Euclid. # A text-book of Euclid's Elements : for the use of schools : Books I-VI and XI online

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diculars are drawn to the sides the sum of the perpendiculars is equal

to n times the radius of the inscribed circle. ''

11. The sum of the perpendiculars drawn from the vertices of a

regular polygon of n sides on any straight line is equal to n times the

perpendicular drawn from the centre of the inscribed circle.

12. The area of a cyclic quadrilateral is independent of the order

in which the sides are placed in the circle.

13. Of all quadrilaterals which can be formed of four straight

lines of given length, that which is cyclic has the maximum area.

14. Of all polygons of a given number of sides, which may be

inscribed in a given circle, that which is regular has the maximum

area and the maximum perimeter.

15. Of all polygons of a given number of sides circumscribed

about a given circle, that which is regular has the minimum area and

the minimum perimeter.

16. Given the vertical angle of a triangle in position and magni-

tude, and the sum of the sides containing it: find the locus of the

centre of the circumscribed circle.

17. P is any point on the circumference of a circle circumscribed

about an equilateral triangle ABC: shew that PA-+ PB-'-f PC- ia

constant.

BOOK V.

Book V. treats of Ratio and Proportion.

INTRODUCTORY.

The first four books of Euclid deal with the absolute equality

or inequality of Geometrical magnitudes. In the Fifth Book

magnitudes are compared by considering their ratio, or relative

greatness.

The meaning of the words ratio and proportion in their

simplest arithmetical sense, as contained in the following defini-

tions, is probably familiar to the student :

The ratio of one number to another is the multiple, part, or

parts that the first number is of the second; and it may therefore be

measured by the fraction of vihich the first number is the numerator

and the second the denominator.

Four numbers are in proportion when the ratio of the first to

the second is equal to that of the third to the fourth.

But it will be seen that these definitions are inapplicable to

Geometrical magnitudes for the following reasons :

(1) Pure Geometry deals only with concrete magnitudes, re-

presented by diagrams, but not referred to any common imit in

terms of which they are measured : in other words, it makes

no use of number for the purpose of comparison between different

magnitudes.

(2) It commonly happens that Geometrical magnitudes of

the same kind are incommensurable, that is, they are such that

it is impossible to express them exactly in terms of some common

unit.

For example, we can make comparison between the side and

diagonal of a square, and we may form an idea of their relative great-

ness, but it can be shewn that it is impossible to divide either of them

into equal parts of which the other contains an exact number. And

as the magnitudes we meet with in Geometry are more often incom-

mensurable than not, it is clear that it would not always be possible

to exactly represent such magnitudes by numbers, even if reference to

a common unit were not foreign to the principles of Euclid.

It is therefore necessary to establish the Geometrical Theory

of Proportion on a basis quite independent of Arithmetical

principles. This is the aim of Euclid's Fifth Book.

286 EUCLID'S ELEMENTS.

We shall employ the following notation.

Capital letters, A, B, C,.., will be used to denote the magnitudes

themselves, not nny numerical or algehraical vieamres of them, and

small letters, m, n, p,... will be used to denote whole numbers. Also

it will be assumed that multiplication, in the sense of repeated

addition, can be applied to any magnitude, so that 7U . A or wA will

denote the magnitude A taken vi times.

The symbol > will be used for the words greater than, and < for

le$s than. â€¢

Definitions.

1. A greater magnitude is said to be a multiple of a

less, wlien the greater contains the less an exact number of

times.

2. A less magnitude is said to be a submultiple of a

greater, wlien the less is contained an exact number of

times in the greater.

The following properties of multiples will be assumed as self-evident.

(1) 7uA > = or < 7/zB according as A > = or < B ; and

conversely.

(2) ?HA + wB + ...=?H(A + B-f-...).

(3) If A > B, then vik - mB=m (A - B).

(4) wA + nA+... = (m + Ji+...) A.

. (5) If m > n, then mA - nk â€” {m - 7/) A.

(6) ni . 7iA = run . A = nm . A = tz . 7/i A.

3. The Ratio of one magnitude to another of the same

kind is the relation which the first bears to the second in

respect of quantuplicity.

The ratio of A to B is denoted thus, A : B; and A is

called the antecedent, B the consequent of the ratio.

The term quantuplicity denotes the capacity of the first magnitude

to contain the second with or without remainder. If the magnitudes

are commensurable, their quantuplicity may be expressed numerically

by observing what multiples of the two magnitudes are equal to one

another.

Thus if A -ma, and B = na, it follows that nA â€” niB. In this case

A = â€” B, and the quantuplicity of A with respect to B is the arith-

metical fraction - .

11

DEFINITIONS. 287

But if the magnitudes are incommensurable, no multiple of the

first can be equal to any multiple of the second, and therefore the

quantuplicity of one with respect to the other cannot exactly be

expressed numerically: in this case it is determined by examining

how the multii)les of one magnitude are distributed among the

multiples of the other.

Thus, let all the multiples of A be formed, the scale extending ad

infinitum; also let all the multiples of B be formed and placed in their

proper order of magnitude among the multiples of A. This forms the

relative scale of the two magnitudes, and the quantuplicity of A with

respect to B is estimated by examining how the multiples of A are

distributed among those of B in their relative scale.

' In other words, the ratio of A to B is known, if for all integral

values of m we know the multiples ?iB and {Â« + !) B between which

otA lies.

In the case of two given magnitudes A and B, the relative scale of

multiples is definite, and is different from that of A to C, if C differs

from B by any magnitude however small.

For let D be the difference between B and C ; then however small

D may be, it will be possible to find a number m such that mD>A.

In this case, wB and mC would differ by a magnitude greater than A,

and therefore could not lie between the same two multiples of A ; so

that after a certain point the relative scale of A and B would differ

from that of A and C,

[It is worthy of notice that we can always estimate the arithmetical

ratio of two incommensurable magnitudes loithin any required degree

of accuracy.

For suppose that A and B are incommensurable ; divide B into w

equal parts each equal to jS, so that B = mj3, where m is an integer.

