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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And because KO, NP are equimultiples of BE, BF,

therefore, if from KO, NP there be taken KH, NM, which are

likewise equimultiples of BE, DF,

the remainders KO, MP are either equal to BE, DF, or equi-

multiples of them. (v. 6.)

First, let HO, 3IP be equal to BE, DF:

then because AE is to EB, as CF to FD, (hyp.)

and that GK, XiVare equimulti^^les of AE, CF;

therefore GK is to EB, as ZiV to FD : (v. 4. Cor.)

but HO is equal to EB, and MP to FD ;

wherefore GK is to HO, as ZJV to 3IP ;

therefore if GK be greater than HO, ZN is greater than MP ; (v. A.)

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

But let HO, MP be equimultiples of EB, FD.

Then because ^^ is to EB, as CFlo FD, (h}^.)

Â» G K H O L N M P

E B C FD

and that of AE, CFare taken equimultiples GK, ZN;

and of EB, FD, the equimultiples HO, MP ;

if GK be greater than HO, ZN is greater than MP ;

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

which was likewise shewn in the preceding case.

But if GH be greater than KO,

taking KH from both, GK is greater than JTO ; (l. ax. 5.)

wherefore also ZN is greater than 3IP ;

and consequently adding NM to both,

Z3I is greater than NP : (l. ax. 4.)

therefore, if GHhe greater than IÂ£0,

ZM is greater than NP.

In like manner it may be shewn, that if GHhe equal to KO,

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

And in the case in which KO is not greater than KH,

it has been shewn that GHis always greater than KO,

and likewise Z3I greater than NP :

but GH, Z3I are any equimultiples whatever of AB, CD, (constr.)

and KO, NP are any whatever of BE, DF;

therefore, as ^-S is to BE, so is CD to DF. (v. def. 5.)

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

BOOK V. PROP. XIX. 225

PROPOSITION XIX. THEOREM.

If a whole magnitude be to a whole, as a magnitude taken from the first

is to a magnitude taken from the other; the remainder shall be to the

remainder as the ivhole to the whole.

Let the whole ^ J5 be to the whole CD, as AH a, magnitude taken

from ^^ is to CF a magnitude taken from CD.

Then the remainder FB shall be to the remainder FD, as the whole

AÂ£ to the whole CD.

A E B

C F D

Because AB is to CD, as AF to CF:

therefore alternately, BA is to AF, as DC to CF: (v. 16.)

and because if magnitudes taken jointly be proportionals, they are

also proportionals, when taken separately; (v. 17.)

therefore, as BF is to FA, so is DF to FC;

and alternately, as JBF is to DF, so is FA to FC:

but, as AF to CF, so, by the hypothesis, is AF to CD;

therefore also FF the remainder is to the remainder DF, as the whole

AB to the whole CD. (v. 11.)

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

Cor. â€” If the whole be to the whole, as a magnitude taken from

the first is to a magnitude taken from the other; the remainder shall

likewise be to the remainder, as the magnitude taken from the first

to that taken from the other. The demonstration is contained in the

preceding.

PROPOSITION E. THEOREM.

If four magnitudes be proportionals, they are also proportionals by con-

version ; that is, the first is to its excess above the second, as the third to its

excess above the fourth.

Let AB be to BF, as CD to DF.

Then BA shall be to AF, as DC to CF.

F I)

Because AB is to BF, as CD to DF,

therefore by division, AF is to FB, as C'-Fto FD-, (v. 17.)

and by inversion, BF is to FA, as Di^is to CF-, (v. B.)

wherefore, by composition, BA is to AF, as DC is to CF. (v. 18.)

If therefore four, &c. Q.E.D.

l5

226 "EucLm's elements.

PROPOSITION XX. THEOREM.

7/ there be three magnitudes, and other three, tvhich^ taken two and two, have

the same ratio ; then if the first be greater than the third, the fourth shall be

greater than the sixth ; and if equal, equal', and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three, which

taken two and two have the same ratio,

viz. as A is to B, so is D to E;

and as B to C, so is E to F.

