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quent of one ratio, and the antecedent of the next ; whereas in other

proportionals this is not the case.

A series of numbers or Algebraical quantities in continued proportion,

is called a Geometrical progression^ from the analogy they bear to a series

of Geometrical magnitudes in continued proportion.

Def. A. The term compound ratio was devised for the purpose of

avoiding circumlocution, and no difficulty can arise in the use of it, if

its exact meaning be strictly attended to.

With respect to the Geometrical measures of compound ratios, three

straight lines may measure the ratio of four, as in Prop. 23, Book vi.

For Kio L measures the ratio oi BC to CO, and L to M measures the

ratio oi DC to CE; and the ratio of K toiV/is that which is said to be

compounded of the ratios of K to L, and L to M, which is the same as the

ratio which is compounded of the ratios of the sides of the parallelograms.

Both duplicate and triplicate ratio are species of compound ratio.

Duplicate ratio is a ratio compounded of two equal ratios ; and in the

case of three magnitudes which are continued proportionals, means the

ratio of the first to a third proportional to the first and second.

Triplicate ratio, in the same manner, is a ratio compounded of three

equal ratios ; and in the case of four magnitudes which are continued

proportionals, the triplicate ratio of the first to the second means the

ratio of the first to a fourth proportional to the first, second, and third

magnitudes. Instances of the composition of three ratios, and of tripli-

cate ratio, will be found in the eleventh and twelfth books.

The product of the fractions which represent or measure the ratios

NOTES TO BOOK V. 243

of numbers, corresponds to the composition of Geometrical ratios of

magnitudes.

It has been shewn that the ratio of two numbers is represented by a

fraction whereof the numerator is the antecedent, and the denominator

the consequent of the ratio ; and if the antecedents of two ratios be

multiplied together, as also the consequents, the new ratio thus formed

is said to be compounded of these two ratios ; and in the same manner,

if there be more than two. It is also obvious, that the ratio compounded

of two equal ratios is equal to the ratio of the squares of one of the ante-

cedents to its consequent ; also when there are three equal ratios, the

ratio compounded of the three ratios is equal to the ratio of the cubes of

any one of the antecedents to its consequent. And further, it may be

observed, that when several numbers are continued proportionals, the

ratio of the first to the last is equal to the ratio of the product of all the

antecedents to the product of all the consequents.

It may be here remarked, that, though the constructions of the pro-

positions in Book v are exhibited by straight lines, the enunciations are

ex]>ressed of magnitude in general, and are equally true of angles,

triangles, parallelograms, arcs, sectors, &c.

The two following axioms may be added to the four Euclid has given.

Ax. 5. A part of a greater magnitude is greater than the same part

of a less magnitude.

Ax. 6. That magnitude of which any part is greater than the same

part of another, is greater than that other magnitude.

The learner must not forget that the capital lettersy used generally by

Euclid in the demonstrations of the fifth Book, represent the magnitudes,

not any numerical or Algebraical measures of them : sometimes however

the magnitude of a line is represented in the usual way by two letters

which are placed at the extremities of the line.

Prop. I. Algebraically.

Let each of the magnitudes A, B, C, &c. be equimultiples of as many

a, b, c, &c.

that is, let A = m times a = ma,

B = m times b = mb,

C â€” m times c â€” mc, &c.

First, if there be two magnitudes equimultiples of two others.

Then A + B â€” ma + mb = m (a + 6) = m times (a + 6),

Hence A + B is the same multiple of {a + b), as A is of a, or B of 6.

Secondly, if there be three magnitudes equimultiples of three others,

then A + B + C = ma + mb + mc = m {a + b + c)

= m times (o + 6 + c),

Hence A + B + Cis the same multiple of (a + b + c);

as A is of a, B of 6, and C of c.

Similarly, if there were four, or any number of magnitudes.

Therefore, if any number of magnitudes be equimultiples of as many,

each of each ; what multiple soever, any one is of its part, the same

multiple shall the first magnitudes be of all the other.

Prop. II. Algebraically.

Let A^ the first magnitude, be the same multiple of a^ the second,

as A^ the third, is of a^ the fourth ; and A^, the fifth the same multiple

of a, the second, as A^ the sixth, is of ai the fourth.

m2

244 Euclid's elements..

That is, let /4j = m times a^ = ma^,

Ai = m times a^ = ma^,

Af,= n times a^ = wa^,

Jq = n times a^ â€” na^,

Then by addition, A^ + A^ = ma^ + naz = {m-\-n) a, = (rn + n) times a^,

and Ai-^ A^ = ma^ + wa^ = (m + w) a^ = (m + w) times a^.

Therefore A^ + ^^ is the same multiple of a^^ as A^ + /^^ is of a^.

That is, if the first magnitude be the same multiple of the second, as

the third is of the fourth, &c.

Cor. If there be any number of magnitudes A-^^ A^, A^, &c. multiples

of another a, such that A^ = ma, A^ = na, A^ = pa, &c.

And as many others JSj, B^, B^, &c. the same multiples of another b,

such that ^1 = mb, B^ = nb, B^ = pb, &c.

Then by addition, A^ + A^ + A3 + &c. = ma + na + pa + &c.

= (m + n + p + &c.) a={m + n+p + &c.) times a :

and Bi + B^ + B^ + &c. = 7nb + nb + pb + &c. = {m + n -\- p + &c.) b

= (m + n + p + &c.) times b :

that is A^-\-Ar, + A^ + &c. is the same multiple of a that

^1 + -Bg + ^3 + &c. is of b.

Prop. in. Algebraically.

Let Ai the first magnitude, be the same multiple of a^ the second,

as A3 the third, is of a^ the fourth,

that is, let Ay^ = m times a^ = ma^,

and As = m times a^ = ma^.

If these equals be each taken n times,

then nAi = mna^ = mn times o^,

and nA^ = vma^ = mn times a^,

or nA^y nA^ each contain a^,, a^ respectively mn times.

Wherefore n//j, 7iA^ the equimultiples of the first and third, are

respectively equimultiples of a^ and a^, the second and fourth.

Prop. IV. Algebraically.

Let ^1, flg, ^3, a^, be proportionals according to the Algebraical

definition :

that is, let A^ : g.^ : : Jg : a^

then â€” = -^ ,

multiply these equals by â€” , w and n being any integers,

or mAy : wa^ : : mA^ : waj.

That is, if the first of four magnitudes has the same ratio to the

second which the third has to the fourth ; then any equimultiples what-

ever of the first and third shall have the same ratio to any equimultiples

of the second and fourth.

NOTES TO BOOK V. 245

.lie Corollary is contained in the proposition itself :

for if n be unity, then mj^ : a^ :: niJ^ : a^:

and if m be unity, also Ai : na^ :: ^s'. na^.

Prop. V. Algebraically.

Let Ji be the same multiple of aâ€ž

that Jg a part of J^ is of a^, a part of ai.

Then Ji â€” J^ is the same multiple of a^^ â€” o^ as J^ is of a, :

For let ^, = m times a^ = ma^^

and A%^ m times Â«8 = ^^ag,

then y^i â€” ^2 = '"^1 ~ "'^2 = m (Â«! - ag) = m times {a^ - Og),

that is Ji - ^2 is ^^^ ^^^ s^Â°^^ multiple of (aj - a^) as ^^ is of a^.

