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

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the double of the fourth :

but if the first be greater than the second,

the double of the first is greater than the double of the second ;

â€¢wherefore also the double of the third is greater than the double of

the fourth ;

therefore the third is greater than the fourth :

in like manner if the first be equal to the second, or less than it,

the third can be proved to be equal to the foui-th, or less than it.

Therefore, if the first, &c. q.e.d.

PROPOSITION B. THEOREM.

If four viagniludes are proportionals, they are proportionals also when

taken i7iversely.

Let A be to B, as C is to D.

Then also inversely, B shall be to -4, as JD to C.

A B c D

Take of B and D any equimultiples whatever E and F)

and of A and C any equimultiples whatever G and H.

First, let E be greater than G, then G is less than E:

and because A is to B, as C is to D, (hyp.)

and of A and C, the first and third, G and H are equimultiples ;

and of B and D, the second and fourth, E and i^are equimultiples ;

and that G is less than E, therefore ^is less than F-, (v. def. 5.)

that is, F is greater than H',

if, therefore, E be greater than Gf

F is greater than H;

in like manner, if E be equal to (?,

jPmay be shewn to be equal to -H";

and if less, less ;

but E, Ff are any equimultiples whatever of ^ and D, (constr.)

and (r, if any whatever of A and C;

therefore, as ^ is to ^, so is D to C. (v. def. 5.)

Therefore, if four magnitudes, &c, Q. E. D.

PROPOSITION C. THEOREM.

If the first he the same multiple of the second, or the same part of itf that

the third is of the fourth ; the first ts to the second, as the third is to the

fourth.

Let the first A be the same multiple of the second B,

that the third C is of the fourth D.

BOOK V. PROP. D. 213

Then A shall be to J5 as C is to D.

Take of -4 and Cany equimultiples whatever E and F-,

and of B and D any equimultiples whatever G^ and //.

Then, because A is the same multiple of B that C is of Z) ; (hyp.)

and that E is the same multiple of ^, that Pis of C; (constr.)

therefore E is the same multiple of B, that i^is of jD ; (v. 3.)

that is, E and i^are equimultiples of J5 and D:

but G and H are equimultiples of B and D ; (constr.)

therefore, if P be a greater multiple of B than G is of B,

jP is a greater multiple of i) than JT is of Z) ;

that is, if E be greater than G^

F is greater than ZT:

in like manner, if E be equal to G, or less than it,

F may be shewn to be equal to H, or less than it,

but E, F are equimultiples, any whatever, of A, C; (constr.)

and G, EC any equimultiples whatever of ^, Z) ;

therefore A is to B, as C is to D. (v. def. 5.)

Next, let the first A be the same part of the second B, that the

third C is of the fourth E.

Then A shall be to B, as C is to E,

For since A is the same part of B that C is of D,

therefore B is the same multiple of A, that Z) is of C:

wherefore, by the preceding case, B is to A, as E is to C;

and therefore inversely, -4 is to B, as C is to E. (v. B.)

Therefore, if the first be the same multiple, &c. q.e.d.

PROPOSITION D. THEOREK.

If the first be to the second as the third to the fourth, and if the fir it be a

multiplexor a part of the second ; the third is the same multiple, or the same

part of the fourth.

Let ^ be to 5 as Cisio E:

and first, let ^ be a multiple of B.

Then C shall be the same multiple of E.

A B C D

Take E e^ual to Ay

and whatever multiple -4 or jS is of B, make F the same multiple

of E:

then, because A is to B, as Cis to E ; (hyp.)

and of B the second, and E the fourth, equimultiples have been

taken, E and F;

therefore A is to E, as C to F: (v. 4. Cor.)

but A is equal to E, (constr.)

therefore C is equal to F: (v. a.)

214: Euclid's elements.

and Fh the same multiple of D, that A is of ^; (constr.)

therefore C is the same multiple of D, that A is of JB.

Next, let A the first be a part of JB the second.

Then C the third shall be the same part of D the fourth.

.Because A is to J3, as C is to D ; (hyp.)

then, inversely, JB is to A, as JD to C: (v. B.)

A B C D

but A is a part of Bf therefore ^ is a multiple of A : (hyp.)

therefore, by the preceding case, D is the same multiple of (7;

that is, C is the same part of D, that A is of B.

Therefore, if the first, &c. Q. E. D.

PROPOSITION VII. THEOREM.

Equal magnitudes have the same ratio to the same magnitude : and the same

has the same ratio to equal magnitudes.

Let A and JB be equal magnitudes, and C any other.

Then A and J3 shall each of them have the same ratio to C :

and C shall have the same ratio to each of the magnitudes A and B.

D E F

Take of -dt and JB any equimultiples whatever D and .E,

and of C any multiple whatever F.

Then, because D is the same multiple of A, that F is of i?, (consti*.)

and that A is equal to JB : (hyp.)

therefore J> is equal to F; (v. ax. 1.)

therefore, if D be greater than F, F is greater than F;

and if equal, equal ; if less, less :

but Z), F are any equimultiples of ^, F, (constr.)

and jPis any multiple of C;

therefore, as A is to C, so is F to C. (v. def. 5.)

Likewise C shall have the same ratio to A, that it has to F.

For having made the same construction,

F may in like manner be shewn to be equal to F;

therefore, if F be greater than Z>,

it is likewise greater than F;

and if equal, equal ; if less, less ;

but i^is any multiple whatever of C,

and JD, F are any equimultiples whatever of ^, F;

therefore, C is to ^ as C is to F. (v. def. 5.)

Therefore, equal magnitudes, &c. Q. E. D.

PROPOSITION VIII. THEOREM.

Of two unequal magnitudes, the greater has a greater ratio to any other

magnitude than the less has : and the same magnitude has a greater ratio to the

less of two other magnitudes, than it has to the greater.

Let AB, BChe two unequal magnitudes, of which AB is the greater,

and let F be any other magnitude.

BOOK V. PROP. vni.

215

Then^^ shall have a greater ratio to D than J5(7has to Di

and D shall have a greater ratio to jSCthan it has to AB,

Fig. 1.

G B

L K H D

Fig. 2.

G B

L K PI D

Fig. 3.

