Euclid. # Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools online

. **(page 24 of 38)**

Online Library → Euclid → Euclid's Elements of geometry, the first six books, chiefly from the text of Dr. Simson, with explanatory notes; a series of questions on each book ... Designed for the use of the junior classes in public and private schools → online text (page 24 of 38)

Font size

greatest magnitude which will measure both of them, it is called the

greatest common measure of the two magnitudes : also when two magni-

tudes 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 pro-

duce 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 through-

out the whole of any particular process. Euclid has not fixed the magni-

tude of any unit of length, nor made reference to any unit of measure of

lines, surfaces, or volumes. Hence arises an essential diff'erence 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.

EUCLID S ELEMENTS.

All whole numbers therefore are commensurable ; for unity is their

common measure r 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 de-

nominator 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 what-

ever 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 repre-

sented by any two numbers.

The terra 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^ine 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. Ao'yos- icTTL duo fxtytduiv ofioysvcvu i] Kara 'TrrjXi/cdxjjTa tt/oo?

d\Xi]Xa iroid o-xÂ£<Tts. By this definition of ratio is to be understood the con-

ception 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,

Aoyov BX^'-i' -Jrpo'i d\.\r\\a /jLtyidi] Xg'ysTat, a duuuTai Tro/WaTrA.acria^d/ifj/a

d\Xi]Xu)v vTTEpsx^i-i^' " Magnitudes are said to have a ratio to one another,

which, when they are multiplied, can exceed one another ;" in other w^ords,

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 ex])rfcssions " ratio

of equality", and â€¢' 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.

KOTES TO BOOK V. 237

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 re-

mainder, or a remainder le|| 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 pro-

positions demonstrated in the Fifth Book respecting the equality or i7ie-

quality of two or more ratios, are demonstrated independently of any know-

ledge 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 it.)

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 how-

ever two advantages ; first, it enables us to pass from one kind of magni-

tude 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 tico 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

238 Euclid's elements.

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 ex-

pressed 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 afractioriy

which will be a measrire of the ratio of the two numbers, and therefore of

the two quantities. If 12 be divided by 4, the quotient is 3, which mea-

Bures 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 con-

sider two other lines Â£Fand G// 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 respec-

tively, 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 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 pro-

portionals 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 :

for, since a contains 6, q times \ - = q;

o

c

and since c contains d, q times ; - = <7 :

d

therefore â€¢; = -, which is the fundamental equation upon which all the

b d

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 i: c i d.

Then 3 = 5;

Multiply these equals by bd,

ahd chd

â€¢*â€¢ ~b' ^~d*

or, ad = bCy

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,

NOTES TO BOOK V. 239

such that ad = bcy

a c

then dividing these equals by bd^ therefore - =

h d

and a '. h :'. c '. d^

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

the fourth.

If c = 6, then ad = b^; and conversely if ad = b^ : then - = -, .

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 con-

tained 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 magni-

tudes 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 be-

tween 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. i. Ax. 8). The preceding

definitions (3, 4) only restrict the conception of ratio within certain limits,

240 Euclid's elements.

but lay down no test for comparison, or tlie 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 de-

finition 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 he 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 â€” 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 pro-

portionality. However, when no hypothesis respecting commensur-

ability is involved, the contemporaneous existence of the three cases

(â€¢' greater, greater; equal, equal ; less, less") must be deduced from the

hypothetical conditions under which the magnitudes exist, to render the

criterion valid.

With re$pect 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, icill be equal ; and if less, will be less, in any case which

may be taken.

The Ai-ithmetical definition of proportion in Book vii, Def. 20, even

if it were equally general with the Geometrical definition in Book v, Def.

6, 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 th^

magnitudes are incommensurable, any multiples whatever of the first and

second mwsi 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

tAvo 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 /card TrjjXtKOTTjra, according to quantity, but

NOTES TO BOOK V. S41

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 de-

duced from the multiples of magnitudes are too general to furnish a

numerical measure of ratios, being all independent of the commensura-

bility 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 ca7i express the ratios of Geo-

metrical magnitudes, when they are commensurable. Similar relations

( f incommensurable magnitudes may be expressed by irrational numbers,

li' the Algebraical expressions for such numbers may be assumed and

emploved in the same manner as rational numbers. The irrational

expressions being considered the exact and definite, though undeter-

mined, values of the ratios, to which a series of rational numbers may

successively approximate.

Though two incommensurable magnitudes have not an assignable numeÂ«

rical 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 irra-

tional 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 Geome-

trically 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 V 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 V 2 10 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

m the ratio of the irrational number V5 â€” 1 to 3 â€” VS. 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 V 3 to v/ 2.

Thus it will be found that the ratios of all incommensurable magni-

tudes 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 : â€” 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, Avhen BC and DE are parallel, the line

BD has to the line D^, the same ratio that the line CE has to the line

M

242

AE ? There is no purpose on the part of Euclid, to assign either of these

ratios in tmmbers: 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 numeri-

cal 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 arti-

ficial difficulty, or even recognising its existence. Had he assumed a

standard unit of length, he would have involved the subject in numeri-

cal 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 p\ire Geometry ; into

Algebraic Geometry it essentially enters, and indeed constitutes the funda-

mental 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. XI includes Def. x. as three magnitudes may be continued pro-

portionals, as well as four or more than four. In continued proportionals,

all the terms except the first and last, are made successively the conse-

greatest common measure of the two magnitudes : also when two magni-

tudes 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 pro-

duce 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 through-

out the whole of any particular process. Euclid has not fixed the magni-

tude of any unit of length, nor made reference to any unit of measure of

lines, surfaces, or volumes. Hence arises an essential diff'erence 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.