If ^ be greater than C, D shall be greater than JP;

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

A B C

D E F

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

and that the greater has to the same magnitude a greater ratio than

the less has to it; (v. 8.)

therefore A has to ^ a greater ratio than C has to B :

but as D is to E, so is ^ to J?; (hyp.)

therefore D has to JG" a greater ratio than Cto ^: (v. 13.)

and because B is to C, as E to F,

by inversion, C is to B, as i^ is to ^ : (v. B.)

and D was shewn to have to E 2i greater ratio th^^ Cto B:

therefore D has to ^ a greater ratio than Fto E: (v. 13. Cor.)

but the magnitude which has a greater ratio than another to the same

magnitude, is the greater of the two; (v. 10.)

therefore Z> is greater than jP.

Secondly, let A be equal to C.

Then J) shall be equal to F.

A B C-

D E F-

Because A and Care equal to one another,

A is to B, as Cisto B: (v. 7.)

but A is to J?, as D to ^; (hyp.)

and Cis to ^, as i^to j&; (hyip.)

wherefore E is to E, as Fto E; (v. 11. and v. B.)

and therefore JD is equal to F (v. 9.)

Next, let A be less than C.

Then E shall be less than F,

A B C

D E F-

For C is greater than A ;

and as was shewn in the first case, Cis to B, as Fto E,

and in like manner, ^ is to ^, as J5^ to D ;

therefore JPis greater than D, by the first case;

that is, jD is less tlian F.

Therefore, if there be three, &c. Q. E. D.

I

BOOK V. PROP. XXI. 227

PROPOSITION XXI. THEOREM.

Jf there he three magnitudes^ and other three^ which have the same ratio

taken two and two, but in a cross order ; then if the first inagnitude he

greater than the third, the fourth, shall be greater than the sixth ; and if

equally equal; and if less, less.

Let A, B, Che three magnitudes, and D, E, jP other three, which

have the same ratio, taken two and two, but in a cross order,

yiz. as A i% Xo B so is E to F,

and as B is to C, so is D to E.

If ^ be greater than C, D shall be greater than F\

and if equal, equal j and if less, less.

A B C-

D E F-

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

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

but as E to F, %oh Ato B; (hyp.)

therefore E has to jP a greater ratio than Cto B: (v. 13.)

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

by inversion, C is to B, as E to I):

and E was shewn to have to i^ a greater ratio than C has to B ;

therefore E has to F a greater ratio than E has to E : (V. 13. Cor.)

but the magnitude to which the same has a greater ratio than it has

to another, is the less of the two : (v. 10.)

therefore F is less than E ;

that is, jD is greater than F.

Secondly, Let A be equal to C;

E shall be equal to F,

A B C-

D E F

Because A and C are equal,

A is to B, as CistoB: (v. 7.)

but A is to B, as E to F-, (hyp.)

and C is to B, as E to E;

wherefore E is to F, as EtoE; fv. 11.)

and therefore E is equal to F. (v. 9.)

Next, let A be less than C:

E shall be less than F.

A B C-

D E F-

I

For C is greater than A ;

and as was shewn, C is to B, as E to D,

and in like manner B is to A, as F to E ;

therefore F is greater than E, by case first j

that is, E is less than F.

Therefore, if there be three, &c. q.e.d.

228 Euclid's elements.

PROPOSITION XXII. THEOREM.

If there he any number of magnitudes ^ and as many others ^ which taken

two and two in order, have the same ratio ; the first shall have to the last of

the first magnitudes, the same ratio which the first has to the last of the

others. N.B. This is usually cited by the words " ex aequali," or " ex

aequo."

First, let there be three magnitudes A, JB, C, and as many others

D, E, F, which taken two and two m order, have the same ratio,

that is, such that A is to JB, as D to JE;

and as Â£ is to C, so is JE to F.

Then A shall be to C, as D to F.

G K M

A B C

D E F-

H L â€” N-

Take of A and D any equimultiples whatever G and H;

and of JB and ^ any equimultiples whatever X and L ;

and of Cand jPany whatever 3/ and N:

then because A is to J?, as JD to F,

and that G, Hare equimultiples of A, I>,

and X, L equimultiples of B, F;

therefore as G is to F, so is ^ to X : (v. 4.)

for the same reason, K is to il/as L to iV:

and because there are three magnitudes G, K, 3f, and other three

JI, L, N, which two and two, have the same ratio ,-

therefore if G be greater than M, H is greater than Nj

and if equal, equal ; and if less, less ; (v. 20.)

but G, Hare any equimultiples whatever of ^, JD,

and M, Nave any equimultiples whatever of C, F; (constr.)