E

- A

C-

G B

L K D

If the magnitude which is not the greater of the two A C, CB, be

not less than D.

take EF, FG, the doubles of ^C, CB, (as in fig. 1.)

but if that which is not the greater of the two A C, CB, be less than D.

(as in fig. 2 and 3.) this magnitude can be multiplied, so as to

become greater than D, whether it be A C, or CB.

Let it be multiplied until it become greater than D,

and let the other be multiplied as often ;

and let FF be the multiple thus taken of A C,

and FG the same multiple of CB :

therefore FF and FG are each of them greater than D :

and in every one of the cases,

take JTthe double of Z>, Xits triple, and so on,

till the multiple of Z> be that which first becomes greater than FG:

let L be that multiple of J) which is first greater than FG,

and X the multiple of Z> which is next less than L.

en because Z is the multiple of Z>, which is the first that becomes

greater than FG,

the next preceding multiple K is not greater than FG;

that is, FG is not less than K:

and since FF is the same multiple of A C, that FG is of CB ; (constr.)

therefore FG is the same multiple of CB, that FG is of AB ; (v. 1.)

that is, FG and FG are equimultiples of AB and CB ;

and since it was shewn, that FG is not less than K,

and, by the construction, FF is greater than D ;

therefore the whole FG is greater than X and D together :

but K together with D is equal to Z ; (constr.)

therefore FG is greater than Z;

but FG is not greater than L : (constr.)

and FG, FG were proved to be equimultiples of AB, BC]

and Z is a multiple of D ; (constr.)

therefore AB has to Z) a greater ratio than ZChas to D. (v. def. 7.)

Also D shall have io BC a greater ratio than it has to AB,

216 Euclid's elements.

For having made the same construction,

it may be shewn in like manner, that L is greater than FGi

but that it is not greater than UG :

and Z is a multiple of D ; (constr.)

and FG, EG were jiroved to be equimultiples of CB, AB :

therefore D has to CB a greater ratio than it has to AB. (v. def. 7.)

Wherefore, of two unequal magnitudes, &c. Q.e.d.

PROPOSITION IX. THEOREM.

Magnitudes which have the same ratio to the same magnitude are equal to

one another : and those to which the same m,agnitude has the same ratio are

equal to one another.

Let, A, B have each of them the same ratio to C,

Then A shall be equal to B.

A D

c F

B E

For, if they are not equal, one of them must be greater than the other :

let A be the greater :

then, by what was shewn in the preceding proposition,

there are some equimultiples of A and B, and some multiple of C, such,

tliat the multiple of ^ is greater than the multiple of C,

but the multiple of jB is not greater than that of C,

let these multiples be taken ;

and let D, E be the equimultiples of A, B,

and i^the multiple of C,

such that D may be greater than F, but F not greater than F.

Then, because ^ is to C as B is to C, (hyp.)

and of A, B, are taken equimultiples, 1), F,

and of Cis taken a multiple F;

and that D is greater than jP;

therefore F is also greater than F: (v. def. 5.)

but F is not greater than F; (constr.) which is impossible :

therefore A and B are not unequal ; that is, they are equal.

Next, let C have the same ratio to each of the magnitudes A and B.

Then A shall be equal to B.

For, if they are not equal, one of them must be greater than the other :

let A be the greater :

therefore, as was shewn in Prop. VIII.

there is some multiple F of C,

and some equimultiples F and Z>,of B and A such,

that F is greater than F, but not greater than F :

and because C is to B, as C is to A, (hyp.)

and that F the multiple of the fii-st, is greater than F the multiple of

the second;

therefore F the multiple of the third, is greater than F the multiple

of the fourth: (v. def. 5.)

BOOK V. PROP. X, XT. 217

but jPis not greater than D (hyp.) ; which is impossible ;

therefore A is equal to B.

Wherefore, magnitudes which, &c. Q.E.D.

PROPOSITION X. THEOREM.

That magnitude icJiich has a greater ratio than another has unto the

same magnitude^ is the greater of the two ; and that magnitude to which the

same has a greater ratio than it has unto another magnitude^ is the less

of the two.

Let A have to Ca greater ratio than B has to C;

then A shall be greater than B.

D-

C F

B E

For, because A has a greater ratio to C, than B has to C,

there are some equimultiples of A and B,

and some multiple of C such, (v. def. 7.)

that the multiple of ^ is greater than the multiple of C,

but the multiple of B is not greater than it :

let them be taken ;

and let D, JE'be the equimultiples of ^, B, and jPthe multiple of C;

such, that D is greater than F, but E is not greater than F :

* therefore D is greater than E :

and, because E and E are equimultiples of A and B,

and that E is greater than E ;

therefore A is greater than B. (v. ax. 4.)

Next, let Chave a greater ratio to B than it has to A

Then B shall be less than A.

For there is some multiple F of C, (v. def. 7.)

and some equimultiples E and E of B and A, such

that i^is greater than E, but not greater than E :

therefore E is less than E :

and because E and E are equimultiples of B and A,

and that E is less than D,

therefore B is less than A. (V. ax. 4.)

Therefore, that magnitude, &c. Q. e. d.

PROPOSITION XI. THEOREM.

Ratios that are the same to the same ratio, are the same to one another.

Let ^ be to jB as C is to D ;

and as C to E, so let E be to F.

Then A shall be to B, as E to F.

G

â€” H

K â€”

A

B

D

F

L

M

- N

Take of -4, C, E, any equimultiples whatever G, H, X;

L

218 Euclid's elements.

and of B, D, F any equimultiples whatever L, M, K.

Therefore, since -4 is to ^ as C to D,

and G, H are taken equimultiples of A, (7,

and L, M, ofB,D',

if G be greater than L, His greater than 3f;

and if equal, equal ; and if less, less. (v. def. 5.)

Again, because C is to I), as JE is to F,

and H, K are taken equimultiples of C, JS;

and 31, N, of D, F;

if ^be greater than 31, K is greater than N]

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

but if G be greater than L,

it has been shewn that His greater than 3Â£',

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

therefore, if G be greater than L,

JTis greater than JV; and if equal, equal ; and if less, less :

and (r, JSTare any equimultiples whatever of ^, E ;

and L, N any whatever of B, F;

therefore, as A is to B, so is E to F. (v. def. 5.)