EUCLID S ELEMENTS.

All whole numbers therefore are commensurable ; for unity is their

common measure r 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 de-

nominator 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 what-

ever 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 repre-

sented by any two numbers.

The terra 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^ine 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. Ao'yos- icTTL duo fxtytduiv ofioysvcvu i] Kara 'TrrjXi/cdxjjTa tt/oo?

d\Xi]Xa iroid o-xÂ£<Tts. By this definition of ratio is to be understood the con-

ception 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,

Aoyov BX^'-i' -Jrpo'i d\.\r\\a /jLtyidi] Xg'ysTat, a duuuTai Tro/WaTrA.acria^d/ifj/a

d\Xi]Xu)v vTTEpsx^i-i^' " Magnitudes are said to have a ratio to one another,

which, when they are multiplied, can exceed one another ;" in other w^ords,

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 ex])rfcssions " ratio

of equality", and â€¢' 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.

KOTES TO BOOK V. 237

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 re-

mainder, or a remainder le|| 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 pro-

positions demonstrated in the Fifth Book respecting the equality or i7ie-

quality of two or more ratios, are demonstrated independently of any know-

ledge 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 it.)

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 how-

ever two advantages ; first, it enables us to pass from one kind of magni-

tude 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 tico 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

238 Euclid's elements.

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 ex-

pressed 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 afractioriy

which will be a measrire of the ratio of the two numbers, and therefore of

the two quantities. If 12 be divided by 4, the quotient is 3, which mea-

Bures 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 con-

sider two other lines Â£Fand G// 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 respec-

tively, 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 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 pro-

portionals 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 :

for, since a contains 6, q times \ - = q;

o

c

and since c contains d, q times ; - = <7 :

d

therefore â€¢; = -, which is the fundamental equation upon which all the

b d

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 i: c i d.

Then 3 = 5;

Multiply these equals by bd,

ahd chd

â€¢*â€¢ ~b' ^~d*

or, ad = bCy

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,

NOTES TO BOOK V. 239

such that ad = bcy

a c

then dividing these equals by bd^ therefore - =

h d

and a '. h :'. c '. d^

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

the fourth.

If c = 6, then ad = b^; and conversely if ad = b^ : then - = -, .

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 con-

tained 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 magni-

tudes 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 be-

tween 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. i. Ax. 8). The preceding

definitions (3, 4) only restrict the conception of ratio within certain limits,

240 Euclid's elements.

but lay down no test for comparison, or tlie 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 de-

finition 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 he 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 â€” 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 pro-

portionality. However, when no hypothesis respecting commensur-

ability is involved, the contemporaneous existence of the three cases

(â€¢' greater, greater; equal, equal ; less, less") must be deduced from the

hypothetical conditions under which the magnitudes exist, to render the

criterion valid.

With re$pect 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, icill be equal ; and if less, will be less, in any case which

may be taken.

The Ai-ithmetical definition of proportion in Book vii, Def. 20, even

if it were equally general with the Geometrical definition in Book v, Def.

6, 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 th^

magnitudes are incommensurable, any multiples whatever of the first and

second mwsi 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

tAvo 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 /card TrjjXtKOTTjra, according to quantity, but

NOTES TO BOOK V. S41

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 de-

duced from the multiples of magnitudes are too general to furnish a

numerical measure of ratios, being all independent of the commensura-

bility 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 ca7i express the ratios of Geo-

metrical magnitudes, when they are commensurable. Similar relations

( f incommensurable magnitudes may be expressed by irrational numbers,

li' the Algebraical expressions for such numbers may be assumed and

emploved in the same manner as rational numbers. The irrational

expressions being considered the exact and definite, though undeter-

mined, values of the ratios, to which a series of rational numbers may

successively approximate.

Though two incommensurable magnitudes have not an assignable numeÂ«

rical 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 irra-

tional 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 Geome-

trically 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 V 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 V 2 10 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

m the ratio of the irrational number V5 â€” 1 to 3 â€” VS. 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 V 3 to v/ 2.

Thus it will be found that the ratios of all incommensurable magni-

tudes 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 : â€” 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, Avhen BC and DE are parallel, the line

BD has to the line D^, the same ratio that the line CE has to the line

M

242

AE ? There is no purpose on the part of Euclid, to assign either of these

ratios in tmmbers: 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 numeri-

cal 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 arti-

ficial difficulty, or even recognising its existence. Had he assumed a

standard unit of length, he would have involved the subject in numeri-

cal 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 p\ire Geometry ; into

Algebraic Geometry it essentially enters, and indeed constitutes the funda-

mental 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. XI includes Def. x. as three magnitudes may be continued pro-

portionals, as well as four or more than four. In continued proportionals,

all the terms except the first and last, are made successively the conse-

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38