Also suppose j8 is contained in A more than n times and less than

(n+1) times; then

A n8 ^ (w+l)/3

=r > -^ and < ^ ^ ,

B m/J m^

that is, _- lies between â€” and ;

B m m

A n 1

so that -^ differs from - by a quantity less than â€” . And since we

can choose /3 (our unit of measurement) as small as we please, m can

be made as great as we please. Hence â€” can be made as small as we

m

please, and two integers n and m can be found whose ratio will express

that of a and b to any required degree of accuracy.]

288 EUCLID'S ELEMENTS.

4. The ratio of one magnitude to another is equal to

that of a third magnitude to a fourth, when if any equi-

multiples whatever of the antecedents of the ratios are

taken, and also any equimultiples whatever of the con-

sequents, the multiple of one antecedent is greater than,

e(jual to, or less than that of its consequent, according as

the multiple of the other antecedent is greater than, equal

to, or less than tliat of its consequent.

Thus the ratio A to B is equal to that of C to D when

mC > = 0T <nD according as mA > = or < ?iB, whatever whole

numbers m and n may be.

Again, let m be any whole number whatever, and n another whole

number determined in such a way that either mA is equal to nB, or

7/iA lies between 728 and (n + 1) B ; then the definition asserts that the

ratio of A to B is equal to that of C to D if mC â€” wD when mA = ?iB;

or if wC lies between wD and (n-i-1) D when jnA lies between wB and

(n-l-l)B.

In other words, the ratio of A to B is equal to that of C to D when

the multiples of A are distributed among those of B in the same

manner as the multiples of C are distributed among those of D.

5. When the ratio of A to B is equal to that of C to D

the four magnitudes are called proportionals. Tliis is ex-

pressed by saying " fKis toE as C is to D", and tlie proportion

is written

A : B : : C : D,

or A : B =- C : D.

A and D are called the extremes, B and C tlie means: also

D is said to be a fourth proportional to A, B, and C.

Two terms in a proportion are said to be homologous

when they are both antecedents, or both consequents of the

ratios.

[It will be useful here to compare the algebraical and geometrical

definitions of proportion, and to shew that each may be deduced from

the other.

According to the geometrical definition A, B, C, D are in propor-

tion, when 7uC> = <:wD according as rÂ»A> = <:nB, 7/1 and n being

any positive integers whatever.

According to the algebraical definition A, B, C, D are in proportion

, A C

when rc^ = -jz .

B D

DEFINITIONS. 289

(i) To deduce the geometrical definition of proportion from

the algebraical definition.

Since s = ni i ^7 niultiplying both sides by â€” , we obtain

mA mC

nB~nD'

hence from the nature of fractions,

mC> = <wD according as mA> = <7tB,

which is the geometrical test of proportion.

(ii) To deduce the algebraical definition of proportion from

the geometrical definition.

Given that mC> = <nD according as ?nA> = <nQ, to prove

AC

B~b'

A C

If - is not equal to â€” , one of them must be the greater.

Suppose -= > â€” ; then it will be possible to find some fraction â€”

t> U VI

which lies between them, n and m being positive integers.

Hence â– ;:;>- (1) ;

B VI ^ '

and ;r < â€” â€¢. ...(2).

D VI ^ '

From (1), 7nA>wB;

from (2), 7?iC<7iD;

and these contradict the hypothesis.

AC AC

Therefore ^ and _. are not unequal ; that is, 0=7:; which proves

the proposition.]

6. The ratio of one magnitude to another is greater

than that of a third magnitude to a fourtli, when it is

possible to lind equimultiples of the antecedents and equi-

multiples of the consequents such that while the multiple

of the antecedent of the first ratio is greater than, or equal

to, that of its consequent, the multiple of the aittecedent

of the second is not greater, or is less, than th^t of its

consequent.

H. E. lit

290 Kill. IDS i;i.k.mi;m>.

This definition asserts that if whole numbers m and n can be found

such that while 7?iA is greater than wB, mC is not greater than ?iD,

or while mA = ;<B, viC is less than ?tD, then the ratio of A to B is

greater than that of C to D.

7. If A is equal to B, the ratio of A to B is called a

ratio of equality.

If A is greater than B, the ratio of A to B is called a

ratio of greater inequality.

If A is less than B, the ratio of A to B is called a ratio

of less inequality.

8. Two ratios are said to be reciprocal when the ante-

cedent and consequent of one are the consequent and ante-

cedent of the other respectively ; thus B : A is the reciprocal

of A : B.

9. Three magnitudes of the same kiiid are said to be

proportionals, when the ratio of the first to the second is

equal to that of the second to the third.

Thus Aj B, C are proportionals if

A : B : : B : C.

B is called a mean proportional to A and C, and C is

called a third proportional to A and B.

10. Three or more magnitudes are said to be in con-

tinued proportion when the ratio of the first to the second

is equal to that of the second to the third, and the ratio of

the second to the third is equal to that of the third to the

fourth, and so on.

11. When there are any number of magnitudes of the

same kind, the first is said to have to the last the ratio

compounded of the ratios of the first to the second, of the

second to .the third, and so on up to the ratio of the last

but one to the last magnitude.

For example, if A, B, C, D, E be magnitudes of the same

kind, A : E is the ratio compounded of the ratios A : B,

B : C, C : D, and D : E.

DEFINITIONS. 291

This is sometimes expressed by the following notation:

/A : B

'^ â€¢ ^~]C : D

I D : E.

12. If there are any number of ratios, and a set of

magnitudes is taken such that the ratio of the first to the

second is equal to the first ratio, and the ratio of the second

to the third is equal to the second ratio, and so on, then

the first of the set of magnitudes is said to have to the

last the ratio compounded of the given ratios.

Thus, if A : B, C : D, E : F be given ratios, and if P, Q,

R, S be magnitudes taken so that

P : Q : : A : B,

Q : R :: C : D,

R : S :: E : F;

A : B

then P : S -^ -( C : D

E : F.