Wherefore, ratios that, &c. Q. E. D.

PROPOSITION XII. THEOREM.

If any number of magnitudes he proportionals, as one of the antecedents

is to its consequent, so shall all the antecedents taken together be to all the

consequents.

Let any number of magnitudes A, B, C, D, E, F, be proportionals :

that is, as A is to B, so C to D, and E to F.

Then as A is to B, so shall A, C, E together, be to B, D, JP together.

G H K

A C E

B D F

L M N

Take of ^, C, E any equimultiples whatever G, H, K;

and of B, D, F any equimultiples whatever, L, 31, N.

Then, because A is to B, as C is to Z), and as E to F;

and that G, H, K are equimultiples of ^, C, E,

and L, M, N, equimultiples of B, D, F;

therefore, if G be greater than L,

His greater than 31, and ^ greater than iV;

and if equal, equal ; and if less, less : (v. def. 5.)

wherefore if G be greater than L,

then G, H, Z" together, are greater than L, 3f, iV together;

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

but G, and G,Hf ^together, are any equimultiples of ^, and A, C,

E together ;

because if there be any number of magnitudes equimultiples of

as many, each of each, whatever multiple one of them is of its part,

the same multiple is the whole of the whole : (v. 1.)

BOOK V. PROP. XIII. 219

for the same reason X, and L, 31, iVare any equimultiples of ^, and

B, D, F:

therefore as A is to JB, so are A, C, E together to B, D, i^ together.

(V. def. 5.)

"Wherefore, if any number, &c. Q. E. D.

PROPOSITION XIII. THEOREM.

If the first has to the second the same ratio which the third has to the

fourth, but the third to the fourth, a greater ratio than the fifth has to the

sixth ; the first shall also have to the second a greater ratio than the fifth

has to the sixth.

Let A the first have the same ratio to B the second, which C the

third has to D the fourth, but C the third a greater ratio to D the

fourth, than E the fifth has to F the sixth.

Then also the first A shall have to the second B, a greater ratio

than the fifth F has to the sixth F.

M G II

A C E

B D F

N K L

Because Chas a greater ratio to D, than E to F,

there are some equimultiples of C and F, and some of D and F, such

that the multiple of C is greater than the multiple of D, but the mul-

tiple of F is not greater than the multiple of F: (v. def. 7.)

let these be taken,

and let G, Hhe equimultiples of C, F,

and K, X equimultiples of Z), F, such that G may be greater than K,

but H not greater than L :

and whatever multiple G is of C, take Jf the same multiple of A ;

and whatever multiple K is of JD, take iV^ the same multiple of B :

then, because A is to B, as Cto D, (hyp.)

and of A and C, M and G are equimultiples ;

and of B and D, iV and jK" are equimultiples ;

therefore, if Jf be greater than N, G is greater than K;

and if equal, equal ; and if less, less ; (V. def. 5.)

but G is greater than K; (constr.)

therefore M is greater than N:

but XT is not greater than L : (constr.)

and M, XT are equimultiples of ^, F ;

and N, L equimultiples oi B, F;

therefore A has a greater ratio to B, than F has to F. (V. def. 7.)

Wherefore, if the first, &c. Q.E.D.

Cor. And if the first have a greater ratio to the second, than the

third has to the fourth, but the third the same ratio to the fourth,

which the fifth has to the sixth ; it may be demonstrated, in like

manner, that the first has a greater ratio to the second, than the fifth

has to the sixth.

l2

220 Euclid's elements.

PROPOSITION XIV. THEOREM.

If the first has the same ratio to the second which the third has to the fourth ;

then, if the first be greater than the third, the second shall be greater than the

fourth ; and if equal, equal ; and if less, less.

Let the first A have the same ratio to the second B "which the

third C has to the fourth D.

li A be greater than C, B shall be greater than D. (fig. 1.)

1. 2. 3.

A A A

B B- B

C C C

D-

Because A is greater than C, and B is any other magnitude,

A has to -S a greater ratio than C has to B : (v. 8.)

but, as A is to B, so is C io D; (hyp.)

therefore also C has to Z) a greater ratio than C has to B : (v. 13.)

but of two magnitudes, that to which the same has the greater ratio,

is the less : (v. 10.) â€¢

therefore D is less than B;

that is, B is greater than JD,

Secondly, if A be equal to (7, (fig, 2.)

then B shall be equal to D.

For A is to B, as C, that is, Ato D:

therefore B is equal to D. (v. 9.)

Thirdly, if A be less than C, (fig. 3.)

then B shall be less than 1).

For C is greater than A ;

and because C is to D, as A is to B,

therefore D is greater than B, by the first case ;

that is, B is less than D.

Therefore, if the fii-st, &c. q. e. d.

PROPOSITION XV. THEOREM.

Magnitudes have the same ratio to one another which their equimultiples

have.

Let -^^ be the same multiple of C, that DBis of F.

Then C shall be to J^, as AB is to DB.

AGHB DKLE ^

Because AB is the same multiple of C, that DJS is of JP;^

there are as many magnitudes in AB equal to C, as there are in DE

equal to F:

let ^^ be divided into magnitudes, each equal to C, viz. A G, GH, HB;

BOOK V. PROP. XVI. 221

and DJE into magnitudes, each equal to F, viz. DK, KL, LE:

then the number of the first AG, Gil, HB, is equal to the number

of the last DK, KL, LE:

and because A G, GH, HB are all equal,

and that DK, KL, LE, are also equal to one another i

therefore ^6? is to DK, as GH lo KL, and as HB to LE: (v. 7.)

but as one of the antecedents is to its consequent, so are all the

antecedents together to all the consequents together, (v. 12.)

wherefore, as A G is to DK, so is A B to DE :

but AG is equal to 6^ and DK to F:

therefore, as C is to F, so is AB to DE.

Therefore, magnitudes, &c. Q.E.D.

PROPOSITION XVI. THEOREM.

If four magnitudes of the same kind be proportionals, they shall also be

proportionals when taken alternately.

Let A, B, C, Dhe four magnitudes of the same kind, which are

proportionals, viz. as A to B, so C to D.

