must be used; but the ratio of two magnitudes may be measured by every thing which has the property of quantity. Two straight lines will measure the ratio of two triangles, or parallelograms (vI. 1. fig.): and two triangles, or two parallelograms will measure the ratio of two straight lines. It would manifestly be absurd to speak of the line as measuring the triangle, or the triangle measuring the line. (See notes on Book II.)

The ratio of any two quantities depends on their relative and not their absolute magnitudes; and it is possible for the absolute magnitude of two quantities to be changed, and their relative magnitude to continue the same as before; and thus, the same ratio may subsist between two given magnitudes, and any other two of the same kind.

In this method of measuring Geometrical ratios, the measures of the ratios are the same in number as the magnitudes themselves. It has however two advantages; first, it enables us to pass from one kind of magnitude to another, and thus, independently of any numerical measure, to institute a comparison between such magnitudes as cannot be directly compared with one another: and secondly, the ratio of two magnitudes of the same kind may be measured by two straight lines, which form a simpler measure of ratios than any other kind of magnitude.

But the simplest method of all would be, to express the measure of the ratio of two magnitudes by one; but this cannot be done, unless the two magnitudes are commensurable. If there be two lines, one of which AB contains 12 units of any length, and the other CD contains 4 units of the same length; then the ratio of the line AB to the line CD, is the same as the ratio of the number 12 to 4. Thus, two numbers may represent the ratio of two lines when the lines are commensurable. In the same manner, two numbers may represent the ratio of two angles, two surfaces, or two solids.

Thus, the ratio of any two magnitudes of the same kind may be expressed by two numbers, when the magnitudes are commensurable. By this means, the consideration of the ratio of two magnitudes is changed to the consideration of the ratio of two numbers, and when one number is divided by the other, the quotient will be a single number, or a fraction, which will be a measure of the ratio of the two numbers, and therefore of the two quantities. If 12 be divided by 4, the quotient is 3, which measures the ratio of the two numbers 12 and 4. Again, if besides the ratio of the lines A B and CD which contain 12 and 4 units respectively, we consider two other lines EF and GH which contain 9 and 3 units respectively; it is obvious that the ratio of the line EF to GH is the same as the ratio of the number 9 to the number 3. And the measure of the ratio of 9 to 3 is 3. That is, the numbers 9 and 3 have the same ratio as the numbers 12 and 4.

But this is a numerical measure of ratio, and can only be applied strictly when the antecedent and consequent are to one another as one number to another.

And generally, if the two lines AB, CD contain a and b units respectively, and q be the quotient which indicates the number of times the number b is contained in a, then q is the measure of the ratio of the two numbers a and b and if EF and GA contain c and d units, and the number d be contained q times in c: the number a has to b the same ratio as the number c has to d.

This is the numerical definition of proportion, which is thus expressed in Euclid's Elements, Book VII, definition 20. "Four numbers are proportionals when the first

is the same multiple of the second, or the same part or parts of it, as the third is of the fourth." This definition of the proportion of four numbers, leads at once to an equation:

α с

therefore

Չ

whch is the fundamental equation upon which all the reasonings b d

on the proportion of numbers depend.

If four numbers be proportionals, the product of the extremes is equal to the product of the means.

For if a, b, c, d be proportionals, or a : b :: cd,

that is, the product of the extremes is equal to the product of the means.

And conversely, If the product of the two extremes be equal to the product of the two means, the four numbers are proportionals.

For if a, b, c, d be four quantities, such that ad = bc,

or the first number has the same ratio to the second, as the third has to the fourth.

α

b

If c = b, then ad = b2; and conversely if ad = 63: then

b

d

These results are analogous to Props. 16 and 17 of the Sixth Book.

Def. v. This definition lays down a criterion by which two ratios may be known to be equal, or four magnitudes proportionals, without involving any inquiry respecting the four quantities, whether the antecedents of the ratios contain, or are contained in, their consequents exactly; or whether there are any magnitudes which measure the terms of the two ratios. The criterion only requires, that the relation of the equimultiples expressed should hold good, not merely for any particular multiples, as the doubles or trebles, but for any multiples whatever, whether large or small.