1 3. When three magnitudes are proportionals, the first

is said to have to the third the duplicate ratio of that

which it has to the second.

Thus if A : B : : B : C,

then A is said to have to C the duplicate ratio of that whioii

it has to B.

Since ^ â€¢ ^ ~ 1 B â€¢ C

it is clear that the ratio compounded of two equal ratios is the dupli-

cate ratio of either of them.

14. When four magnitudes are in continued proportion,

the first is said to have to the fourth the triplicate ratio of

that which it has to the second.

It may be shewn as above that the ratio compounded of three equal

ratios is the triplicate ratio of any one of them.

19-2

292 Euclid's elements.

Although an algebraical treatment of ratio and proportion when

applied to geometrical magnitudes cannot be considered exact, it will

perhaps be useful here to summarise in algebraical form the principal

theorems of proportion contained in Book V. The student will then

perceive that its leading propositions do not introduce new ideas, but

merely supply rigorous jjroofs, based on the geometrical definition ol'

proportion, of results already familiar in the study of Algebra.

We shall only here give those propositions which are afterwards

referred to in Book VI. It will be seen that in their algebraical form

many of them are so simple that they hardly require proof.

Summary of Pkincipal Tjiuorems of Book V.

Proposition 1.

Ratios which are equal to the same ratio are equal to one another.

That is, if A : B=rX : Y and C : D==X : Y;

then A : B = C : D.

I'JIOPOSITION y.

If four magnitudes are proportionals, they are also proportionals

when taken inversely.

That is, if A : B = C : D,

then B:A=D:C.

This inference is referred to as invertendo or inversely.

Proposition 4,

(i) Equal magnitudes have the same ratio to the same magnitude.

For if A = B,

then A : C=B : C.

(ii) The same magnitude has the same ratio to equal magnitudes.

For if A=B,

then C : A = C : B.

SUMMARY OF PRINCIPAL THEOREMS OF BOOK V. 293

Proposition 6.

(i) Magnitudes which have the same ratio to the name magnitude

are equal to one another.

That is, if A:C=B;C,

then A=B.

(ii) Those magnitudes to ichich the same magnitude has the same

ratio are equal to one another.

That is, if

C: A = C: B,

in

A=B.

Proposition 8,

Magnitudes have the same ratio to one another which their equi-

multiples have.

That is, A : B=mA : mB,

where vi is any whole number.

Proposition 11.

If four magnitudes of the same kind are proportionals, they are also

proportionals when taken altei'natehj .

If

A : B = C :

: D,

then shall

A :C = B :

; D.

For since

A C

B~D'

.â€¢. multiplying

by

B

C

, we

A B

haveg.-.

C

B

C

that is,

A B

c~b'

A : C-=B : D.

This inference is referred to as alternando or alternately.

2!>4 EUCLID'S ELEMENTft.

Proposition 12.

If any number of magnitudes of the same kind are proportionals,

then as one of the antecedents is to its consequent, so is the sum of the

antecedents to the sum of the consequents.

Let A : B = C : D = E : F = ...;

then shall A: B = A + C+E+... : B + D + F+....

ACE

For put each of the equal ratios b Â» ^ j â– ? >â€¢â€¢â– equal to k ;

D D r

then A = Bk, C=Dk, E=Fk,...

â– A + C+E+... _ Bk+Dk + Fk+..._ A _ C _ E_

â€¢â€¢B + D + F+...~ B+D + F + ... -^~B~D~F"-'

/. A: B = A + C+E + ... : B+D + F+....

This inference is sometimes referred to as addendo.

Proposition 13.

(i) If four magnitudes are proportionals, the sum of the first and

second is to the second as the sum of the third and fourth is to the fourth.

Let

A:B = C: D,

then shall

A+B : B=C+D

For since

A C

B~ D'

â€¢â€¢â€¢^-^l=

that is,

A+B C+D

B ~ b

or

A+B: B=C+D

D.

This inference is referred to as componendo.

(ii) If four magnitudes are proportionals, the difference of the first

and second is to the second as the difference of the third and fourth is to

the fourth.

That is, if A: B = C : D,

then A~B:BpC~D:D.

The proof is similar to that of the former case.

This inference is referred to as dividendo.

SUMMARY OF l'RI]?fClPAL THEOREMS OF BOOK Y. 295

Proposition 14.

If there are two sets of magnitudes, such that the first is to the

second of the first set as the first to the second of the other set, and the

second to the third of the first set as the second to the third of the other,

and so on to the last magnitude : then the first is to the last of the first

set as the first to the last of the other.

First let there be three magnitudes, A, B, C, of one set, and three,

P, Q, R, of another set,

and let

A: B=P:Q,

and

B :C = Q: R ;

then shall

A : C=P : R.

For since

A P ^ B

B = Q'^"^C =

A B P Q

â€¢ â€¢ B â€¢ C Q â€¢ R '

that is,

A P

C-R'

or

A : C=P : R.

Similarly if

A: B=P:Q,

B:C = Q: R,

=

L : M=:Y : Z;

it can be proved that A : M = P : Z .

This inference is referred to as ex sequall.

Corollary.

If

A:B=P:Q,

and

B:C = R : P;

then shall

A : C = R : Q.

For since

A P , B R

-:=^-,and^=p

A B P R

â€¢ BC~Q" P'

A R

â€¢ C Q'

or

A :C = R : Q.

296 KUCJ.ID's KLEMKNT8.

Proposition 15.

;/â€¢

A : B = C : D,

(Did

E : B = F : D;

then shall

A + E : B = C+F : D.

For since

AC n E F

B = D'""^B = D'

A+E C+F

â€¢ B ~ D '

that) is,

A+E: B = C+F: D.

Proposition 16.

If two ratios are equal, their duplicate ratios are equal; and:*

conversely.

Let A : B = C : D;

then shall the duplicate ratio of A : B be equal to the duplicate ratio

of C : D.