Prop. VI. Algebraically.

Let Ji, ^g be equimultiples respectively of a^, a, two others,

that is, let A^ =m times Oj = ma^,

-^2 = w times a^ = ma^^

Also if B^ a part of ^^ = n times a^ = wa^,

and B^ a part of Ac^ = n times a2 = ^^2*

Then by taking equals from equals,

.*. A-^ â€” B^= ma^ â€” na^ = {m â€” n) a^ = (m â€” n) times a^,

A2 â€” B^ = ma^ â€” na^, = (m â€” n) a^ = (m â€” n) times a^ :

that is, the remainders A-^ â€” B^, A^ â€” B^ are equimultiples of fli, aj,

respectively.

And if m - w = 1, then A^ â€” B^ â€” a^, and A^ â€” B2 = a^i

or the remainders are equal to a^, a^ respectively.

rop. A. Algebraically,

Let Au a^, A 3, a^ be proportionals,

or A^ : a.^:: A^: a^,

And since the fraction â€” is equal to â€” , the following relations

a, a^

only can subsist between A^ and a,; and between A^ and a^,

First, if Ai be greater than a^; then A^ is also greater than a^:

Secondly, if A^ be equal to a^ ; then A^ is also equal to a^ :

'I'hirdly, if Ai be less than a^ ; then ^3 is also less than a^ :

A A

Otherwise, the fraction â€” could not be equal to the fraction ~ .

a, a^

Prop. B. Algebraically.

Let A^, ttg, ^j,, a^ be proportionals,

or. ^1 : a, :: /^3 ; ff^,

Then shall a^ : Ai : : a^ : ^3 .

For since Ai i a^ :: A^ : a^

A\ A?.

I^>

246 Euclid's elements.

and if 1 be divided by each of these equals,

and therefore a^: A^ : : a^ : /I3.

Prop. c. " This is frequently made use of by geometers, and is necessary

to the oth and 6th Propositions of the 10th Book. Clavius, in his notes

subjoined to the 8th def. of Book 5, demonstrates it only in numbers, by

help of some of the propositions of the 7th Book ; in order to demonstrate

the property contained in the 5th definition of the 5th Book, when applied

to numbers, from the property of proportionals contained in the 20th def.

of the 7th Book : and most of the commentators judge it difficult to prove

that four magnitudes which are proportionals according to the 20th def.

of the 7th Book, are also proportionals according to the 5th def. of the

6th Book. But this is easily made out as follows ;

First, if A, B, C, Z), be four magnitudes, such that A is the same

multiple, or the same part of JB, which C is of D :

Then A, B, C, D, are proportionals:

this is demonstrated in proposition (c).

Secondly, if AB contain the same parts of CD that EF does of GH ;

in this case likewise AB is to CD, as EF to GH.

A B E F

C K D G L II

Let CKhe a part of CD, and GL the same part of GH;

and let AB be the same multiple of CK, that EF is of GL :

therefore, by Prop, c, of Book v, AB is to CK, as EF to GL :

and CDy GH, are equimultiples of CK, GL, the second and fourth ;

wherefore, by Cor. Prop. 4, Book v, AB is to CD, as EF to GH.

And if four magnitudes be proportionals according to the 5th def. of Book v,

they are also proportionals according to the 20th def. of Book vii.

First, if A be to ^, as CtoD;

then if A be any multiple or part of B, C is the same multiple or

part of D, by Prop, d. Book v.

Next, ifAB be to CD, as EF to GH:

then if AB contain any part of CD, EF contains the same part of GH :

for let CKhe a part of CD, and GL the same part of GH,

and let AB be a multiple of CK:

EF is the same multiple of GL :

take M the same multiple of GL that AB is of CK;

therefore, by Prop, c, Book v, A Bis to CK, as M to GL :

and CD, GH, are equimultiples of CK, GL ;

wherefore, by Cor. Prop. 4, Book v, AB is to CD, as M to GH.

And, by the hvpothesis, AB is to CD, as EF to GH;

therefore 3/is equal to EF by Prop. 9, Book v, ^

and consequently, EF is the same multiple of GL that AB is of CK.

J

NOTES TO BOOK V.

247

This is the method by which Simson shews that the Geometrical

definition of proportion is a consequence of the Arithmetical definition,

and conversely.

It may however be shewn by employingthe equation -r = -j , and taking

ma, tnc any equimultiples of a and c the first and third, and nb, nd any

equimultiples of b and d the second and fourth.

And conversely, it may be shewn ex absurdo, that if four quantities

are proportionals according to the fifth definition of the fifth book of

Euclid, they are also proportionals according to the Algebraical definition.

The student must however bear in mind, that the Algebraical defini-

nition is not equally applicable to the Geometrical demonstrations con-

tained in the sixth, eleventh, and twelfth Books of Euclid, where the

Geometrical definition is employed. It has been before remarked, that Geo-

metry is the science oi magnitude and not oi number ; and though a sum and

a difierence of two magnitudes can be represented Geometrically, as well

as a multiple of any given magnitude, there is no method in Geometry

whereby the quotient of two magnitudes of the same kind can be ex-

pressed. The idea of a quotient is entirely foreign to the principles of

the Fifth Book, as are also any distinctions of magnitudes as being com-

mensurable or incommensurable. As Euclid in Books viiâ€” x has treated

of the properties of proportion according to the Arithmetical definition

and of their application to Geometrical magnitudes ; there can be no

doubt that his intention was to exclude all reference to numerical mea-

sures and quotients in his treatment of the doctrine of proportion in the

Fifth Book ; and in his applications of that doctrine in the sixth, eleventh

I and twelfth books of the Elements.

Prop. C. Algebraically.

Let A-^, ag, ^,, a^ be four magnitudes.

First let ^^ = via^ and A^ â€” ma^ :

Then A^^ : a^:: A^ '. a^.

For since A^

and A^

Hence -J: =r -^

Secondly.

and A^ : a^ : : A^ : a^.

-r ^ 1 , , 1

Let A, = â€” aâ€ž, and A., â€” â€” a,

m * m *

Then, as before,

a^ m

A A,

Hence â€” = â€”

tto a.

1 - A., 1

â€” , and â€” i = â€” ;

and Ai'. a^:: A^ '. a^.

D. Algebraically.

Let ^1, r?g, Ay a^ be proportionals,

or A^ : a, : : A^'. a^.

248

First let A^ be a multiple of a,, or A^ =m times a^ = maj.

Then shall A^ = wa^,

For since A^ : a^i: A^: a^f

* * Â«g a* '

but since -4i = ma^^

mao Ao A^

. . â€” = â€” - y or m = â€” ^ ,

and /^3 = ma^,

Therefore the third A^ is the same multiple of a^ the fourth.

Secondly. If ^^ = â€” a^, then shall A^ = â€” a^.

Fori

since

ill

!

1

^1

1

i^i =

-Â«'2Â»

.*.

t

m

aa

OT

^o

1

1

. '5

, and A^

=

â€” flf,

a.

m

m

â€¢wherefore, the third A^ is the same part of the fourth tti.

Prop. VII. is so ob-vious that it may be considered axiomatic. Also

Prop. VIII. and Prop. ix. are so simple and obvious, as not to require

algebraical proof.

Prop. X. Algebraically.