They shall also be proportionals when taken alternately :

that is, A shall be to C, as B to D.

E

A

â€” G

C

B

D .

F

H

Take of A and B any equimultiples whatever E and F:

and of C and D take any equimultiples whatever G and H.

And because E is the same multiple of A, that i^is of B,

and that magnitudes have the same ratio to one another which

their equimultiples have; (v. 15.)

therefore A is to B, as ^ is to i^:

but as ^ is to J5 so is Cto D\ (hyp.)

wherefore as C is to D, so is E to F: (v. 11.)

again, because G, H are equimultiples of C, D,

therefore as Cis to D, so is G to H: (v. 15.)

but it was proved that as C is to D, so is E to F;

therefore, as E is to F, so is G to H. (v. 11.)

But when four magnitudes are proportionals, if the first be greater

than the third, the second is greater than the fourth :

and if equal, equal; if less, less ; (v. 14.)

therefore, if E be greater than G, F likewise is greater than H-,

and if equal, equal ; if less, less :

and ^, i*^are any equimultiples whatever of A,B ; (constr.)

and G, H any whatever of C, D :

therefore A is to C, as B to D. (v. def. 5.)

If then four magnitudes, &c. Q. e. d.

i

222 Euclid's elements.

PROPOSITION XVII. THEOREM.

If magnitudes, taken jointly, be proportionals, they shall also be pro-

portionals when taken separately : that is, if two magnitudes together have

to one of them, the same ratio which two others have to one of these, the

remaining one of the first two shall have to the other the same ratio which

the remaining one of the last two has to the other of these.

Let AJS, BE, CD, DF be the magnitudes, taken jointly which

are proportionals ;

that is, as AB to JBU, so let CD be to DF.

Then they shall also be proportionals taken separately,

viz. as ^^ to FB, so shall Ci^be to FD.

GHK X LMNP

E B C FD

Take of AF, FB, CF, FD any equimultiples whatever GS, HK,

LM,MN'.

and again, of FB, FD take any equimultiples whatever KX, NF.

Then because GH\^ the same multiple of AF, that HKis of FB,

therefore G'^is the same multiple of AF, that GKis of AB ; (v. 1.)

but G^^is the same multiple of ^^, that L3f is of CF:

therefore GKh the same multiple of AB, that Z3/is of CF.

Again, because ZMh the same multiple of CF, that 3IN is of FD;

therefore ZM is the same multiple of CF, that ZiVis of CD: (v. 1.)

but LMw&s shewn to be the same multiple of CF, that GX is of AB;

therefore GX is the same multiple of AB, that ZiV^is of CD;

that is, GX, ZN are equimultiples of AB, CD.

Next, because HX is the same multiple of FB, that MN is of FD ;

and that XX is also the same multi{)le of FB, that NP is of FD ;

therefore HX is the same multiple of FB, that MP is of FD. (v. 2.)

And because AB is to BF, as CD is to DF, (hyp.)

and that of AB and CD, GX and ZN am equimultiples,

and of FB and FD,IIX and 3IP are equimultiples ;

therefore if GX be greater than HX, then ZN is greater than MP j

and if equal, equal : and if less, less : (v. def. 5.)

but if GHhe greater than XX,

then, by adding the common part HX to both,

GX is greater than HX; (l. ax. 4.)

wherefore also ZN is greater than 3IP j

and by taking away 3/iV from both,

Z3Iis greater than XP: (i. ax. 5.)

therefore, if GH be greater than XXf

ZM is greater than XP.

In like manner it may be demonstrated,

that if GH be equal to JTX,

Z3f is equal to XP ; and if less, less :

but GH, ZM are any equimultiples whatever of AF, CF, (constr.)

and XX, XP are any whatever of FB, FD :

therefore, as AF is to FB, so is CFto FD. (v. def. 5.)

If then magnitudes, &c. Q. E. D.

BOOK V. PROP, xviir. 223

PROPOSITION XVIII. THEOREM.

7f magnitudes, taken separately, be proportionals, they shall also be

proportionals when taken jointly : that is, if the first be to the second, as

the third to the fourth, the first and second together shall be to the second,

as the third and fourth together to the fourth.

Let AE, EB, CF, FD be proportionals ;

that is, as AE to EB, so let Ci^be to FD. ^

Then they shall also be proportionals when taken jointly;

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

G KOH L NPM

A E B C F D

Take of AB, BE, CD, DF any equimultiples whatever GH, HE,

LM, MN',

and again, oi BE, DFtoke any equimuhiples whatever KG, NP:

and because KO, NP are equimultiples of BE, DF;

and that KH, NM are likewise equimultiples of BE, DF-,

therefore if KO, the multiple of BE, be greater than EEl^, which

is a multiple of the same BE,

then NP, the multiple of DF, is also greater than N3I, the mul-

tiple of the same DF ;

and if KO be equal to KII, â€¢

NP is equal to NM; and if less, less.

First, let KO be not greater than KII;

therefore NP is not greater than N3I:

and because GH, UK, are equimultiples of AB, BE,

and that AB is greater than BE,

therefore Gil is greater than UK ; (v. ax. 3.)

but KO is not greater than KII;

therefore GH is greater than KO,

In like manner it may be shewn, that L3I is greater than NP.

Therefore, if KO be not greater than KH,

then GH, the multiple of AB, is always greater than KO, the

multiple of BE ;

and likewise LM, the multiple of CD, is greater than NP, the

multiple of DF.

Next, let KO be greater than KH;

therefore, as has been shewn, NP is greater than N3I,

G K HO LNMP

E B C F D

And because the whole GH is the same multiple of the whole

AB, that HK is oi' BE,

therefore the remainder GK is the same multiple of the remainder

AE that GH is of AB, (v. 5.)

which is the same that LM is of CD.

224: Euclid's elements.

In like manner, because LMk the same multiple of CD, that MJV

is of BJE,

therefore the remainder LN is the same multiple of the remainder

CF, that the whole ZM is of the whole CJD : (v. 5.)

but it was shewn that LM is the same multiple of CD, that GK

isof^^;

therefore GK is the same multiple of AE, that ZN is of CF;

that is, GK, iiV are equimultiples of AF, CF.

but if the first be greater than the second,

the double of the first is greater than the double of the second ;

â€¢wherefore also the double of the third is greater than the double of

the fourth ;

therefore the third is greater than the fourth :

in like manner if the first be equal to the second, or less than it,

the third can be proved to be equal to the foui-th, or less than it.