Let X be a third proportional to A, B ;

so that A : B=B : X;

B A

X "

-B'

B A

x* b"

A

"b*

A

B

A

A2

x"

^B^'

that is,

But A : X is the duplicate ratio of A : B ;

.-. the duplicate ratio of A : B = A2 : B"^.

But since A : B = C : D ;

AC

â– â– B ~ D '

â– â– â€¢ B2-D-"

or A-' : B2 = C2 : D^;

that is, the duplicate ratio of A : B=:the duplicate ratio of C : D.

Conversely, let the duplicate ratio of A : B be equal to the dupli-

cate ratio of C : D ;

then shall

A : B = C: D,

for since

A2 : B2 = C2 : D^,

.. A : B = C : D.

PROOFS OF THE PROPOSITIONS OF BOOK V. 297

Proofs op the Propositions of Book V. derived from

THE GEOMETRICAL DEFINITION OP PROPORTION.

Ohs. The Propositions of Book V. are all theorems.

Proposition 1.

Ratios which are equal to the same ratio are equal to one

another.

Let A : B :: P : Q, and also C : D :: P : Q; then shall

A : B :: C : D.

For it is evident that two scales or arrangements of

multiples which agree in every respect with a third scale,

will agree with one another.

Proposition 2.

If tivo raiios are equal, the antecedsnt of the second is

greater than, equal to, or less than its consequent according

as the antecedent of the first is greater than, equal to, or less

than its consequent.

Let

A : B :: C : D,

then

C > = or < D,

accordinor

as A > = or < B.

This follows at once from Def. 4, by taking m and

each equal to unity.

298 euclid's eleme^jts.

Proposition 3.

If two ratios are equals their reciprocal ratios are eqnal.

Let A : B : : C : D,

then shall B : A : : D : C.

For, by hypothesis, the multiples of A are distributed

among those of B in the same manner as the multiples of

C are among those of D ;

therefore also, the multiples of B are distributed among

those of A in the same manner as the multiples of D are

among those of C.

That is, B : A :: D : C.

Note. This proposition is sometimes enunciated thus

If four magnitudes are proportionaU, they are also â€¢proportionals

wlien taken inversely,

and the inference is referred to as invertendo or inversely.

Proposition 4.

Equal Ttiagnitudes liave the saine raiio to the same niag-

nitvde; atid the same magnitude has the same ratio to equal

magnitudes.

Let A, B, C be three magnitudes of the same kind, and

let A be equal to B;

then shall A : C : : B : C

and C : A :: C : B.

Since A = B, their multiples are identical and therefore

are distributed in the same way among the multiples of C.

.-. A : C : : B : C, J)e/. L

.'. also, invertendo, C : A :: C : B. v. 3.

proofs of the propositions of book v. 299

/ Proposition 5.

/

Of two unequal magnitudes^ the greater has a greater

ratio to a third magnitude than the less has; and the same

magnitude hafs a greater ratio to the, less of two magiiitudes

than it has m the greater.

First,/ let A be > B;

then shall A : C be > B : C.

Since A > B, it will be possible to find m such that mk

exceeds mB by a magnitude greater than C;

hence if mlK lies between nO and {71 + 1)C, ??iB < nO:

and if mA = tiC, then mB < nQ,\

/. A : C > B : C. Bef 6.

Secondly, let B be < A ;

then shall C : B be > C : A.

For taking m and n as before,

TiC > mB, while nC is not > mIK ;

.-. C : B > C : A. Def 6.

Proposition 6.

Mag7iitudes tvhich have the same ratio to the same mag-

nitude are equal to one another; and those to vjhich the same

magnitude has the same ratio are equal to one a7iother.

First, let A : C : : B : C ;

then shall A = B.

For if A > B, then A : C > B : C,

and if B > A, then B : C > A : C, v. 5.

which contradict the hypothesis;

.-. A= B.

300 Euclid's elements.

Secondh/, let C : A :: C : B;

then shall A = B.

Because C : A : : C : B,

.*., invertendo, A : C :: B : C, V. 3.

A-B,

by the first part of the proof.

Proposition 7.

That magnitude which has a greater ratio than another

has to the same viagnitude is the greater of the two-, and

tliat maxjnitude to which the same has a greater ratio than it

has to another magnitude is the less of the two.

First, let A : C be > B : C;

then shall A be > B.

For if A = B, then A : C : : B : C, v. 4.

which is contrary to the hypothesis.

And if A < B, then A : C < B : C ; v. 5.

which is contrary to the hypothesis;

.'. A>B.

Secondly, let C : A be > C : B;

then shall A be < B.

For if A- B, then C : A :: C : B, v. 4.

which is contrary to the hypothesis.

And if A > B, then C : A < C : B : \. 5.

which is contrary to the hypothesis;

.-. A<B.

PKOOFS OF THE PROrOSITIOjS'ib OF BOOK V. 301

Proposition 8.

Magnitudes have the same ratio to one another which

their equimultij^les have.

Let A, B be two magnitudes;

then shall A : B :: mA : mB.

If 2^^ Q. be any two whole numbers,

then m . pA> -^ or < m . ^-B

according as pA > â€” or <:qB.

But m .pfii=p . m/K, and 7)i . qB â€” q . iiiB;

.'. p . QiiA > = or <:q . r/iB

according as pA > ^ or < 5'B;

.'. A : B :: mA : mB. I)ef. 4.

CoK. Let A : B :: C : D.

Then since A : B : : mA : mB,

and C : D :: nC : nD;

:. iiiA : mB :: nC : nD. v. 1.

Proposition 9.

1/ tivo 7'atios are equal, and any equimultiples of the

antecedents and aUo of the consequents are taken, the multiple

of the first antecedent has to that of its consequent the same

ratio as the multiple of the other antecedent has to that of its

Let A : B :: C : D;

then shall mA : nB :: mC : nD.