Let A^ have a greater ratio to a, than A^ has to a.

Then A^ > A^.

For the ratio of A^ to a is represented by â€” ,

and the ratio of A^ to a is represented by â€” ,

. A^ Ao

and since â€” > â€” ^ ;

a a

It follows that A^ > A.^.

Secondly. Let a have to A^ a greater ratio than a has to Ai.

Then A^< A^.

For the ratio of Â« : ^3 is represented by â€” ,

^3

a

and the ratio of a : ^j is represented by â€” ,

A\

, . a a

and since -7- > -r *

A^ A^

dividing these unequals by a,

Jl i_

and multiplying these unequals by A^.A^,

:. Af > Aa,

or A3 < A^.

NOTES TO BOOK V

249

Prop. XI. Algebraically.

Let the ratio of A^ : a^he the same as the ratio of -^3 : a^,

and the ratio of ^^ : a^ be the same as the ratio of A^ : a^.

Then the ratio of ^1, : a,^ shall be the same as the ratio of A^ : a^

Jb'or since Ai :

ao'.iJ^'.

Az

and since J3

: a, : : ^j.

. Â£3

^5

Hence ^

Â«2

and Ai : a^

: : ^5 : ao-

Prop. XII. Algebraically.

Let ^1, a2, ylg, ^4, y/5, ttg be proportionals,

so that A^ : a2 \: A^ : Ui :: A,^

Then shall A^ : a^:'. J^ -\- A-a + A^ :\

Por since Ai : a^:: As : a^ :: J,

'2+ Â«4

^3

And V -!- =

^1

Â£3

Hence ^1 (oj + a^ + a^) = Â«2 (^1 + ^3 + ^s)' by addition,

and dividing these equals by a^ {a^ + 04 + Oq)*

â€¢ ^ _ ^1 + ^3 + ^5 .

^2 ~ Â«2 + "^4 + "0

and J, : (72 : : ^1 + ^3 + J5 : a2 + ^4 -j- a^.

Prop. XIII. Algebraically.

Let ^1, rt2>^3' "4' A' <^6' b^ six magnitudes, such that A^ : rt2 :: -^s : <^n

but that the ratio of A^ : Â«4 is greater than the ratio of A-^ : r/^.

Then the ratio of A^ : a^ shall be greater than the ratio of A.^ : a^.

Por since ^, '. a^'.i A^x Â«4 .*. â€” = â€” 5

but since ^3 : 04 > ^5 : a^

^1

Hence â€” i

Â«9

>^-^

3

Â«4 Â«6

That is, the ratio of ^4, : a^ is greater than the ratio of A^ : a^.

Prop. XIV. Algebraically.

Let /I J, ^2, ^s, Â«4 be proportionals,

Then if A^ > A^y then Oj > <^4Â» ^^^ if equal, equal ; and if less, less.

Por since A^: a^i\ A^ : a^.

"A

M5

250

Multiply these equals by â€” ;

" A^ 04*

and because these fractions are always equal,

if A^ be > ^3, then a^ must be greater than a^,

for if a.2 were not greater than a^^

oi , , . , , . A,

the fraction - could not be equal to

_4 .

A J

"3

which would be contrary to the hypothesis.

In the same manner,

if Ai be = ^3, then a^ must be equal to 04,

and if A^ be < A^, a^ must be less than 04.

Hence, therefore, if &c.

Prop. XV. Algebraically.

Let ylj, ffa be any magnitudes of the same kind,

Then ^4, : o.^y. mA : ma^ ;

mA^ and mÂ«2 being any equimultiples of ^1 and a^.

For ^^ = ^ ,

. ^2 ^2

and since the numerator and denominator of a fraction may be mul-

tiplied by the same number without altering the value of the fraction,

A^ niA^

and ^1 : 02 : : mAi : ma^.

Prop. XVI. Algebraically.

Let ^â€ž ^2, ^3, <t^ be four magnitudes of the same kind, which are

proportionals,

A^'. a^'.'. Aq'. 04.

Then these shall be proportionals when taken alternately, that is,

Ai : ^, : : 02 â€¢ Â«4-

For since Ai : a.^ â€¢ â€¢ -^3 * Â«4 Â»

then -'=â€”Â».

Â«2 Â«4

Multiply these equals by -~ ,

â€¢â€¢ ^, a*'

and ^j : ^3 : : a2 : ^4.

Prop. xviT. Algebraically.

Let A^ + flg' Â«2' -^^^ + ^4' ''^4 ^^ proportionals,

then ^1, Â«2, A^, a^ shall be proportionals.

For since A^-^ a^i a^ : : ^3 + 04 : G4 ^

â€¢ ^1 +^ 3 _ ^8 + "4 ,

"a, a4

A , A.,

or - t + 1 = â€”^ 4. 1,

NOTES TO BOOK V. 251

and taking 1 from each of these equals,

â€¢ -il â€” _

and Ai : az :: A3 : a^.

Prop. xTiii, is the converse of Prop. xvii.

The following is Euclid's indirect demonstration.

Let AE, EB, CF, FD be proportionals,

that is, as AE to EB, so let CF be to FD :

then these shall be proportionals also when taken jointly :

that is, as JB to BE, so shall CD be to DF.

Q F D

For if the ratio oi AB to BE be not the same as the ratio of CD to DF;

the ratio of AB to BE is either greater than, or less than the ratio of

CD to DF.

Pirst, let AB have to BE a less ratio than CD has to DF ;

and let DQ be taken so that AB has to BE the same ratio as CD to DQ:

and since magnitudes when taken jointly are proportionals,

they are also proportionals when taken separately ; (v. 17.)

therefore AE has to EB the same ratio as CQ to QD ;

but, by the hypothesis, AE has to EB the same ratio as CF to FD ;

therefore the ratio of CQ to QD is the same as the ratio oiCFto FD. (v. 11.)

And when four magnitudes are proportionals, if the first be greater than

the second, the third is greater than the fourth ; and if equal, equal ; and

if less, less ; (v. 14.) but CQ is less than CF,

therefore QD is less than FD ; which is absurd.

Wherefore the ratio of AB to BE is not less than the ratio of CD to DF;

that is, AB has the same ratio to BE as CD has to DF.

Secondly. By a similar mode of reasoning, it may likewise be shewn,

that AB has the same ratio to BE as CD has to DF, if JB be assumed to

have to BE ?i greater ratio than CD has to DF.

Prop, xviii. Algebraically.

Let Ai : ffg â€¢â€¢ -^3 '. cl^.

Then Ai + a^ : a^ :: Ag + a^ : tti.

For since Ai : a^ :: A^ : a^,

and adding 1 to each of these equals ;

A^ As .

â€¢â– â€¢ TT + 1 = T + ^Â»

A^ + Â«2 A3+ ai

or, â€” i = ,

and A^ + a^ : a^:: A^+ a^: a^.

Prop. XIX. Algebraically.

Let the whole A^ have the same ratio to the whole A,^,

as a\ taken from the first, is to a^ taken from the second,

that is, let Ai : A^:: a^: a^.

Then Ai â€” ui : A^ â€” a^:: Ai i Ai.

352

#

Euclid's elements.

