or the first number has the same ratio to the second, as the third has to the fourth. a If cb, then ad = and conversely if ad = 62: then = b d These results are analogous to Props. 16 and 17 of the Sixth Book. Sometimes a proportion is defined to be the equality of two ratios. Def. vIII declares the meaning of the term analogy or proportion. The ratio of two lines, two angles, two surfaces or two solids, means nothing more than their relative magnitude in contradistinction to their absolute magnitudes; and a similitude or likeness of ratios implies, at least, the two ratios of the four magnitudes which constitute the analogy or proportion. Def. Ix states that a proportion consists in three terms at least; the meaning of which is, that the second magnitude is repeated, being made the consequent of the first, and the antecedent of the second ratio. It is also obvious that when a proportion consists of three magnitudes, all three are of the same kind. Def. vi appears only to be a further explanation of what is implied in Def. vIII. Def. v. Proportion having been defined to be the similitude of ratios, or more properly, the equality or identity of ratios, the fifth 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 shew 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 Geometrical 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 ir. 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 distinction between Geometrical ratio and Arithmetical ratio; the former is a comparision κatà wŋλikótηta, according to quantity, but the latter, according to quotity. The former gives an undetermined, though definite measure, in magnitudes; but the latter attempts to give the exact value in numbers. The fifth book exhibits no method whereby two magnitudes may be determined to be commensurable, and the Geometrical conclusions deduced from the multiples of magnitudes are too general to furnish a numerical measure of ratios, being all independent of the commensurability or incommensurability of the magnitudes themselves. It is the numerical ratio of two magnitudes which will more certainly discover whether they are commensurable or incommensurable, and hence, recourse must be had to the forms and properties of numbers. All numbers and fractions are either rational or irrational. It has been seen that rational numbers and fractions can express the ratios of Geometrical magnitudes, when they are commensurable. Similar relations of incommensurable magnitudes may be expressed by irrational numbers, if the Algebraical expressions for such numbers may be assumed and employed in the same manner as rational numbers. The irrational expressions being considered the exact and definite, though undetermined, values of the ratios, to which a series of rational numbers may successively approximate. Though two incommensurable magnitudes have not an assignable numerical ratio to one another, yet they have a certain definite ratio to one another, and two other magnitudes may have the same ratio as the first two and it will be found, that, when reference is made to the numerical value of the ratios of four incommensurable magnitudes, the same irrational number appears in the two ratios. The sides and diagonals of squares can be shewn to be proportionals, and though the ratio of the side to the diagonal is represented Geometrically by the two lines which form the side and the diagonal, there is no rational number or fraction which will measure exactly their ratio. If the side of a square contain a units, the ratio of the diagonal to the side is numerically as √2 to 1; and if the side of another square contain b units, the ratio of the diagonal to the side will be found to be in the ratio of √2 to 1. Again, the two parts of any number of lines which may be divided in extreme and mean ratio will be found to be respectively in the ratio of the irrational number √5-1 to 3 √5. Also, the ratios of the diagonals of cubes to the diagonals of one of the faces will be found to be in the irrational or incommensurate ratio of √3 to √ 2. Thus it will be found that the ratios of all incommensurable magnitudes which are proportionals do involve the same irrational numbers, and these may be used as the numerical measures of ratios in the same manner as rational numbers and fractions. It is not however to such enquiries, nor to the ratios of magnitudes when expressed as rational or irrational numbers, that Euclid's doctrine of proportion is legitimately directed. There is no enquiry into what a ratio is in numbers, but whether in diagrams formed according to assigned conditions, the ratios between certain parts of the one are the same as the ratios between corresponding parts of the other. Thus, with respect to any two squares, the question that properly belongs to pure Geometry is: whether the diagonals of two squares have the same ratio as the sides of the squares? Or whether the side of one square has to its diagonal, the same ratio as the side of the other square has to its diagonal? Or again, whether in Euc. vi. 2, when BC and DE are parallel, the line BD has to the line DA, the same ratio that the line CE has to the line M AE? There is no purpose on the part of Euclid, to assign either of these ratios in numbers: but only to prove that their universal sameness is inevitably a consequence of the original conditions according to which the diagrams were constituted. There is, consequently, no introduction of the idea of incommensurables: and indeed, with such an object as Euclid had in view, the simple mention of them would have been at least irrelevant and superfluous. If however it be attempted to apply numerical considerations to pure geometrical investigations, incommensurables will soon be apparent, and difficulties will arise which were not foreseen. Euclid, however, effects his demonstrations without creating