Therefore, if the first, &c. q.e.d.

PROPOSITION B. THEOREM.

If four viagniludes are proportionals, they are proportionals also when

taken i7iversely.

Let A be to B, as C is to D.

Then also inversely, B shall be to -4, as JD to C.

A B c D

Take of B and D any equimultiples whatever E and F)

and of A and C any equimultiples whatever G and H.

First, let E be greater than G, then G is less than E:

and because A is to B, as C is to D, (hyp.)

and of A and C, the first and third, G and H are equimultiples ;

and of B and D, the second and fourth, E and i^are equimultiples ;

and that G is less than E, therefore ^is less than F-, (v. def. 5.)

that is, F is greater than H',

if, therefore, E be greater than Gf

F is greater than H;

in like manner, if E be equal to (?,

jPmay be shewn to be equal to -H";

and if less, less ;

but E, Ff are any equimultiples whatever of ^ and D, (constr.)

and (r, if any whatever of A and C;

therefore, as ^ is to ^, so is D to C. (v. def. 5.)

Therefore, if four magnitudes, &c, Q. E. D.

PROPOSITION C. THEOREM.

If the first he the same multiple of the second, or the same part of itf that

the third is of the fourth ; the first ts to the second, as the third is to the

fourth.

Let the first A be the same multiple of the second B,

that the third C is of the fourth D.

BOOK V. PROP. D. 213

Then A shall be to J5 as C is to D.

Take of -4 and Cany equimultiples whatever E and F-,

and of B and D any equimultiples whatever G^ and //.

Then, because A is the same multiple of B that C is of Z) ; (hyp.)

and that E is the same multiple of ^, that Pis of C; (constr.)

therefore E is the same multiple of B, that i^is of jD ; (v. 3.)

that is, E and i^are equimultiples of J5 and D:

but G and H are equimultiples of B and D ; (constr.)

therefore, if P be a greater multiple of B than G is of B,

jP is a greater multiple of i) than JT is of Z) ;

that is, if E be greater than G^

F is greater than ZT:

in like manner, if E be equal to G, or less than it,

F may be shewn to be equal to H, or less than it,

but E, F are equimultiples, any whatever, of A, C; (constr.)

and G, EC any equimultiples whatever of ^, Z) ;

therefore A is to B, as C is to D. (v. def. 5.)

Next, let the first A be the same part of the second B, that the

third C is of the fourth E.

Then A shall be to B, as C is to E,

For since A is the same part of B that C is of D,

therefore B is the same multiple of A, that Z) is of C:

wherefore, by the preceding case, B is to A, as E is to C;

and therefore inversely, -4 is to B, as C is to E. (v. B.)

Therefore, if the first be the same multiple, &c. q.e.d.

PROPOSITION D. THEOREK.

If the first be to the second as the third to the fourth, and if the fir it be a

multiplexor a part of the second ; the third is the same multiple, or the same

part of the fourth.

Let ^ be to 5 as Cisio E:

and first, let ^ be a multiple of B.

Then C shall be the same multiple of E.

A B C D

Take E e^ual to Ay

and whatever multiple -4 or jS is of B, make F the same multiple

of E:

then, because A is to B, as Cis to E ; (hyp.)

and of B the second, and E the fourth, equimultiples have been

taken, E and F;

therefore A is to E, as C to F: (v. 4. Cor.)

but A is equal to E, (constr.)

therefore C is equal to F: (v. a.)

214: Euclid's elements.

and Fh the same multiple of D, that A is of ^; (constr.)

therefore C is the same multiple of D, that A is of JB.

Next, let A the first be a part of JB the second.

Then C the third shall be the same part of D the fourth.

.Because A is to J3, as C is to D ; (hyp.)

then, inversely, JB is to A, as JD to C: (v. B.)

A B C D

but A is a part of Bf therefore ^ is a multiple of A : (hyp.)

therefore, by the preceding case, D is the same multiple of (7;

that is, C is the same part of D, that A is of B.

Therefore, if the first, &c. Q. E. D.

PROPOSITION VII. THEOREM.

Equal magnitudes have the same ratio to the same magnitude : and the same

has the same ratio to equal magnitudes.

Let A and JB be equal magnitudes, and C any other.

Then A and J3 shall each of them have the same ratio to C :

and C shall have the same ratio to each of the magnitudes A and B.

D E F

Take of -dt and JB any equimultiples whatever D and .E,

and of C any multiple whatever F.

Then, because D is the same multiple of A, that F is of i?, (consti*.)

and that A is equal to JB : (hyp.)

therefore J> is equal to F; (v. ax. 1.)

therefore, if D be greater than F, F is greater than F;

and if equal, equal ; if less, less :

but Z), F are any equimultiples of ^, F, (constr.)

and jPis any multiple of C;

therefore, as A is to C, so is F to C. (v. def. 5.)

Likewise C shall have the same ratio to A, that it has to F.

For having made the same construction,

F may in like manner be shewn to be equal to F;

therefore, if F be greater than Z>,

it is likewise greater than F;

and if equal, equal ; if less, less ;

but i^is any multiple whatever of C,

and JD, F are any equimultiples whatever of ^, F;

therefore, C is to ^ as C is to F. (v. def. 5.)

Therefore, equal magnitudes, &c. Q. E. D.

PROPOSITION VIII. THEOREM.

Of two unequal magnitudes, the greater has a greater ratio to any other

magnitude than the less has : and the same magnitude has a greater ratio to the

less of two other magnitudes, than it has to the greater.

Let AB, BChe two unequal magnitudes, of which AB is the greater,

and let F be any other magnitude.

BOOK V. PROP. vni.

215

Then^^ shall have a greater ratio to D than J5(7has to Di

and D shall have a greater ratio to jSCthan it has to AB,

Fig. 1.

G B

L K H D

Fig. 2.

G B

L K PI D

Fig. 3.