For since Ai : A^:i ai : a^,

A.y

Multiplying these equals by â€” ,

ai

. Ai A^ rti Ai ^

A.2 Â«! Â«2 <^1

or â€” i = -^ ,

and subtracting 1 from eacb of these equals,

A-^ â€” Cli -^2 ~ ^2

or,

^2

and multiplying these equals by â– - â€” "^ â€” Â»

A2 â€” ttg

A^ â€” flj ffj

but 4-' = ?-'

â€¢^2 Â«s

A, â€” a, -4

and Ai - a^ : A^- a^ :: A^ : A^,

Cor. If J 1 : /i 2 â€¢ â€¢ <*i â€¢ ^2Â»

Then Ay - a, : ^ 2 "~ ^2 : â€¢ "1 = %> is found proved in the preceding

process.

Prop. E. Algebraically.

~ )t Ay : u^

Then shall A^ : A^ â€” a^ :: ^3 : ^3 â€” a^.

Â«4i

subtracting 1 from each of these equals,

. ^1 1 _^3 1

â€¢ . A â€” â€” -i,

or

A^ â€” Cln -/* g â€” 0^4

ttj

bI.t-^ = -^

Dividing the latter by the former of these equals,

Ai Ai â€” Oo A-^ A^ â€” ttj.

Uo (Zo

or X â€” ; = X -; ,

02 Ai â€” a^ a^ A3 â€” a^'

NOTES TO BOOK V. 253

or -J = -z â€” ;

and ^1 : -4i â€” ttj : : -^3 : -43 â€” a^.

Prop. XX. Algebraically.

Let An A2, A2 be three magnitudes, and a^, a,, ag, other three,

such that Ai : Ao :: a^: a2t

and Ao : A^ : : Oo : ciz :

if ^1 > ^3, then shall a^ > a^t

and if equal, equal ; and if less, less,

Suice Ai'. A2 :: ai : a2, .*. -r- = - ,

also since A^: A^ : : Â«3 : Â«3. .'. -j- = - Â»

A2 ^3 ^

and multiplying these equals,

^ Ai A2 _ai Oxi

or â€” = â€” ,

^3 Â«3

A a

and since the fraction â€” ^ is equal to â€” ;

A Â«3

and that ^^ > ^3 :

It follows that tti is > a^.

In the same way it may be shewn

that if A^ = A^, then a^ = a^; and if Ai be < A^, then ^i < a^.

Prop. XXI. Algebraically.

Let Ai, A2, A^y be three magnitudes,

and Â«i, a2> ^3 three others,

such that Ai : A2 : : a^ ; 03,

and A2 : A^ : : ay : ch.

If ^1 > A2, then shall a^ > a^, and if equal, equal ; and if less, less.

Por smce A^: A^ :: a^: a^, .'. -r=~ >

A^ aâ€ž

Ji ' A A . ^2 Â«1

and since Ao : Ao : : a^ : ao, . . -â€” = .

A.^ Â«3

Multiplying these equals,

A-l At) Qny Ctrl

â€¢ â€¢ ^ ^ â€” â€” X â€” J

i.Â«.2 ^3 fl/3 0^2

or -^ = -^ ;

^3 Â«3

and since the fraction ~ is equal to â€” ,

and that A^ > A^,

It follows that also a^ > ag.

Similarly, it may be shewn, that if A^ = ^13, then aj

and if A-^ < -^g, also #Â«v < Â«3.

254

EUCLID S ELEMENTS.

Prop. XXII. Algebraically.

Let Ay A2, A^ be tbree magnitudes,

and Â«!, a2Â» Â«3 other three,

such that ^1 : ^2 : : Â«! : o.^,

and A^: A^'.: a^i a^.

Then shall A^: A^'.i a^: a^.

For since A^-, A^'.'. a^\ a^, /. _i = ^ ,

and since ^, : ^3 : : ag â€¢ Â«3Â» .". â€” = â€” ^ .

Multiply these equals,

, ^1 Ao, a^ do

â€¢ â€¢ T X -7- = â€” X -

^2 ^3 Â«3 Â«3

â– Â»

^1 ^1

or â€” = - .

^3 Â«3*

and ^1 : ^3 : : ttj ; Og

,.

if there be four magnitudes, and other four such, that

A'A-'-ar-ch,

A^ : A^ :: a^ : a^,

A : A : : Â«3 : a^.

Then shaU A^ : ^4 : : a^

:Â«4.

For since Aii J.2 : : a^ : Og, .*.

^1

A.r

Â«1

^3 : ^3 : : 02 : ag, .-.

A2

^3

^4 04

Multiplying these equals,

Ai Ao

^3 ^3

Â£3 ^ Â«i

^4 Â«2

x^x

Â«3

Â«3

or

^1 Â«!

Ai Ui*

and Ai

:Ai::a^

:Â«4,

Rnd similarly,

if there were more

than foui

â€¢ magnil

tudes.

Prop.

XXIII.

, Algebraically.

Let ^1, ^2Â» -^3

be three i

magnitudes.

and a^, a,,

, 03 other three,

such that Ai : A2 :: ^

03:03.

and ^2 :

; ^3 : : Â«! :

03.

Â«

Then shall A^iA^w

oi rog.

For since A-^ : A^

; : 03 : 053,

= ^,

and since ^42 : ^3 : : Oj : Og,

NOTES TO BOOK V.

.^55

Multiplying these equals,

.*. -^ X -^ = - X

Ao A^ ^3

Â«1

a,*

A^ Â«i

or -i = -^ ,

^3 <h

and Ai'. A^:: tti :

Og.

If there were four magnitudes, and other four,

such that Ai'. A^:: a^,

: Â«4.

A2 : A^ : : ch,

:Â«3,

A.^iAi'.-.ai

roj.

Then shall also A^: A^:

: ai

: aj.

For

since A^ \ Ao'.: a^x a^y ,'.

^1

^3

~a4'

^3 : ^3 : : 03 : 03, /.

^2

^3

~a3'

^3 : ^4:: Â«! : 03 .-.

^3

^4

Â«2'

Multiplying these equals,

A^ Ao Ao tto

/, -1 X -^ X â€” ^ = -3 X

Az A3 Ai ^4

Â«3 ^ Â«1

or -r = â€” >

^4 Â«4

/. Ai : A^ :: oi : a^,

and similarly, if there he more than four magnitudes.

Prop. XXIV. Algebraically.

Let ^1 : ^3 : : ^3 : 0^4,

and A^ : a^ :; A^ : 04,

Then shall ^^ + ^5 : ag : : ^3 + ^^ : a^.

For since A, : a^ :: ^3 : 04, .*. â€” = â€” ^,

% Â«4

a^ ct-x

Divide the former by the latter of these equals,

, -^1 . -^5 ^3 . A^

and since A^: 02 :: A^: 04,

03

do A.

Oo,

Oft

Â«4

Â«4

^3 Â«4

Â«4 -^f

^3

adding 1 to each of these equals,

A

+ 1

or

A

+ 1.

^3 + ^6

256

and ^ = ^.

Multijjly these equals together,

. ^1 + ^5 Ab _ Aq + ^6 :^6

^1 + ^5 _ -^3 + -^.

or â€” â€” = 6.

% ^4

and .*. ^1 + ^5 : % : : ^3 + ^6 5 '*4Â«

Cor. 1, Similarly may be shewn, that

Ai â€” Ai : a^:: A^ -^ Aa : a^.