Let p, q be any two whole numbers,

then because A : B : : C : D,

pm . C > = or <.qn. D

according as pm . A > = or <.qn . B, Def A.

that is, p . mC > = or <q .nD,

according as p . mA > â€” or <:q .7iB ;

.'. in A : 7iB :: mC : 7iD. Def 4.

to n times the radius of the inscribed circle. ''

11. The sum of the perpendiculars drawn from the vertices of a

regular polygon of n sides on any straight line is equal to n times the

perpendicular drawn from the centre of the inscribed circle.

12. The area of a cyclic quadrilateral is independent of the order

in which the sides are placed in the circle.

13. Of all quadrilaterals which can be formed of four straight

lines of given length, that which is cyclic has the maximum area.

14. Of all polygons of a given number of sides, which may be

inscribed in a given circle, that which is regular has the maximum

area and the maximum perimeter.

15. Of all polygons of a given number of sides circumscribed

about a given circle, that which is regular has the minimum area and

the minimum perimeter.

16. Given the vertical angle of a triangle in position and magni-

tude, and the sum of the sides containing it: find the locus of the

centre of the circumscribed circle.

17. P is any point on the circumference of a circle circumscribed

about an equilateral triangle ABC: shew that PA-+ PB-'-f PC- ia

constant.

BOOK V.

Book V. treats of Ratio and Proportion.

INTRODUCTORY.

The first four books of Euclid deal with the absolute equality

or inequality of Geometrical magnitudes. In the Fifth Book

magnitudes are compared by considering their ratio, or relative

greatness.

The meaning of the words ratio and proportion in their

simplest arithmetical sense, as contained in the following defini-

tions, is probably familiar to the student :

The ratio of one number to another is the multiple, part, or

parts that the first number is of the second; and it may therefore be

measured by the fraction of vihich the first number is the numerator

and the second the denominator.

Four numbers are in proportion when the ratio of the first to

the second is equal to that of the third to the fourth.

But it will be seen that these definitions are inapplicable to

Geometrical magnitudes for the following reasons :

(1) Pure Geometry deals only with concrete magnitudes, re-

presented by diagrams, but not referred to any common imit in

terms of which they are measured : in other words, it makes

no use of number for the purpose of comparison between different

magnitudes.

(2) It commonly happens that Geometrical magnitudes of

the same kind are incommensurable, that is, they are such that

it is impossible to express them exactly in terms of some common

unit.

For example, we can make comparison between the side and

diagonal of a square, and we may form an idea of their relative great-

ness, but it can be shewn that it is impossible to divide either of them

into equal parts of which the other contains an exact number. And

as the magnitudes we meet with in Geometry are more often incom-

mensurable than not, it is clear that it would not always be possible

to exactly represent such magnitudes by numbers, even if reference to

a common unit were not foreign to the principles of Euclid.

It is therefore necessary to establish the Geometrical Theory

of Proportion on a basis quite independent of Arithmetical

principles. This is the aim of Euclid's Fifth Book.

286 EUCLID'S ELEMENTS.

We shall employ the following notation.

Capital letters, A, B, C,.., will be used to denote the magnitudes

themselves, not nny numerical or algehraical vieamres of them, and

small letters, m, n, p,... will be used to denote whole numbers. Also

it will be assumed that multiplication, in the sense of repeated

addition, can be applied to any magnitude, so that 7U . A or wA will

denote the magnitude A taken vi times.

The symbol > will be used for the words greater than, and < for

le$s than. â€¢

Definitions.

1. A greater magnitude is said to be a multiple of a

less, wlien the greater contains the less an exact number of

times.

2. A less magnitude is said to be a submultiple of a

greater, wlien the less is contained an exact number of

times in the greater.

The following properties of multiples will be assumed as self-evident.

(1) 7uA > = or < 7/zB according as A > = or < B ; and

conversely.

(2) ?HA + wB + ...=?H(A + B-f-...).

(3) If A > B, then vik - mB=m (A - B).

(4) wA + nA+... = (m + Ji+...) A.

. (5) If m > n, then mA - nk â€” {m - 7/) A.

(6) ni . 7iA = run . A = nm . A = tz . 7/i A.

3. The Ratio of one magnitude to another of the same

kind is the relation which the first bears to the second in

respect of quantuplicity.

The ratio of A to B is denoted thus, A : B; and A is

called the antecedent, B the consequent of the ratio.

The term quantuplicity denotes the capacity of the first magnitude

to contain the second with or without remainder. If the magnitudes

are commensurable, their quantuplicity may be expressed numerically

by observing what multiples of the two magnitudes are equal to one

another.

Thus if A -ma, and B = na, it follows that nA â€” niB. In this case

A = â€” B, and the quantuplicity of A with respect to B is the arith-

metical fraction - .

11

DEFINITIONS. 287

But if the magnitudes are incommensurable, no multiple of the

first can be equal to any multiple of the second, and therefore the

quantuplicity of one with respect to the other cannot exactly be

expressed numerically: in this case it is determined by examining

how the multii)les of one magnitude are distributed among the

multiples of the other.

Thus, let all the multiples of A be formed, the scale extending ad

infinitum; also let all the multiples of B be formed and placed in their

proper order of magnitude among the multiples of A. This forms the

relative scale of the two magnitudes, and the quantuplicity of A with

respect to B is estimated by examining how the multiples of A are

distributed among those of B in their relative scale.

' In other words, the ratio of A to B is known, if for all integral

values of m we know the multiples ?iB and {Â« + !) B between which

otA lies.

In the case of two given magnitudes A and B, the relative scale of

multiples is definite, and is different from that of A to C, if C differs

from B by any magnitude however small.

For let D be the difference between B and C ; then however small

D may be, it will be possible to find a number m such that mD>A.

In this case, wB and mC would differ by a magnitude greater than A,

and therefore could not lie between the same two multiples of A ; so

that after a certain point the relative scale of A and B would differ

from that of A and C,

[It is worthy of notice that we can always estimate the arithmetical

ratio of two incommensurable magnitudes loithin any required degree

of accuracy.

For suppose that A and B are incommensurable ; divide B into w

equal parts each equal to jS, so that B = mj3, where m is an integer.