E

- A

C-

G B

L K D

If the magnitude which is not the greater of the two A C, CB, be

not less than D.

take EF, FG, the doubles of ^C, CB, (as in fig. 1.)

but if that which is not the greater of the two A C, CB, be less than D.

(as in fig. 2 and 3.) this magnitude can be multiplied, so as to

become greater than D, whether it be A C, or CB.

Let it be multiplied until it become greater than D,

and let the other be multiplied as often ;

and let FF be the multiple thus taken of A C,

and FG the same multiple of CB :

therefore FF and FG are each of them greater than D :

and in every one of the cases,

take JTthe double of Z>, Xits triple, and so on,

till the multiple of Z> be that which first becomes greater than FG:

let L be that multiple of J) which is first greater than FG,

and X the multiple of Z> which is next less than L.

en because Z is the multiple of Z>, which is the first that becomes

greater than FG,

the next preceding multiple K is not greater than FG;

that is, FG is not less than K:

and since FF is the same multiple of A C, that FG is of CB ; (constr.)

therefore FG is the same multiple of CB, that FG is of AB ; (v. 1.)

that is, FG and FG are equimultiples of AB and CB ;

and since it was shewn, that FG is not less than K,

and, by the construction, FF is greater than D ;

therefore the whole FG is greater than X and D together :

but K together with D is equal to Z ; (constr.)

therefore FG is greater than Z;

but FG is not greater than L : (constr.)

and FG, FG were proved to be equimultiples of AB, BC]

and Z is a multiple of D ; (constr.)

therefore AB has to Z) a greater ratio than ZChas to D. (v. def. 7.)

Also D shall have io BC a greater ratio than it has to AB,

216 Euclid's elements.

For having made the same construction,

it may be shewn in like manner, that L is greater than FGi

but that it is not greater than UG :

and Z is a multiple of D ; (constr.)

and FG, EG were jiroved to be equimultiples of CB, AB :

therefore D has to CB a greater ratio than it has to AB. (v. def. 7.)

Wherefore, of two unequal magnitudes, &c. Q.e.d.

PROPOSITION IX. THEOREM.

Magnitudes which have the same ratio to the same magnitude are equal to

one another : and those to which the same m,agnitude has the same ratio are

equal to one another.

Let, A, B have each of them the same ratio to C,

Then A shall be equal to B.

A D

c F

B E

For, if they are not equal, one of them must be greater than the other :

let A be the greater :

then, by what was shewn in the preceding proposition,

there are some equimultiples of A and B, and some multiple of C, such,

tliat the multiple of ^ is greater than the multiple of C,

but the multiple of jB is not greater than that of C,

let these multiples be taken ;

and let D, E be the equimultiples of A, B,

and i^the multiple of C,

such that D may be greater than F, but F not greater than F.

Then, because ^ is to C as B is to C, (hyp.)

and of A, B, are taken equimultiples, 1), F,

and of Cis taken a multiple F;

and that D is greater than jP;

therefore F is also greater than F: (v. def. 5.)

but F is not greater than F; (constr.) which is impossible :

therefore A and B are not unequal ; that is, they are equal.

Next, let C have the same ratio to each of the magnitudes A and B.

Then A shall be equal to B.

For, if they are not equal, one of them must be greater than the other :

let A be the greater :

therefore, as was shewn in Prop. VIII.

there is some multiple F of C,

and some equimultiples F and Z>,of B and A such,

that F is greater than F, but not greater than F :

and because C is to B, as C is to A, (hyp.)

and that F the multiple of the fii-st, is greater than F the multiple of

the second;

therefore F the multiple of the third, is greater than F the multiple

of the fourth: (v. def. 5.)

BOOK V. PROP. X, XT. 217

but jPis not greater than D (hyp.) ; which is impossible ;

therefore A is equal to B.

Wherefore, magnitudes which, &c. Q.E.D.

PROPOSITION X. THEOREM.

That magnitude icJiich has a greater ratio than another has unto the

same magnitude^ is the greater of the two ; and that magnitude to which the

same has a greater ratio than it has unto another magnitude^ is the less

of the two.

Let A have to Ca greater ratio than B has to C;

then A shall be greater than B.

D-

C F

B E

For, because A has a greater ratio to C, than B has to C,

there are some equimultiples of A and B,

and some multiple of C such, (v. def. 7.)

that the multiple of ^ is greater than the multiple of C,

but the multiple of B is not greater than it :

let them be taken ;

and let D, JE'be the equimultiples of ^, B, and jPthe multiple of C;

such, that D is greater than F, but E is not greater than F :

* therefore D is greater than E :

and, because E and E are equimultiples of A and B,

and that E is greater than E ;

therefore A is greater than B. (v. ax. 4.)

Next, let Chave a greater ratio to B than it has to A

Then B shall be less than A.

For there is some multiple F of C, (v. def. 7.)

and some equimultiples E and E of B and A, such

that i^is greater than E, but not greater than E :

therefore E is less than E :

and because E and E are equimultiples of B and A,

and that E is less than D,

therefore B is less than A. (V. ax. 4.)

Therefore, that magnitude, &c. Q. e. d.

PROPOSITION XI. THEOREM.

Ratios that are the same to the same ratio, are the same to one another.

Let ^ be to jB as C is to D ;

and as C to E, so let E be to F.

Then A shall be to B, as E to F.

G

â€” H

K â€”

A

B

D

F

L

M

- N

Take of -4, C, E, any equimultiples whatever G, H, X;

L

218 Euclid's elements.

and of B, D, F any equimultiples whatever L, M, K.

Therefore, since -4 is to ^ as C to D,

and G, H are taken equimultiples of A, (7,

and L, M, ofB,D',

if G be greater than L, His greater than 3f;

and if equal, equal ; and if less, less. (v. def. 5.)

Again, because C is to I), as JE is to F,

and H, K are taken equimultiples of C, JS;

and 31, N, of D, F;

if ^be greater than 31, K is greater than N]

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

but if G be greater than L,

it has been shewn that His greater than 3Â£',

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

therefore, if G be greater than L,

JTis greater than JV; and if equal, equal ; and if less, less :

and (r, JSTare any equimultiples whatever of ^, E ;

and L, N any whatever of B, F;

therefore, as A is to B, so is E to F. (v. def. 5.)