Prop. XXV. Algebraically.

proportionals this is not the case.

A series of numbers or Algebraical quantities in continued proportion,

is called a Geometrical progression^ from the analogy they bear to a series

of Geometrical magnitudes in continued proportion.

Def. A. The term compound ratio was devised for the purpose of

avoiding circumlocution, and no difficulty can arise in the use of it, if

its exact meaning be strictly attended to.

With respect to the Geometrical measures of compound ratios, three

straight lines may measure the ratio of four, as in Prop. 23, Book vi.

For Kio L measures the ratio oi BC to CO, and L to M measures the

ratio oi DC to CE; and the ratio of K toiV/is that which is said to be

compounded of the ratios of K to L, and L to M, which is the same as the

ratio which is compounded of the ratios of the sides of the parallelograms.

Both duplicate and triplicate ratio are species of compound ratio.

Duplicate ratio is a ratio compounded of two equal ratios ; and in the

case of three magnitudes which are continued proportionals, means the

ratio of the first to a third proportional to the first and second.

Triplicate ratio, in the same manner, is a ratio compounded of three

equal ratios ; and in the case of four magnitudes which are continued

proportionals, the triplicate ratio of the first to the second means the

ratio of the first to a fourth proportional to the first, second, and third

magnitudes. Instances of the composition of three ratios, and of tripli-

cate ratio, will be found in the eleventh and twelfth books.

The product of the fractions which represent or measure the ratios

NOTES TO BOOK V. 243

of numbers, corresponds to the composition of Geometrical ratios of

magnitudes.

It has been shewn that the ratio of two numbers is represented by a

fraction whereof the numerator is the antecedent, and the denominator

the consequent of the ratio ; and if the antecedents of two ratios be

multiplied together, as also the consequents, the new ratio thus formed

is said to be compounded of these two ratios ; and in the same manner,

if there be more than two. It is also obvious, that the ratio compounded

of two equal ratios is equal to the ratio of the squares of one of the ante-

cedents to its consequent ; also when there are three equal ratios, the

ratio compounded of the three ratios is equal to the ratio of the cubes of

any one of the antecedents to its consequent. And further, it may be

observed, that when several numbers are continued proportionals, the

ratio of the first to the last is equal to the ratio of the product of all the

antecedents to the product of all the consequents.

It may be here remarked, that, though the constructions of the pro-

positions in Book v are exhibited by straight lines, the enunciations are

ex]>ressed of magnitude in general, and are equally true of angles,

triangles, parallelograms, arcs, sectors, &c.

The two following axioms may be added to the four Euclid has given.

Ax. 5. A part of a greater magnitude is greater than the same part

of a less magnitude.

Ax. 6. That magnitude of which any part is greater than the same

part of another, is greater than that other magnitude.

The learner must not forget that the capital lettersy used generally by

Euclid in the demonstrations of the fifth Book, represent the magnitudes,

not any numerical or Algebraical measures of them : sometimes however

the magnitude of a line is represented in the usual way by two letters

which are placed at the extremities of the line.

Prop. I. Algebraically.

Let each of the magnitudes A, B, C, &c. be equimultiples of as many

a, b, c, &c.

that is, let A = m times a = ma,

B = m times b = mb,

C â€” m times c â€” mc, &c.

First, if there be two magnitudes equimultiples of two others.

Then A + B â€” ma + mb = m (a + 6) = m times (a + 6),

Hence A + B is the same multiple of {a + b), as A is of a, or B of 6.

Secondly, if there be three magnitudes equimultiples of three others,

then A + B + C = ma + mb + mc = m {a + b + c)

= m times (o + 6 + c),

Hence A + B + Cis the same multiple of (a + b + c);

as A is of a, B of 6, and C of c.

Similarly, if there were four, or any number of magnitudes.

Therefore, if any number of magnitudes be equimultiples of as many,

each of each ; what multiple soever, any one is of its part, the same

multiple shall the first magnitudes be of all the other.

Prop. II. Algebraically.

Let A^ the first magnitude, be the same multiple of a^ the second,

as A^ the third, is of a^ the fourth ; and A^, the fifth the same multiple

of a, the second, as A^ the sixth, is of ai the fourth.

m2

244 Euclid's elements..

That is, let /4j = m times a^ = ma^,

Ai = m times a^ = ma^,

Af,= n times a^ = wa^,

Jq = n times a^ â€” na^,

Then by addition, A^ + A^ = ma^ + naz = {m-\-n) a, = (rn + n) times a^,

and Ai-^ A^ = ma^ + wa^ = (m + w) a^ = (m + w) times a^.

Therefore A^ + ^^ is the same multiple of a^^ as A^ + /^^ is of a^.

That is, if the first magnitude be the same multiple of the second, as

the third is of the fourth, &c.

Cor. If there be any number of magnitudes A-^^ A^, A^, &c. multiples

of another a, such that A^ = ma, A^ = na, A^ = pa, &c.

And as many others JSj, B^, B^, &c. the same multiples of another b,

such that ^1 = mb, B^ = nb, B^ = pb, &c.

Then by addition, A^ + A^ + A3 + &c. = ma + na + pa + &c.

= (m + n + p + &c.) a={m + n+p + &c.) times a :

and Bi + B^ + B^ + &c. = 7nb + nb + pb + &c. = {m + n -\- p + &c.) b

= (m + n + p + &c.) times b :

that is A^-\-Ar, + A^ + &c. is the same multiple of a that

^1 + -Bg + ^3 + &c. is of b.

Prop. in. Algebraically.

Let Ai the first magnitude, be the same multiple of a^ the second,

as A3 the third, is of a^ the fourth,

that is, let Ay^ = m times a^ = ma^,

and As = m times a^ = ma^.

If these equals be each taken n times,

then nAi = mna^ = mn times o^,

and nA^ = vma^ = mn times a^,

or nA^y nA^ each contain a^,, a^ respectively mn times.

Wherefore n//j, 7iA^ the equimultiples of the first and third, are

respectively equimultiples of a^ and a^, the second and fourth.

Prop. IV. Algebraically.

Let ^1, flg, ^3, a^, be proportionals according to the Algebraical

definition :

that is, let A^ : g.^ : : Jg : a^

then â€” = -^ ,

multiply these equals by â€” , w and n being any integers,

or mAy : wa^ : : mA^ : waj.

That is, if the first of four magnitudes has the same ratio to the

second which the third has to the fourth ; then any equimultiples what-

ever of the first and third shall have the same ratio to any equimultiples

of the second and fourth.

NOTES TO BOOK V. 245

.lie Corollary is contained in the proposition itself :

for if n be unity, then mj^ : a^ :: niJ^ : a^:

and if m be unity, also Ai : na^ :: ^s'. na^.

Prop. V. Algebraically.

Let Ji be the same multiple of aâ€ž

that Jg a part of J^ is of a^, a part of ai.

Then Ji â€” J^ is the same multiple of a^^ â€” o^ as J^ is of a, :

For let ^, = m times a^ = ma^^

and A%^ m times Â«8 = ^^ag,

then y^i â€” ^2 = '"^1 ~ "'^2 = m (Â«! - ag) = m times {a^ - Og),

that is Ji - ^2 is ^^^ ^^^ s^Â°^^ multiple of (aj - a^) as ^^ is of a^.