Also suppose j8 is contained in A more than n times and less than

(n+1) times; then

A n8 ^ (w+l)/3

=r > -^ and < ^ ^ ,

B m/J m^

that is, _- lies between â€” and ;

B m m

A n 1

so that -^ differs from - by a quantity less than â€” . And since we

can choose /3 (our unit of measurement) as small as we please, m can

be made as great as we please. Hence â€” can be made as small as we

m

please, and two integers n and m can be found whose ratio will express

that of a and b to any required degree of accuracy.]

288 EUCLID'S ELEMENTS.

4. The ratio of one magnitude to another is equal to

that of a third magnitude to a fourth, when if any equi-

multiples whatever of the antecedents of the ratios are

taken, and also any equimultiples whatever of the con-

sequents, the multiple of one antecedent is greater than,

e(jual to, or less than that of its consequent, according as

the multiple of the other antecedent is greater than, equal

to, or less than tliat of its consequent.

Thus the ratio A to B is equal to that of C to D when

mC > = 0T <nD according as mA > = or < ?iB, whatever whole

numbers m and n may be.

Again, let m be any whole number whatever, and n another whole

number determined in such a way that either mA is equal to nB, or

7/iA lies between 728 and (n + 1) B ; then the definition asserts that the

ratio of A to B is equal to that of C to D if mC â€” wD when mA = ?iB;

or if wC lies between wD and (n-i-1) D when jnA lies between wB and

(n-l-l)B.

In other words, the ratio of A to B is equal to that of C to D when

the multiples of A are distributed among those of B in the same

manner as the multiples of C are distributed among those of D.

5. When the ratio of A to B is equal to that of C to D

the four magnitudes are called proportionals. Tliis is ex-

pressed by saying " fKis toE as C is to D", and tlie proportion

is written

A : B : : C : D,

or A : B =- C : D.

A and D are called the extremes, B and C tlie means: also

D is said to be a fourth proportional to A, B, and C.

Two terms in a proportion are said to be homologous

when they are both antecedents, or both consequents of the

ratios.

[It will be useful here to compare the algebraical and geometrical

definitions of proportion, and to shew that each may be deduced from

the other.

According to the geometrical definition A, B, C, D are in propor-

tion, when 7uC> = <:wD according as rÂ»A> = <:nB, 7/1 and n being

any positive integers whatever.

According to the algebraical definition A, B, C, D are in proportion

, A C

when rc^ = -jz .

B D

DEFINITIONS. 289

(i) To deduce the geometrical definition of proportion from

the algebraical definition.

Since s = ni i ^7 niultiplying both sides by â€” , we obtain

mA mC

nB~nD'

hence from the nature of fractions,

mC> = <wD according as mA> = <7tB,

which is the geometrical test of proportion.

(ii) To deduce the algebraical definition of proportion from

the geometrical definition.

Given that mC> = <nD according as ?nA> = <nQ, to prove

AC

B~b'

A C

If - is not equal to â€” , one of them must be the greater.

Suppose -= > â€” ; then it will be possible to find some fraction â€”

t> U VI

which lies between them, n and m being positive integers.

Hence â– ;:;>- (1) ;

B VI ^ '

and ;r < â€” â€¢. ...(2).

D VI ^ '

From (1), 7nA>wB;

from (2), 7?iC<7iD;

and these contradict the hypothesis.

AC AC

Therefore ^ and _. are not unequal ; that is, 0=7:; which proves

the proposition.]

6. The ratio of one magnitude to another is greater

than that of a third magnitude to a fourtli, when it is

possible to lind equimultiples of the antecedents and equi-

multiples of the consequents such that while the multiple

of the antecedent of the first ratio is greater than, or equal

to, that of its consequent, the multiple of the aittecedent

of the second is not greater, or is less, than th^t of its

consequent.

H. E. lit

290 Kill. IDS i;i.k.mi;m>.

This definition asserts that if whole numbers m and n can be found

such that while 7?iA is greater than wB, mC is not greater than ?iD,

or while mA = ;<B, viC is less than ?tD, then the ratio of A to B is

greater than that of C to D.

7. If A is equal to B, the ratio of A to B is called a

ratio of equality.

If A is greater than B, the ratio of A to B is called a

ratio of greater inequality.

If A is less than B, the ratio of A to B is called a ratio

of less inequality.

8. Two ratios are said to be reciprocal when the ante-

cedent and consequent of one are the consequent and ante-

cedent of the other respectively ; thus B : A is the reciprocal

of A : B.

9. Three magnitudes of the same kiiid are said to be

proportionals, when the ratio of the first to the second is

equal to that of the second to the third.

Thus Aj B, C are proportionals if

A : B : : B : C.

B is called a mean proportional to A and C, and C is

called a third proportional to A and B.

10. Three or more magnitudes are said to be in con-

tinued proportion when the ratio of the first to the second

is equal to that of the second to the third, and the ratio of

the second to the third is equal to that of the third to the

fourth, and so on.

11. When there are any number of magnitudes of the

same kind, the first is said to have to the last the ratio

compounded of the ratios of the first to the second, of the

second to .the third, and so on up to the ratio of the last

but one to the last magnitude.

For example, if A, B, C, D, E be magnitudes of the same

kind, A : E is the ratio compounded of the ratios A : B,

B : C, C : D, and D : E.

DEFINITIONS. 291

This is sometimes expressed by the following notation:

/A : B

'^ â€¢ ^~]C : D

I D : E.

12. If there are any number of ratios, and a set of

magnitudes is taken such that the ratio of the first to the

second is equal to the first ratio, and the ratio of the second

to the third is equal to the second ratio, and so on, then

the first of the set of magnitudes is said to have to the

last the ratio compounded of the given ratios.

Thus, if A : B, C : D, E : F be given ratios, and if P, Q,

R, S be magnitudes taken so that

P : Q : : A : B,

Q : R :: C : D,

R : S :: E : F;

A : B

then P : S -^ -( C : D

E : F.

1 3. When three magnitudes are proportionals, the first

is said to have to the third the duplicate ratio of that

which it has to the second.