Wherefore, ratios that, &c. Q. E. D.

PROPOSITION XII. THEOREM.

If any number of magnitudes he proportionals, as one of the antecedents

is to its consequent, so shall all the antecedents taken together be to all the

consequents.

Let any number of magnitudes A, B, C, D, E, F, be proportionals :

that is, as A is to B, so C to D, and E to F.

Then as A is to B, so shall A, C, E together, be to B, D, JP together.

G H K

A C E

B D F

L M N

Take of ^, C, E any equimultiples whatever G, H, K;

and of B, D, F any equimultiples whatever, L, 31, N.

Then, because A is to B, as C is to Z), and as E to F;

and that G, H, K are equimultiples of ^, C, E,

and L, M, N, equimultiples of B, D, F;

therefore, if G be greater than L,

His greater than 31, and ^ greater than iV;

and if equal, equal ; and if less, less : (v. def. 5.)

wherefore if G be greater than L,

then G, H, Z" together, are greater than L, 3f, iV together;

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

but G, and G,Hf ^together, are any equimultiples of ^, and A, C,

E together ;

because if there be any number of magnitudes equimultiples of

as many, each of each, whatever multiple one of them is of its part,

the same multiple is the whole of the whole : (v. 1.)

BOOK V. PROP. XIII. 219

for the same reason X, and L, 31, iVare any equimultiples of ^, and

B, D, F:

therefore as A is to JB, so are A, C, E together to B, D, i^ together.

(V. def. 5.)

"Wherefore, if any number, &c. Q. E. D.

PROPOSITION XIII. THEOREM.

If the first has to the second the same ratio which the third has to the

fourth, but the third to the fourth, a greater ratio than the fifth has to the

sixth ; the first shall also have to the second a greater ratio than the fifth

has to the sixth.

Let A the first have the same ratio to B the second, which C the

third has to D the fourth, but C the third a greater ratio to D the

fourth, than E the fifth has to F the sixth.

Then also the first A shall have to the second B, a greater ratio

than the fifth F has to the sixth F.

M G II

A C E

B D F

N K L

Because Chas a greater ratio to D, than E to F,

there are some equimultiples of C and F, and some of D and F, such

that the multiple of C is greater than the multiple of D, but the mul-

tiple of F is not greater than the multiple of F: (v. def. 7.)

let these be taken,

and let G, Hhe equimultiples of C, F,

and K, X equimultiples of Z), F, such that G may be greater than K,

but H not greater than L :

and whatever multiple G is of C, take Jf the same multiple of A ;

and whatever multiple K is of JD, take iV^ the same multiple of B :

then, because A is to B, as Cto D, (hyp.)

and of A and C, M and G are equimultiples ;

and of B and D, iV and jK" are equimultiples ;

therefore, if Jf be greater than N, G is greater than K;

and if equal, equal ; and if less, less ; (V. def. 5.)

but G is greater than K; (constr.)

therefore M is greater than N:

but XT is not greater than L : (constr.)

and M, XT are equimultiples of ^, F ;

and N, L equimultiples oi B, F;

therefore A has a greater ratio to B, than F has to F. (V. def. 7.)

Wherefore, if the first, &c. Q.E.D.

Cor. And if the first have a greater ratio to the second, than the

third has to the fourth, but the third the same ratio to the fourth,

which the fifth has to the sixth ; it may be demonstrated, in like

manner, that the first has a greater ratio to the second, than the fifth

has to the sixth.

l2

220 Euclid's elements.

PROPOSITION XIV. THEOREM.

If the first has the same ratio to the second which the third has to the fourth ;

then, if the first be greater than the third, the second shall be greater than the

fourth ; and if equal, equal ; and if less, less.

Let the first A have the same ratio to the second B "which the

third C has to the fourth D.

li A be greater than C, B shall be greater than D. (fig. 1.)

1. 2. 3.

A A A

B B- B

C C C

D-

Because A is greater than C, and B is any other magnitude,

A has to -S a greater ratio than C has to B : (v. 8.)

but, as A is to B, so is C io D; (hyp.)

therefore also C has to Z) a greater ratio than C has to B : (v. 13.)

but of two magnitudes, that to which the same has the greater ratio,

is the less : (v. 10.) â€¢

therefore D is less than B;

that is, B is greater than JD,

Secondly, if A be equal to (7, (fig, 2.)

then B shall be equal to D.

For A is to B, as C, that is, Ato D:

therefore B is equal to D. (v. 9.)

Thirdly, if A be less than C, (fig. 3.)

then B shall be less than 1).

For C is greater than A ;

and because C is to D, as A is to B,

therefore D is greater than B, by the first case ;

that is, B is less than D.

Therefore, if the fii-st, &c. q. e. d.

PROPOSITION XV. THEOREM.

Magnitudes have the same ratio to one another which their equimultiples

have.

Let -^^ be the same multiple of C, that DBis of F.

Then C shall be to J^, as AB is to DB.

AGHB DKLE ^

Because AB is the same multiple of C, that DJS is of JP;^

there are as many magnitudes in AB equal to C, as there are in DE

equal to F:

let ^^ be divided into magnitudes, each equal to C, viz. A G, GH, HB;

BOOK V. PROP. XVI. 221

and DJE into magnitudes, each equal to F, viz. DK, KL, LE:

then the number of the first AG, Gil, HB, is equal to the number

of the last DK, KL, LE:

and because A G, GH, HB are all equal,

and that DK, KL, LE, are also equal to one another i

therefore ^6? is to DK, as GH lo KL, and as HB to LE: (v. 7.)

but as one of the antecedents is to its consequent, so are all the

antecedents together to all the consequents together, (v. 12.)

wherefore, as A G is to DK, so is A B to DE :

but AG is equal to 6^ and DK to F:

therefore, as C is to F, so is AB to DE.

Therefore, magnitudes, &c. Q.E.D.

PROPOSITION XVI. THEOREM.

If four magnitudes of the same kind be proportionals, they shall also be

proportionals when taken alternately.

Let A, B, C, Dhe four magnitudes of the same kind, which are

proportionals, viz. as A to B, so C to D.