Prop. VI. Algebraically.

Let Ji, ^g be equimultiples respectively of a^, a, two others,

that is, let A^ =m times Oj = ma^,

-^2 = w times a^ = ma^^

Also if B^ a part of ^^ = n times a^ = wa^,

and B^ a part of Ac^ = n times a2 = ^^2*

Then by taking equals from equals,

.*. A-^ â€” B^= ma^ â€” na^ = {m â€” n) a^ = (m â€” n) times a^,

A2 â€” B^ = ma^ â€” na^, = (m â€” n) a^ = (m â€” n) times a^ :

that is, the remainders A-^ â€” B^, A^ â€” B^ are equimultiples of fli, aj,

respectively.

And if m - w = 1, then A^ â€” B^ â€” a^, and A^ â€” B2 = a^i

or the remainders are equal to a^, a^ respectively.

rop. A. Algebraically,

Let Au a^, A 3, a^ be proportionals,

or A^ : a.^:: A^: a^,

And since the fraction â€” is equal to â€” , the following relations

a, a^

only can subsist between A^ and a,; and between A^ and a^,

First, if Ai be greater than a^; then A^ is also greater than a^:

Secondly, if A^ be equal to a^ ; then A^ is also equal to a^ :

'I'hirdly, if Ai be less than a^ ; then ^3 is also less than a^ :

A A

Otherwise, the fraction â€” could not be equal to the fraction ~ .

a, a^

Prop. B. Algebraically.

Let A^, ttg, ^j,, a^ be proportionals,

or. ^1 : a, :: /^3 ; ff^,

Then shall a^ : Ai : : a^ : ^3 .

For since Ai i a^ :: A^ : a^

A\ A?.

I^>

246 Euclid's elements.

and if 1 be divided by each of these equals,

and therefore a^: A^ : : a^ : /I3.

Prop. c. " This is frequently made use of by geometers, and is necessary

to the oth and 6th Propositions of the 10th Book. Clavius, in his notes

subjoined to the 8th def. of Book 5, demonstrates it only in numbers, by

help of some of the propositions of the 7th Book ; in order to demonstrate

the property contained in the 5th definition of the 5th Book, when applied

to numbers, from the property of proportionals contained in the 20th def.

of the 7th Book : and most of the commentators judge it difficult to prove

that four magnitudes which are proportionals according to the 20th def.

of the 7th Book, are also proportionals according to the 5th def. of the

6th Book. But this is easily made out as follows ;

First, if A, B, C, Z), be four magnitudes, such that A is the same

multiple, or the same part of JB, which C is of D :

Then A, B, C, D, are proportionals:

this is demonstrated in proposition (c).

Secondly, if AB contain the same parts of CD that EF does of GH ;

in this case likewise AB is to CD, as EF to GH.

A B E F

C K D G L II

Let CKhe a part of CD, and GL the same part of GH;

and let AB be the same multiple of CK, that EF is of GL :

therefore, by Prop, c, of Book v, AB is to CK, as EF to GL :

and CDy GH, are equimultiples of CK, GL, the second and fourth ;

wherefore, by Cor. Prop. 4, Book v, AB is to CD, as EF to GH.

And if four magnitudes be proportionals according to the 5th def. of Book v,

they are also proportionals according to the 20th def. of Book vii.

First, if A be to ^, as CtoD;

then if A be any multiple or part of B, C is the same multiple or

part of D, by Prop, d. Book v.

Next, ifAB be to CD, as EF to GH:

then if AB contain any part of CD, EF contains the same part of GH :

for let CKhe a part of CD, and GL the same part of GH,

and let AB be a multiple of CK:

EF is the same multiple of GL :

take M the same multiple of GL that AB is of CK;

therefore, by Prop, c, Book v, A Bis to CK, as M to GL :

and CD, GH, are equimultiples of CK, GL ;

wherefore, by Cor. Prop. 4, Book v, AB is to CD, as M to GH.

And, by the hvpothesis, AB is to CD, as EF to GH;

therefore 3/is equal to EF by Prop. 9, Book v, ^

and consequently, EF is the same multiple of GL that AB is of CK.

J

NOTES TO BOOK V.

247

This is the method by which Simson shews that the Geometrical

definition of proportion is a consequence of the Arithmetical definition,

and conversely.

It may however be shewn by employingthe equation -r = -j , and taking

ma, tnc any equimultiples of a and c the first and third, and nb, nd any

equimultiples of b and d the second and fourth.

And conversely, it may be shewn ex absurdo, that if four quantities

are proportionals according to the fifth definition of the fifth book of

Euclid, they are also proportionals according to the Algebraical definition.

The student must however bear in mind, that the Algebraical defini-

nition is not equally applicable to the Geometrical demonstrations con-

tained in the sixth, eleventh, and twelfth Books of Euclid, where the

Geometrical definition is employed. It has been before remarked, that Geo-

metry is the science oi magnitude and not oi number ; and though a sum and

a difierence of two magnitudes can be represented Geometrically, as well

as a multiple of any given magnitude, there is no method in Geometry

whereby the quotient of two magnitudes of the same kind can be ex-

pressed. The idea of a quotient is entirely foreign to the principles of

the Fifth Book, as are also any distinctions of magnitudes as being com-

mensurable or incommensurable. As Euclid in Books viiâ€” x has treated

of the properties of proportion according to the Arithmetical definition

and of their application to Geometrical magnitudes ; there can be no

doubt that his intention was to exclude all reference to numerical mea-

sures and quotients in his treatment of the doctrine of proportion in the

Fifth Book ; and in his applications of that doctrine in the sixth, eleventh

I and twelfth books of the Elements.

Prop. C. Algebraically.

Let A-^, ag, ^,, a^ be four magnitudes.

First let ^^ = via^ and A^ â€” ma^ :

Then A^^ : a^:: A^ '. a^.

For since A^

and A^

Hence -J: =r -^

Secondly.

and A^ : a^ : : A^ : a^.

-r ^ 1 , , 1

Let A, = â€” aâ€ž, and A., â€” â€” a,

m * m *

Then, as before,

a^ m

A A,

Hence â€” = â€”

tto a.

1 - A., 1

â€” , and â€” i = â€” ;

and Ai'. a^:: A^ '. a^.

D. Algebraically.

Let ^1, r?g, Ay a^ be proportionals,

or A^ : a, : : A^'. a^.

248

First let A^ be a multiple of a,, or A^ =m times a^ = maj.

Then shall A^ = wa^,

For since A^ : a^i: A^: a^f

* * Â«g a* '

but since -4i = ma^^

mao Ao A^

. . â€” = â€” - y or m = â€” ^ ,

and /^3 = ma^,

Therefore the third A^ is the same multiple of a^ the fourth.

Secondly. If ^^ = â€” a^, then shall A^ = â€” a^.

Fori

since

ill

!

1

^1

1

i^i =

-Â«'2Â»

.*.

t

m

aa

OT

^o

1

1

. '5

, and A^

=

â€” flf,

a.

m

m

â€¢wherefore, the third A^ is the same part of the fourth tti.

Prop. VII. is so ob-vious that it may be considered axiomatic. Also

Prop. VIII. and Prop. ix. are so simple and obvious, as not to require

algebraical proof.

Prop. X. Algebraically.

Let A^ have a greater ratio to a, than A^ has to a.