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

Next, let there be four magnitudes A, B, C, JD,

and other four F, F, G, H, which two and two have the same ratio,

viz. as A is to B, so is F to F;

and as B to C, so i^ to 6^ ;

and as C to D, so G to JEC.

Then A shall be to D, as F to IT.

A.B.C.D

E.F.G.H

Because A, B, Care three magnitudes, and F, F, G other three,

which taken two and two, have the same ratio ;

therefore by the foregoing case, A is to C, as ^ to G i

but C is to D, as G is to H;

wherefore again, by the first case A is to D, as F to H:

and so on, whatever be the number of magnitudes.

Therefore, if there be any number, &c. Q. E. d.

BOOK V. PROP. XXIII. 229

PROPOSITION XXIII. THEOREM.

If there be any number of magnitudes, and as many others, which

taken two and two in a cross ordey, have the same ratio ; the first shall have

to the last of the first magnittides the same ratio which the first has to the

last of the others. N.B. This is usually cited by the words *' ex sequali

in proportione perturbata ;" or " ex aequo perturbato."

First, let there be three magnitudes A, B, C, and other three D,

E, F, which taken two and two in a cross order have the same ratio,

that is, such that A is to B, as JE to F;

and as B is to C, so is D to F.

Then A shall be to C, as D to F.

G H L

A B C

D E F

K M N-

Take of A, B, D any equimultiples whatever G, H, K;

and of C, F, F any equimultiples whatever Z, 31, N:

and because G, If are equimultiples of A, B,

and that magnitudes have the same ratio which their equimultiples

have; (v. 15.)

therefore as A Is to B, so is G to IT:

and for the same reason, as F is to F, so is 31 to N:

but as A is to B, so is F to F; (hyp.)

therefore as G is to //, so is 31 to 3^: (v. 11.)

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

and that II, K are equimultiples of B, D, and L, 31 oi C, F;

therefore as His to L, so is K to 3f: (v. 4.)

and it has been shewn that G is to II, as 31 to JV:

therefore, because there are three magnitudes G, II, L, and other

three K, 31, N, which have the same ratio taken two and two in a

cross order;

if G be greater than L, K is greater than Ni

and if equal, equal ; and if less, less : (V. 21.)

but G, K are any equimultiples whatever of ^, 7); (constr.)

and L, N any whatever of C, F;

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

Next, let there be four magnitudes A, B, C, D, and other four F,

F, G, H, which taken two and two in a cross order have the same

ratio,

viz. A to B, as G to H)

Bto C, asjPto G;

and Cto D, as F to F.

Then A shall be to D, as F to H,

A.B.C.D

E.F.G.H

Because A, B, C are three magnitudes, and F, G, H other three,

which taken two and two in a cross order, have the same ratio ;

230 Euclid's elements.

by the jfirst case, A is to C, as J^ to ^;

but C is to D, as JE is to F;

wherefore again, by the first case, A is to D, as F to H ;

and so on, whatever be the number of magnitudes.

Therefore, if there be any number, &c. Q. E. D.

PROPOSITION XXIV. THEOREM.

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

and the fifth to the second the same ratio which the sixth has to the fourth; the

first and fifth together shall have to the second, the same ratio which the third

and sixth together have to the fourth.

Let AB the first have to Cthe second the same ratio which DS

the third has to F the fourth ;

and let BG the fifth have to C the second the same ratio which

FH the sixth has to F the fourth.

Then A G, the first and fifth together, shall have to C the second,

the same ratio which DH, the third and sixth together, has to F the

fourth.

A B G D E H

Because BGh to C, as EH to F;

by inversion, Cis to BG, as i^to FH: (v. B.)

and because, as AB is to C, so is DF to F; (hyp.)

and as C to BG, so is F to FH;

ex Â«quali, AB is to BG, as FF to FH: (v. 22.)

and because these magnitudes are proportionals when taken separately,

they are likewise proportionals when taken jointly; (v. 18.)

therefore as ^6^ is to GB, so is FH to HF:

but as GB to C, so is HF to F: {hjip.)

therefore, ex sequali, as ^ G^ is to C, so is FH to F. (v. 22.)