They shall also be proportionals when taken alternately :

that is, A shall be to C, as B to D.

E

A

â€” G

C

B

D .

F

H

Take of A and B any equimultiples whatever E and F:

and of C and D take any equimultiples whatever G and H.

And because E is the same multiple of A, that i^is of B,

and that magnitudes have the same ratio to one another which

their equimultiples have; (v. 15.)

therefore A is to B, as ^ is to i^:

but as ^ is to J5 so is Cto D\ (hyp.)

wherefore as C is to D, so is E to F: (v. 11.)

again, because G, H are equimultiples of C, D,

therefore as Cis to D, so is G to H: (v. 15.)

but it was proved that as C is to D, so is E to F;

therefore, as E is to F, so is G to H. (v. 11.)

But when four magnitudes are proportionals, if the first be greater

than the third, the second is greater than the fourth :

and if equal, equal; if less, less ; (v. 14.)

therefore, if E be greater than G, F likewise is greater than H-,

and if equal, equal ; if less, less :

and ^, i*^are any equimultiples whatever of A,B ; (constr.)

and G, H any whatever of C, D :

therefore A is to C, as B to D. (v. def. 5.)

If then four magnitudes, &c. Q. e. d.

i

222 Euclid's elements.

PROPOSITION XVII. THEOREM.

If magnitudes, taken jointly, be proportionals, they shall also be pro-

portionals when taken separately : that is, if two magnitudes together have

to one of them, the same ratio which two others have to one of these, the

remaining one of the first two shall have to the other the same ratio which

the remaining one of the last two has to the other of these.

Let AJS, BE, CD, DF be the magnitudes, taken jointly which

are proportionals ;

that is, as AB to JBU, so let CD be to DF.

Then they shall also be proportionals taken separately,

viz. as ^^ to FB, so shall Ci^be to FD.

GHK X LMNP

E B C FD

Take of AF, FB, CF, FD any equimultiples whatever GS, HK,

LM,MN'.

and again, of FB, FD take any equimultiples whatever KX, NF.

Then because GH\^ the same multiple of AF, that HKis of FB,

therefore G'^is the same multiple of AF, that GKis of AB ; (v. 1.)

but G^^is the same multiple of ^^, that L3f is of CF:

therefore GKh the same multiple of AB, that Z3/is of CF.

Again, because ZMh the same multiple of CF, that 3IN is of FD;

therefore ZM is the same multiple of CF, that ZiVis of CD: (v. 1.)

but LMw&s shewn to be the same multiple of CF, that GX is of AB;

therefore GX is the same multiple of AB, that ZiV^is of CD;

that is, GX, ZN are equimultiples of AB, CD.

Next, because HX is the same multiple of FB, that MN is of FD ;

and that XX is also the same multi{)le of FB, that NP is of FD ;

therefore HX is the same multiple of FB, that MP is of FD. (v. 2.)

And because AB is to BF, as CD is to DF, (hyp.)

and that of AB and CD, GX and ZN am equimultiples,

and of FB and FD,IIX and 3IP are equimultiples ;

therefore if GX be greater than HX, then ZN is greater than MP j

and if equal, equal : and if less, less : (v. def. 5.)

but if GHhe greater than XX,

then, by adding the common part HX to both,

GX is greater than HX; (l. ax. 4.)

wherefore also ZN is greater than 3IP j

and by taking away 3/iV from both,

Z3Iis greater than XP: (i. ax. 5.)

therefore, if GH be greater than XXf

ZM is greater than XP.

In like manner it may be demonstrated,

that if GH be equal to JTX,

Z3f is equal to XP ; and if less, less :

but GH, ZM are any equimultiples whatever of AF, CF, (constr.)

and XX, XP are any whatever of FB, FD :

therefore, as AF is to FB, so is CFto FD. (v. def. 5.)

If then magnitudes, &c. Q. E. D.

BOOK V. PROP, xviir. 223

PROPOSITION XVIII. THEOREM.

7f magnitudes, taken separately, be proportionals, they shall also be

proportionals when taken jointly : that is, if the first be to the second, as

the third to the fourth, the first and second together shall be to the second,

as the third and fourth together to the fourth.

Let AE, EB, CF, FD be proportionals ;

that is, as AE to EB, so let Ci^be to FD. ^

Then they shall also be proportionals when taken jointly;

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

G KOH L NPM

A E B C F D

Take of AB, BE, CD, DF any equimultiples whatever GH, HE,

LM, MN',

and again, oi BE, DFtoke any equimuhiples whatever KG, NP:

and because KO, NP are equimultiples of BE, DF;

and that KH, NM are likewise equimultiples of BE, DF-,

therefore if KO, the multiple of BE, be greater than EEl^, which

is a multiple of the same BE,

then NP, the multiple of DF, is also greater than N3I, the mul-

tiple of the same DF ;

and if KO be equal to KII, â€¢

NP is equal to NM; and if less, less.

First, let KO be not greater than KII;

therefore NP is not greater than N3I:

and because GH, UK, are equimultiples of AB, BE,

and that AB is greater than BE,

therefore Gil is greater than UK ; (v. ax. 3.)

but KO is not greater than KII;

therefore GH is greater than KO,

In like manner it may be shewn, that L3I is greater than NP.

Therefore, if KO be not greater than KH,

then GH, the multiple of AB, is always greater than KO, the

multiple of BE ;

and likewise LM, the multiple of CD, is greater than NP, the

multiple of DF.

Next, let KO be greater than KH;

therefore, as has been shewn, NP is greater than N3I,

G K HO LNMP

E B C F D

And because the whole GH is the same multiple of the whole

AB, that HK is oi' BE,

therefore the remainder GK is the same multiple of the remainder

AE that GH is of AB, (v. 5.)

which is the same that LM is of CD.

224: Euclid's elements.

In like manner, because LMk the same multiple of CD, that MJV

is of BJE,

therefore the remainder LN is the same multiple of the remainder

CF, that the whole ZM is of the whole CJD : (v. 5.)

but it was shewn that LM is the same multiple of CD, that GK

isof^^;

therefore GK is the same multiple of AE, that ZN is of CF;

that is, GK, iiV are equimultiples of AF, CF.

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