Then A^ > A^.

For the ratio of A^ to a is represented by â€” ,

and the ratio of A^ to a is represented by â€” ,

. A^ Ao

and since â€” > â€” ^ ;

a a

It follows that A^ > A.^.

Secondly. Let a have to A^ a greater ratio than a has to Ai.

Then A^< A^.

For the ratio of Â« : ^3 is represented by â€” ,

^3

a

and the ratio of a : ^j is represented by â€” ,

A\

, . a a

and since -7- > -r *

A^ A^

dividing these unequals by a,

Jl i_

and multiplying these unequals by A^.A^,

:. Af > Aa,

or A3 < A^.

NOTES TO BOOK V

249

Prop. XI. Algebraically.

Let the ratio of A^ : a^he the same as the ratio of -^3 : a^,

and the ratio of ^^ : a^ be the same as the ratio of A^ : a^.

Then the ratio of ^1, : a,^ shall be the same as the ratio of A^ : a^

Jb'or since Ai :

ao'.iJ^'.

Az

and since J3

: a, : : ^j.

. Â£3

^5

Hence ^

Â«2

and Ai : a^

: : ^5 : ao-

Prop. XII. Algebraically.

Let ^1, a2, ylg, ^4, y/5, ttg be proportionals,

so that A^ : a2 \: A^ : Ui :: A,^

Then shall A^ : a^:'. J^ -\- A-a + A^ :\

Por since Ai : a^:: As : a^ :: J,

'2+ Â«4

^3

And V -!- =

^1

Â£3

Hence ^1 (oj + a^ + a^) = Â«2 (^1 + ^3 + ^s)' by addition,

and dividing these equals by a^ {a^ + 04 + Oq)*

â€¢ ^ _ ^1 + ^3 + ^5 .

^2 ~ Â«2 + "^4 + "0

and J, : (72 : : ^1 + ^3 + J5 : a2 + ^4 -j- a^.

Prop. XIII. Algebraically.

Let ^1, rt2>^3' "4' A' <^6' b^ six magnitudes, such that A^ : rt2 :: -^s : <^n

but that the ratio of A^ : Â«4 is greater than the ratio of A-^ : r/^.

Then the ratio of A^ : a^ shall be greater than the ratio of A.^ : a^.

Por since ^, '. a^'.i A^x Â«4 .*. â€” = â€” 5

but since ^3 : 04 > ^5 : a^

^1

Hence â€” i

Â«9

>^-^

3

Â«4 Â«6

That is, the ratio of ^4, : a^ is greater than the ratio of A^ : a^.

Prop. XIV. Algebraically.

Let /I J, ^2, ^s, Â«4 be proportionals,

Then if A^ > A^y then Oj > <^4Â» ^^^ if equal, equal ; and if less, less.

Por since A^: a^i\ A^ : a^.

"A

M5

250

Multiply these equals by â€” ;

" A^ 04*

and because these fractions are always equal,

if A^ be > ^3, then a^ must be greater than a^,

for if a.2 were not greater than a^^

oi , , . , , . A,

the fraction - could not be equal to

_4 .

A J

"3

which would be contrary to the hypothesis.

In the same manner,

if Ai be = ^3, then a^ must be equal to 04,

and if A^ be < A^, a^ must be less than 04.

Hence, therefore, if &c.

Prop. XV. Algebraically.

Let ylj, ffa be any magnitudes of the same kind,

Then ^4, : o.^y. mA : ma^ ;

mA^ and mÂ«2 being any equimultiples of ^1 and a^.

For ^^ = ^ ,

. ^2 ^2

and since the numerator and denominator of a fraction may be mul-

tiplied by the same number without altering the value of the fraction,

A^ niA^

and ^1 : 02 : : mAi : ma^.

Prop. XVI. Algebraically.

Let ^â€ž ^2, ^3, <t^ be four magnitudes of the same kind, which are

proportionals,

A^'. a^'.'. Aq'. 04.

Then these shall be proportionals when taken alternately, that is,

Ai : ^, : : 02 â€¢ Â«4-

For since Ai : a.^ â€¢ â€¢ -^3 * Â«4 Â»

then -'=â€”Â».

Â«2 Â«4

Multiply these equals by -~ ,

â€¢â€¢ ^, a*'

and ^j : ^3 : : a2 : ^4.

Prop. xviT. Algebraically.

Let A^ + flg' Â«2' -^^^ + ^4' ''^4 ^^ proportionals,

then ^1, Â«2, A^, a^ shall be proportionals.

For since A^-^ a^i a^ : : ^3 + 04 : G4 ^

â€¢ ^1 +^ 3 _ ^8 + "4 ,

"a, a4

A , A.,

or - t + 1 = â€”^ 4. 1,

NOTES TO BOOK V. 251

and taking 1 from each of these equals,

â€¢ -il â€” _

and Ai : az :: A3 : a^.

Prop. xTiii, is the converse of Prop. xvii.

The following is Euclid's indirect demonstration.

Let AE, EB, CF, FD be proportionals,

that is, as AE to EB, so let CF be to FD :

then these shall be proportionals also when taken jointly :

that is, as JB to BE, so shall CD be to DF.

Q F D

For if the ratio oi AB to BE be not the same as the ratio of CD to DF;

the ratio of AB to BE is either greater than, or less than the ratio of

CD to DF.

Pirst, let AB have to BE a less ratio than CD has to DF ;

and let DQ be taken so that AB has to BE the same ratio as CD to DQ:

and since magnitudes when taken jointly are proportionals,

they are also proportionals when taken separately ; (v. 17.)

therefore AE has to EB the same ratio as CQ to QD ;

but, by the hypothesis, AE has to EB the same ratio as CF to FD ;

therefore the ratio of CQ to QD is the same as the ratio oiCFto FD. (v. 11.)

And when four magnitudes are proportionals, if the first be greater than

the second, the third is greater than the fourth ; and if equal, equal ; and

if less, less ; (v. 14.) but CQ is less than CF,

therefore QD is less than FD ; which is absurd.

Wherefore the ratio of AB to BE is not less than the ratio of CD to DF;

that is, AB has the same ratio to BE as CD has to DF.

Secondly. By a similar mode of reasoning, it may likewise be shewn,

that AB has the same ratio to BE as CD has to DF, if JB be assumed to

have to BE ?i greater ratio than CD has to DF.

Prop, xviii. Algebraically.

Let Ai : ffg â€¢â€¢ -^3 '. cl^.

Then Ai + a^ : a^ :: Ag + a^ : tti.

For since Ai : a^ :: A^ : a^,

and adding 1 to each of these equals ;

A^ As .

â€¢â– â€¢ TT + 1 = T + ^Â»

A^ + Â«2 A3+ ai

or, â€” i = ,

and A^ + a^ : a^:: A^+ a^: a^.

Prop. XIX. Algebraically.

Let the whole A^ have the same ratio to the whole A,^,

as a\ taken from the first, is to a^ taken from the second,

that is, let Ai : A^:: a^: a^.

Then Ai â€” ui : A^ â€” a^:: Ai i Ai.

352

#

Euclid's elements.

