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also, let the ratio of m to p, be that which is compounded of the ratios of m to n, n to o, and o to p, which are the same, each to each, to the remaining other ratios, viz. of M to N, O to P, and to R, Then the ratio of h to I shall be the same to the ratio of m top; or h shall be to l, as m to p.

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and fis to g, as (K to L, that is, as) Z to a;

therefore, ex æquali, e is to g, as Y to a: (v. 22.)
and by the hypothesis, A is to B, that is, S to T, as e to g;
wherefore S is to T, as Y to a; (v. 11.)

and by inversion, Tis to S, as a to Y: (V. B.)
but S is to X, as Y to D; (hyp.)

therefore, ex æquali, Tis to X, as a to d:

also, because h is to k, as (C to D, that is, as) T' to V; (hyp.)
and k is to las (E to F, that is, as)_V to X;
therefore, ex æquali, h is to l, as T to X:

in like manner, it may be demonstrated, that m is to p, as a to d; and it has been shewn, that T is to X, as a to d; therefore h is to l, as m to p. (v. 11.) Q. E.D.

The propositions G and K are usually, for the sake of brevity, expressed in the same terms with propositions Fand H: and therefore it was proper to shew the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers.

IN the first four Books of the Elements are considered, only the absolute equality and inequality of Geometrical magnitudes. The Fifth Book contains an exposition of the principles whereby a more definite comparison may be instituted of the relation of magnitudes, besides their simple equality or inequality.

The doctrine of Proportion is one of the most important in the whole course of mathematical truths, and it appears probable that if the subject were read simultaneously in the Algebraical and Geometrical form, the investigations of the properties, under both aspects, would mutually assist each other, and both become equally comprehensible; also their distinct characters would be more easily perceived.


Def. I, II. In the first Four Books the word part is used in the same sense as we find it in the ninth axiom, "The whole is greater than its part:" where the word part means any portion whatever of any whole magnitude but in the Fifth Book, the word part is restricted to mean that portion of magnitude which is contained an exact number of times in the whole. For instance, if any straight line be taken two, three, four, or any number of times another straight line, by Euc. 1. 3; the less line is called a part, or rather a submultiple of the greater line; and the greater, a multiple of the less line. The multiple is composed of a repetition of the same magnitude, and these definitions suppose that the multiple may be divided into its parts, any one of which is a measure of the multiple. And it is also obvious that when there are two magnitudes, one of which is a multiple of the other, the two magnitudes must be of the same kind, that is, they must be two lines, two angles, two surfaces, or two solids: thus, a triangle is doubled, trebled, &c., by doubling, trebling, &c. the base, and completing the figure. The same may be said of a parallelogram. Angles, arcs, and sectors of equal circles may be doubled, trebled, or any multiples found by Prop. xxvi-xxix, Book III.

Two magnitudes are said to be commensurable when a third magnitude of the same kind can be found which will measure both of them; and this third magnitude is called their common measure: and when it is the greatest magnitude which will measure both of them, it is called the greatest common measure of the two magnitudes: also when two magnitudes of the same kind have no common measure, they are said to be incommensurable. The same terms are also applied to numbers.

Unity has no magnitude, properly so called, but may represent that portion of every kind of magnitude which is assumed as the measure of all magnitudes of the same kind. The composition of unities cannot produce Geometrical magnitude; three units are more in number than one unit, but still as much different from magnitude as unity itself. Numbers may be considered as quantities, for we consider every thing that can be exactly measured, as a quantity.

Unity is a common measure of all rational numbers, and all numerical reasonings proceed upon the hypothesis that the unit is the same throughout the whole of any particular process. Euclid has not fixed the magnitude of any unit of length, nor made reference to any unit of measure of lines, surfaces, or volumes. Hence arises an essential difference between number and magnitude; unity, being invariable, measures all rational numbers; but though any quantity be assumed as the unit of magnitude, it is impossible to assert that this assumed unit will measure all other magnitudes of the same kind.

All whole numbers therefore are commensurable; for unity is their common measure: also all rational fractions proper or improper, are com mensurable; for 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; as, for instance, the side of a square is incommensurable to the diagonal of the square. For, it may be shewn numerically, that if the side of the 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. 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 (vI. 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, (vi. 33. fig.) the arc EN and the angle EHN are equimultiples of the arc EF and the angle EHF.


Def. III. Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητα πρὸς ἄλληλα ποιὰ σχέσις. 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.

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 same 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 two lines AB, CD, 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 AB 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 GH contain c and d units, and the number d be contained 9 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:





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which is the fundamental equation upon which all the

reasonings 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 :: c : d.

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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,

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