This criterion of proportion may be applied to all Geometrical magnitudes which can be multiplied, that is, to all which can be doubled, trebled, quadrupled, &c. But it must be borne in mind, that this criterion does not exhibit a definite measure for either of the two ratios which constitute the proportion, but only, an undetermined measure for the sameness or equality of the two ratios. The nature of the proportion of Geometrical magnitudes neither requires nor admits of a numerical measure of either of the two ratios, for this would be to suppose that all magnitudes are commensurable. Though we know not the definite measure of either of the ratios, further than that they are both equal, and one may be taken as the measure of the other, yet particular conclusions may be arrived at by this method : for by the test of proportionality here laid down, it can be proved that one magnitude is greater than, equal to, or less than another: that a third proportional can be found to two, and a fourth proportional to three straight lines, also that a mean proportional can be found between two straight lines: and

further, that which is here stated of straight lines may be extended to other Geometrical magnitudes.

The fifth definition is that of equal ratios. The definition of ratio itself (defs. 3, 4) contains no criterion by which one ratio may be known to be equal to another ratio: analogous to that by which one magnitude is known to be equal to another magnitude (Euc. 1. Ax. 8). The preceding definitions (3, 4) only restrict the conception of ratio within certain limits, but lay down no test for comparison, or the deduction of properties. All Euclid's reasonings were to turn upon this comparison of ratios, and hence it was competent to lay down a criterion of equality and inequality of two ratios between two pairs of magnitudes. In short, his effective definition is a definition of proportionals.

The precision with which this definition is expressed, considering the number of conditions involved in it, is remarkable. Like all complete definitions the terms (the subject and predicate) are convertible: that is,

(a) If four magnitudes be proportionals, and any equimultiples be taken as prescribed, they shall have the specified relations with respect to "greater, greater," &c.

(b) If of four magnitudes, two and two of the same Geometrical Species, it can be shewn that the prescribed equimultiples being taken, the conditions under which those magnitudes exist, must be such as to fulfil the criterion "greater, greater, &c."; then these four magnitudes shall be proportionals.

It may be remarked, that the cases in which the second part of the criterion ("equal, equal") can be fulfilled, are comparatively few: namely those in which the given magnitudes, whose ratio is under consideration, are both exact multiples of some third magnitude-or those which are called commensurable. When this, however, is fulfilled, the other two will be fulfilled as a consequence of this. When this is not the case, or the magnitudes are incommensurable, the other two criteria determine the proportionality. However, when no hypothesis respecting commensurability is involved, the contemporaneous existence of the three cases ("greater, greater; equal, equal; less, less") must be deduced from the hypothetical conditions under which the magnitudes exist, to render the criterion valid.

With respect to this test or criterion of the proportionality of four magnitudes, it has been objected, that it is utterly impossible to make trial of all the possible equimultiples of the first and third magnitudes, and also of the second and fourth. It may be replied, that the point in question is not determined by making such trials, but by shewing from the nature of the magnitudes, that whatever be the multipliers, if the multiple of the first exceeds the multiple of the second magnitude, the multiple of the third will exceed the multiple of the fourth magnitude, and if equal, will be equal; and if less, will be less, in any case which may be taken.

The Arithmetical definition of proportion in Book VII, Def. 20, even if it were equally general with the Geometrical definition in Book v, Def. 5, is by no means universally applicable to the subject of Geometrical magnitudes. The Geome-trical criterion is founded on multiplication, which is always possible. When the magnitudes are commensurable, the multiples of the first and second may be equal or unequal: but when the magnitudes are incommensurable, any multiples whatever of the first and second must be unequal: but the Arithmetical criterion of proportion is founded on division, which is not always possible. Euclid has not shewn in Book v, how to take any part of a line or other magnitude, or that the two terms of a ratio have a common measure, and therefore the numerical definition could not be strictly applied, even in the limited way in which it may be applied. Number and Magnitude do not correspond in all their relations; and hence the

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