For since Ai : A^:i ai : a^,

A.y

Multiplying these equals by â€” ,

ai

. Ai A^ rti Ai ^

A.2 Â«! Â«2 <^1

or â€” i = -^ ,

and subtracting 1 from eacb of these equals,

A-^ â€” Cli -^2 ~ ^2

or,

^2

and multiplying these equals by â– - â€” "^ â€” Â»

A2 â€” ttg

A^ â€” flj ffj

but 4-' = ?-'

â€¢^2 Â«s

A, â€” a, -4

and Ai - a^ : A^- a^ :: A^ : A^,

Cor. If J 1 : /i 2 â€¢ â€¢ <*i â€¢ ^2Â»

Then Ay - a, : ^ 2 "~ ^2 : â€¢ "1 = %> is found proved in the preceding

process.

Prop. E. Algebraically.

~ )t Ay : u^

Then shall A^ : A^ â€” a^ :: ^3 : ^3 â€” a^.

Â«4i

subtracting 1 from each of these equals,

. ^1 1 _^3 1

â€¢ . A â€” â€” -i,

or

A^ â€” Cln -/* g â€” 0^4

ttj

bI.t-^ = -^

Dividing the latter by the former of these equals,

Ai Ai â€” Oo A-^ A^ â€” ttj.

Uo (Zo

or X â€” ; = X -; ,

02 Ai â€” a^ a^ A3 â€” a^'

NOTES TO BOOK V. 253

or -J = -z â€” ;

and ^1 : -4i â€” ttj : : -^3 : -43 â€” a^.

Prop. XX. Algebraically.

Let An A2, A2 be three magnitudes, and a^, a,, ag, other three,

such that Ai : Ao :: a^: a2t

and Ao : A^ : : Oo : ciz :

if ^1 > ^3, then shall a^ > a^t

and if equal, equal ; and if less, less,

Suice Ai'. A2 :: ai : a2, .*. -r- = - ,

also since A^: A^ : : Â«3 : Â«3. .'. -j- = - Â»

A2 ^3 ^

and multiplying these equals,

^ Ai A2 _ai Oxi

or â€” = â€” ,

^3 Â«3

A a

and since the fraction â€” ^ is equal to â€” ;

A Â«3

and that ^^ > ^3 :

It follows that tti is > a^.

In the same way it may be shewn

that if A^ = A^, then a^ = a^; and if Ai be < A^, then ^i < a^.

Prop. XXI. Algebraically.

Let Ai, A2, A^y be three magnitudes,

and Â«i, a2> ^3 three others,

such that Ai : A2 : : a^ ; 03,

and A2 : A^ : : ay : ch.

If ^1 > A2, then shall a^ > a^, and if equal, equal ; and if less, less.

Por smce A^: A^ :: a^: a^, .'. -r=~ >

A^ aâ€ž

Ji ' A A . ^2 Â«1

and since Ao : Ao : : a^ : ao, . . -â€” = .

A.^ Â«3

Multiplying these equals,

A-l At) Qny Ctrl

â€¢ â€¢ ^ ^ â€” â€” X â€” J

i.Â«.2 ^3 fl/3 0^2

or -^ = -^ ;

^3 Â«3

and since the fraction ~ is equal to â€” ,

and that A^ > A^,

It follows that also a^ > ag.

Similarly, it may be shewn, that if A^ = ^13, then aj

and if A-^ < -^g, also #Â«v < Â«3.

254

EUCLID S ELEMENTS.

Prop. XXII. Algebraically.

Let Ay A2, A^ be tbree magnitudes,

and Â«!, a2Â» Â«3 other three,

such that ^1 : ^2 : : Â«! : o.^,

and A^: A^'.: a^i a^.

Then shall A^: A^'.i a^: a^.

For since A^-, A^'.'. a^\ a^, /. _i = ^ ,

and since ^, : ^3 : : ag â€¢ Â«3Â» .". â€” = â€” ^ .

Multiply these equals,

, ^1 Ao, a^ do

â€¢ â€¢ T X -7- = â€” X -

^2 ^3 Â«3 Â«3

â– Â»

^1 ^1

or â€” = - .

^3 Â«3*

and ^1 : ^3 : : ttj ; Og

,.

if there be four magnitudes, and other four such, that

A'A-'-ar-ch,

A^ : A^ :: a^ : a^,

A : A : : Â«3 : a^.

Then shaU A^ : ^4 : : a^

:Â«4.

For since Aii J.2 : : a^ : Og, .*.

^1

A.r

Â«1

^3 : ^3 : : 02 : ag, .-.

A2

^3

^4 04

Multiplying these equals,

Ai Ao

^3 ^3

Â£3 ^ Â«i

^4 Â«2

x^x

Â«3

Â«3

or

^1 Â«!

Ai Ui*

and Ai

:Ai::a^

:Â«4,

Rnd similarly,

if there were more

than foui

â€¢ magnil

tudes.

Prop.

XXIII.

, Algebraically.

Let ^1, ^2Â» -^3

be three i

magnitudes.

and a^, a,,

, 03 other three,

such that Ai : A2 :: ^

03:03.

and ^2 :

; ^3 : : Â«! :

03.

Â«

Then shall A^iA^w

oi rog.

For since A-^ : A^

; : 03 : 053,

= ^,

and since ^42 : ^3 : : Oj : Og,

NOTES TO BOOK V.

.^55

Multiplying these equals,

.*. -^ X -^ = - X

Ao A^ ^3

Â«1

a,*

A^ Â«i

or -i = -^ ,

^3 <h

and Ai'. A^:: tti :

Og.

If there were four magnitudes, and other four,

such that Ai'. A^:: a^,

: Â«4.

A2 : A^ : : ch,

:Â«3,

A.^iAi'.-.ai

roj.

Then shall also A^: A^:

: ai

: aj.

For

since A^ \ Ao'.: a^x a^y ,'.

^1

^3

~a4'

^3 : ^3 : : 03 : 03, /.

^2

^3

~a3'

^3 : ^4:: Â«! : 03 .-.

^3

^4

Â«2'

Multiplying these equals,

A^ Ao Ao tto

/, -1 X -^ X â€” ^ = -3 X

Az A3 Ai ^4

Â«3 ^ Â«1

or -r = â€” >

^4 Â«4

/. Ai : A^ :: oi : a^,

and similarly, if there he more than four magnitudes.

Prop. XXIV. Algebraically.

Let ^1 : ^3 : : ^3 : 0^4,

and A^ : a^ :; A^ : 04,

Then shall ^^ + ^5 : ag : : ^3 + ^^ : a^.

For since A, : a^ :: ^3 : 04, .*. â€” = â€” ^,

% Â«4

a^ ct-x

Divide the former by the latter of these equals,

, -^1 . -^5 ^3 . A^

and since A^: 02 :: A^: 04,

03

do A.

Oo,

Oft

Â«4

Â«4

^3 Â«4

Â«4 -^f

^3

adding 1 to each of these equals,

A

+ 1

or

A

+ 1.

^3 + ^6

256

and ^ = ^.

Multijjly these equals together,

. ^1 + ^5 Ab _ Aq + ^6 :^6

^1 + ^5 _ -^3 + -^.

or â€” â€” = 6.

% ^4

and .*. ^1 + ^5 : % : : ^3 + ^6 5 '*4Â«

Cor. 1, Similarly may be shewn, that

Ai â€” Ai : a^:: A^ -^ Aa : a^.

Prop. XXV. Algebraically.

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