Thus if A : B : : B : C,

then A is said to have to C the duplicate ratio of that whioii

it has to B.

Since ^ â€¢ ^ ~ 1 B â€¢ C

it is clear that the ratio compounded of two equal ratios is the dupli-

cate ratio of either of them.

14. When four magnitudes are in continued proportion,

the first is said to have to the fourth the triplicate ratio of

that which it has to the second.

It may be shewn as above that the ratio compounded of three equal

ratios is the triplicate ratio of any one of them.

19-2

292 Euclid's elements.

Although an algebraical treatment of ratio and proportion when

applied to geometrical magnitudes cannot be considered exact, it will

perhaps be useful here to summarise in algebraical form the principal

theorems of proportion contained in Book V. The student will then

perceive that its leading propositions do not introduce new ideas, but

merely supply rigorous jjroofs, based on the geometrical definition ol'

proportion, of results already familiar in the study of Algebra.

We shall only here give those propositions which are afterwards

referred to in Book VI. It will be seen that in their algebraical form

many of them are so simple that they hardly require proof.

Summary of Pkincipal Tjiuorems of Book V.

Proposition 1.

Ratios which are equal to the same ratio are equal to one another.

That is, if A : B=rX : Y and C : D==X : Y;

then A : B = C : D.

I'JIOPOSITION y.

If four magnitudes are proportionals, they are also proportionals

when taken inversely.

That is, if A : B = C : D,

then B:A=D:C.

This inference is referred to as invertendo or inversely.

Proposition 4,

(i) Equal magnitudes have the same ratio to the same magnitude.

For if A = B,

then A : C=B : C.

(ii) The same magnitude has the same ratio to equal magnitudes.

For if A=B,

then C : A = C : B.

SUMMARY OF PRINCIPAL THEOREMS OF BOOK V. 293

Proposition 6.

(i) Magnitudes which have the same ratio to the name magnitude

are equal to one another.

That is, if A:C=B;C,

then A=B.

(ii) Those magnitudes to ichich the same magnitude has the same

ratio are equal to one another.

That is, if

C: A = C: B,

in

A=B.

Proposition 8,

Magnitudes have the same ratio to one another which their equi-

multiples have.

That is, A : B=mA : mB,

where vi is any whole number.

Proposition 11.

If four magnitudes of the same kind are proportionals, they are also

proportionals when taken altei'natehj .

If

A : B = C :

: D,

then shall

A :C = B :

; D.

For since

A C

B~D'

.â€¢. multiplying

by

B

C

, we

A B

haveg.-.

C

B

C

that is,

A B

c~b'

A : C-=B : D.

This inference is referred to as alternando or alternately.

2!>4 EUCLID'S ELEMENTft.

Proposition 12.

If any number of magnitudes of the same kind are proportionals,

then as one of the antecedents is to its consequent, so is the sum of the

antecedents to the sum of the consequents.

Let A : B = C : D = E : F = ...;

then shall A: B = A + C+E+... : B + D + F+....

ACE

For put each of the equal ratios b Â» ^ j â– ? >â€¢â€¢â– equal to k ;

D D r

then A = Bk, C=Dk, E=Fk,...

â– A + C+E+... _ Bk+Dk + Fk+..._ A _ C _ E_

â€¢â€¢B + D + F+...~ B+D + F + ... -^~B~D~F"-'

/. A: B = A + C+E + ... : B+D + F+....

This inference is sometimes referred to as addendo.

Proposition 13.

(i) If four magnitudes are proportionals, the sum of the first and

second is to the second as the sum of the third and fourth is to the fourth.

Let

A:B = C: D,

then shall

A+B : B=C+D

For since

A C

B~ D'

â€¢â€¢â€¢^-^l=

that is,

A+B C+D

B ~ b

or

A+B: B=C+D

D.

This inference is referred to as componendo.

(ii) If four magnitudes are proportionals, the difference of the first

and second is to the second as the difference of the third and fourth is to

the fourth.

That is, if A: B = C : D,

then A~B:BpC~D:D.

The proof is similar to that of the former case.

This inference is referred to as dividendo.

SUMMARY OF l'RI]?fClPAL THEOREMS OF BOOK Y. 295

Proposition 14.

If there are two sets of magnitudes, such that the first is to the

second of the first set as the first to the second of the other set, and the

second to the third of the first set as the second to the third of the other,

and so on to the last magnitude : then the first is to the last of the first

set as the first to the last of the other.

First let there be three magnitudes, A, B, C, of one set, and three,

P, Q, R, of another set,

and let

A: B=P:Q,

and

B :C = Q: R ;

then shall

A : C=P : R.

For since

A P ^ B

B = Q'^"^C =

A B P Q

â€¢ â€¢ B â€¢ C Q â€¢ R '

that is,

A P

C-R'

or

A : C=P : R.

Similarly if

A: B=P:Q,

B:C = Q: R,

=

L : M=:Y : Z;

it can be proved that A : M = P : Z .

This inference is referred to as ex sequall.

Corollary.

If

A:B=P:Q,

and

B:C = R : P;

then shall

A : C = R : Q.

For since

A P , B R

-:=^-,and^=p

A B P R

â€¢ BC~Q" P'

A R

â€¢ C Q'

or

A :C = R : Q.

296 KUCJ.ID's KLEMKNT8.

Proposition 15.

;/â€¢

A : B = C : D,

(Did

E : B = F : D;

then shall

A + E : B = C+F : D.

For since

AC n E F

B = D'""^B = D'

A+E C+F

â€¢ B ~ D '

that) is,

A+E: B = C+F: D.

Proposition 16.

If two ratios are equal, their duplicate ratios are equal; and:*

conversely.

Let A : B = C : D;

then shall the duplicate ratio of A : B be equal to the duplicate ratio

of C : D.

Let X be a third proportional to A, B ;

so that A : B=B : X;

B A

X "

-B'

B A

x* b"

A

"b*

A

B

A

A2

x"

^B^'

that is,

But A : X is the duplicate ratio of A : B ;

.-. the duplicate ratio of A : B = A2 : B"^.