this artificial difficulty, or even recognising its existence. Had he assumed a standard unit of length, he would have involved the subject in numerical considerations; and entailed upon the subject of Geometry the almost insuperable difficulties which attach to all such methods. It cannot, however, be too strongly or too frequently impressed upon the learner's mind, that all Euclid's reasonings are independent of the numerical expositions of the magnitudes concerned. That the enquiry as to what numerical function any magnitude is of another, belongs not to Pure Geometry, but to another Science. The consideration of any intermediate standard unit does not enter into pure Geometry; into Algebraic Geometry it essentially enters, and indeed constitutes the fundamental idea. The former is wholly free from numerical considerations; the latter is entirely dependent upon them. Def. vII is analogous to Def. 5, and lays down the criterion whereby the ratio of two magnitudes of the same kind may be known to be greater or less than the ratio of two other magnitudes of the same kind. Def. x1 includes Def. x. as three magnitudes may be continued proportionals, as well as four or more than four. In continued proportionals, all the terms except the first and last, are made successively the consequent of one ratio, and the antecedent of the next; whereas in other proportionals this is not the case. A series of numbers or Algebraical quantities in continued proportion, is called a Geometrical progression, from the analogy they bear to a series of Geometrical magnitudes in continued proportion. Def. A. The term compound ratio was devised for the purpose of avoiding circumlocution, and no difficulty can arise in the use of it, if its exact meaning be strictly attended to. With respect to the Geometrical measures of compound ratios, three straight lines may measure the ratio of four, as in Prop. 23, Book vi. For K to L measures the ratio of BC to CG, and L to M measures the ratio of DC to CE; and the ratio of K to Mis that which is said to be compounded of the ratios of K to L, and L to M, which is the same as the ratio which is compounded of the ratios of the sides of the parallelograms. Both duplicate and triplicate ratio are species of compound ratio. Duplicate ratio is a ratio compounded of two equal ratios; and in the case of three magnitudes which are continued proportionals, means the ratio of the first to a third proportional to the first and second. Triplicate ratio, in the same manner, is a ratio compounded of three equal ratios; and in the case of four magnitudes which are continued proportionals, the triplicate ratio of the first to the second means the ratio of the first to a fourth proportional to the first, second, and third magnitudes. Instances of the composition of three ratios, and of triplicate ratio, will be found in the eleventh and twelfth books. The product of the fractions which represent or measure the ratios of numbers, corresponds to the composition of Geometrical ratios of magnitudes. It has been shewn that the ratio of two numbers is represented by a fraction whereof the numerator is the antecedent, and the denominator the consequent of the ratio; and if the antecedents of two ratios be multiplied together, as also the consequents, the new ratio thus formed is said to be compounded of these two ratios; and in the same manner, if there be more than two. It is also obvious, that the ratio compounded of two equal ratios is equal to the ratio of the squares of one of the antecedents to its consequent; also when there are three equal ratios, the ratio compounded of the three ratios is equal to the ratio of the cubes of any one of the antecedents to its consequent. And further, it may be observed, that when several numbers are continued proportionals, the ratio of the first to the last is equal to the ratio of the product of all the antecedents to the product of all the consequents. It may be here remarked, that, though the constructions of the propositions in Book v are exhibited by straight lines, the enunciations are expressed of magnitude in general, and are equally true of angles, triangles, parallelograms, arcs, sectors, &c. The two following axioms may be added to the four Euclid has given. Ax. 5. A part of a greater magnitude is greater than the same part of a less magnitude. Ax. 6. That magnitude of which any part is greater than the same part of another, is greater than that other magnitude. The learner must not forget that the capital letters, used generally by Euclid in the demonstrations of the fifth Book, represent the magnitudes, not any numerical or Algebraical measures of them: sometimes however the magnitude of a line is represented in the usual way by two letters which are placed at the extremities of the line. Prop. 1. Algebraically. Let each of the magnitudes A, B, C, &c. be equimultiples of as many a, b, c, &c. that is, let A = m times a = ma, Bm times b= mb, Cm times cmc, &c. First, if there be two magnitudes equimultiples of two others, Then A+B = ma + mb = m (a + b) = m times (a + b), Hence A + B is the same multiple of (a + b), as A is of a, or B of b. Secondly, if there be three magnitudes equimultiples of three others, then A + B + C: = ma + mb + mc = m (a + b + c) = m times (a + b + c), Hence A + B + C is the same multiple of (a + b + c); as A is of a, B of b, and C of c. Similarly, if there were four, or any number of magnitudes. Therefore, if any number of magnitudes be equimultiples of as many, each of each; what multiple soever, any one is of its part, the same multiple shall the first magnitudes be of all the other. Prop.. Algebraically. Let A, the first magnitude, be the same multiple of a, the second, as A, the third, is of a, the fourth; and 4, the fifth the same multiple the second, as A, the sixth, is of as the fourth. of a |