But since A : B = C : D ;

AC

â– â– B ~ D '

â– â– â€¢ B2-D-"

or A-' : B2 = C2 : D^;

that is, the duplicate ratio of A : B=:the duplicate ratio of C : D.

Conversely, let the duplicate ratio of A : B be equal to the dupli-

cate ratio of C : D ;

then shall

A : B = C: D,

for since

A2 : B2 = C2 : D^,

.. A : B = C : D.

PROOFS OF THE PROPOSITIONS OF BOOK V. 297

Proofs op the Propositions of Book V. derived from

THE GEOMETRICAL DEFINITION OP PROPORTION.

Ohs. The Propositions of Book V. are all theorems.

Proposition 1.

Ratios which are equal to the same ratio are equal to one

another.

Let A : B :: P : Q, and also C : D :: P : Q; then shall

A : B :: C : D.

For it is evident that two scales or arrangements of

multiples which agree in every respect with a third scale,

will agree with one another.

Proposition 2.

If tivo raiios are equal, the antecedsnt of the second is

greater than, equal to, or less than its consequent according

as the antecedent of the first is greater than, equal to, or less

than its consequent.

Let

A : B :: C : D,

then

C > = or < D,

accordinor

as A > = or < B.

This follows at once from Def. 4, by taking m and

each equal to unity.

298 euclid's eleme^jts.

Proposition 3.

If two ratios are equals their reciprocal ratios are eqnal.

Let A : B : : C : D,

then shall B : A : : D : C.

For, by hypothesis, the multiples of A are distributed

among those of B in the same manner as the multiples of

C are among those of D ;

therefore also, the multiples of B are distributed among

those of A in the same manner as the multiples of D are

among those of C.

That is, B : A :: D : C.

Note. This proposition is sometimes enunciated thus

If four magnitudes are proportionaU, they are also â€¢proportionals

wlien taken inversely,

and the inference is referred to as invertendo or inversely.

Proposition 4.

Equal Ttiagnitudes liave the saine raiio to the same niag-

nitvde; atid the same magnitude has the same ratio to equal

magnitudes.

Let A, B, C be three magnitudes of the same kind, and

let A be equal to B;

then shall A : C : : B : C

and C : A :: C : B.

Since A = B, their multiples are identical and therefore

are distributed in the same way among the multiples of C.

.-. A : C : : B : C, J)e/. L

.'. also, invertendo, C : A :: C : B. v. 3.

proofs of the propositions of book v. 299

/ Proposition 5.

/

Of two unequal magnitudes^ the greater has a greater

ratio to a third magnitude than the less has; and the same

magnitude hafs a greater ratio to the, less of two magiiitudes

than it has m the greater.

First,/ let A be > B;

then shall A : C be > B : C.

Since A > B, it will be possible to find m such that mk

exceeds mB by a magnitude greater than C;

hence if mlK lies between nO and {71 + 1)C, ??iB < nO:

and if mA = tiC, then mB < nQ,\

/. A : C > B : C. Bef 6.

Secondly, let B be < A ;

then shall C : B be > C : A.

For taking m and n as before,

TiC > mB, while nC is not > mIK ;

.-. C : B > C : A. Def 6.

Proposition 6.

Mag7iitudes tvhich have the same ratio to the same mag-

nitude are equal to one another; and those to vjhich the same

magnitude has the same ratio are equal to one a7iother.

First, let A : C : : B : C ;

then shall A = B.

For if A > B, then A : C > B : C,

and if B > A, then B : C > A : C, v. 5.

which contradict the hypothesis;

.-. A= B.

300 Euclid's elements.

Secondh/, let C : A :: C : B;

then shall A = B.

Because C : A : : C : B,

.*., invertendo, A : C :: B : C, V. 3.

A-B,

by the first part of the proof.

Proposition 7.

That magnitude which has a greater ratio than another

has to the same viagnitude is the greater of the two-, and

tliat maxjnitude to which the same has a greater ratio than it

has to another magnitude is the less of the two.

First, let A : C be > B : C;

then shall A be > B.

For if A = B, then A : C : : B : C, v. 4.

which is contrary to the hypothesis.

And if A < B, then A : C < B : C ; v. 5.

which is contrary to the hypothesis;

.'. A>B.

Secondly, let C : A be > C : B;

then shall A be < B.

For if A- B, then C : A :: C : B, v. 4.

which is contrary to the hypothesis.

And if A > B, then C : A < C : B : \. 5.

which is contrary to the hypothesis;

.-. A<B.

PKOOFS OF THE PROrOSITIOjS'ib OF BOOK V. 301

Proposition 8.

Magnitudes have the same ratio to one another which

their equimultij^les have.

Let A, B be two magnitudes;

then shall A : B :: mA : mB.

If 2^^ Q. be any two whole numbers,

then m . pA> -^ or < m . ^-B

according as pA > â€” or <:qB.

But m .pfii=p . m/K, and 7)i . qB â€” q . iiiB;

.'. p . QiiA > = or <:q . r/iB

according as pA > ^ or < 5'B;

.'. A : B :: mA : mB. I)ef. 4.

CoK. Let A : B :: C : D.

Then since A : B : : mA : mB,

and C : D :: nC : nD;

:. iiiA : mB :: nC : nD. v. 1.

Proposition 9.

1/ tivo 7'atios are equal, and any equimultiples of the

antecedents and aUo of the consequents are taken, the multiple

of the first antecedent has to that of its consequent the same

ratio as the multiple of the other antecedent has to that of its

Let A : B :: C : D;

then shall mA : nB :: mC : nD.

Let p, q be any two whole numbers,

then because A : B : : C : D,

pm . C > = or <.qn. D

according as pm . A > = or <.qn . B, Def A.

that is, p . mC > = or <q .nD,

according as p . mA > â€” or <:q .7iB ;

.'. in A : 7iB :: mC : 7iD. Def 4.

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