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

Cor. 1. â€” If the same hypothesis be made as in the proposition, the

excess of the first and fifth shall be to the second, as the excess of the

third and sixth to the fourth. The demonstration of this is the same

with that of the proposition, if division be used instead of composition.

Cor. 2. â€” The proposition holds true of two ranks of magnitudes,

whatever be their number, of which each of the first rank has to the

second magnitude the same ratio that the corresponding one of the

second rank has to a fourth magnitude : as is manifest.

PROPOSITION XXY. THEOREM.

If four magnitudes of the same kind are proportionals, the greatest and

least of them together are greater than the other two together.

Let the four magnitudes AB, CD, F, Fhe proportionals,

viz. AB to CD, as F to F-,

and let AB be the greatest of them, and consequently F the least,

(v. 14. and A.)

BOOK V. PROP. XXV, r. 231

Then AB together with i^ shall be greater than CD together with .Â£'.

A G B c H D

Take A G equal to E, and CII equal to F.

Then because as ^^ is to CD, so is E to F,

and that AG \^ equal to E, and CH equal to F,

therefore AB is to CD, as ^(r to CH: (v. 11, and 7.)

and because AB the whole, is to the whole CD, as -4 6^ is to CII,

likewise the remainder GB is to the remainder II D, as the whole AB

is to the whole CD : (v. 19.)

but AB is greater than CD; (hyp.)

therefore GB is greater than IID ; (v. A.)

and because AG is equal to E, and CII to F;

A G and F together are equal to CH and E together : (l. ax. 2.)

therefore if to the unequal magnitudes GB, HD, of which GB is

the greater, there be added equal magnitudes, viz. to GB the two AG

and F, and CH and E to HD ;

AB and i^ together are greater than CD and E. (1. ax. 4.)

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

PROPOSITION P. THEOREM.

Ratios which are compounded of the same ratios, are the same to one another.

Let A be to B, as D to E-, and B to C, as .E to F.

Then the ratio which is compounded of the ratios of A to B, and B

to C,

which, by the definition of compound ratio, is the ratio of A to C,

shall be the same with the ratio of D to F, which, by the same

definition, is compounded of the ratios of D to E, and E to F.

A.B.C

D.E.F

Because there are three magnitudes A, B, C, and three others Z), E, F,

which, taken two and two, in order, have the same ratio ;

ex sequali, A is to C, as D to F. (v. 22.)

Next, let A be to B, as E to F, and B to Q as Dio E:

A.B.C

D.E.F

therefore, ex aquali in proportione perturhata, (v. 23.)

A is to C, as D to F;

that is, the ratio of A to C, which is compounded of the ratios of

A to B, and B to C, is the same with the ratio of D to F, which is

compounded of the ratios of D to E, and E to F.

And in like manner the proposition may be demonstrated, what-

ever be the number of ratios in either case.

Euclid's elements.

PROPOSITION G. THEOREM.

If several ratios be the same to several ratios, each to each ; the ratio

which is compounded of ratios which are the same to the first ratios, each

to each, shall be the same to the ratio compounded of ratios which are the

same to the other ratios, each to each.

Let A be to J9, as ^ to i?'; and (7 to D, as (? to iT:

and let A be to ^, as jK" to X ; and C to D, as L to M.

Then the ratio of K to 31,

by the definition of compound ratio, is compounded of the ratios

of K to L, and L to 31, which are the same with the ratios of ^ to -B

and C to D.

Again, as JE to F, so let iV be to O ; and as G to //, so let O be to P.

Then the ratio of iV to P is compounded of the ratios of H to O,

and to P, which are the same with the ratios of JE to F, and G to

Hi

and it is to be shewn that the ratio of K to 31, is the seme with

the ratio of iV to P ;

or that K is to 31, as N to P.

A.B.C.D. K.L.xM

E.P.G.H. N.O.P

Because K is to Z, as (A to B, that is, as E to F, that is, as) iV to O ;

and as L to 31, so is (C to D, and so is G to H, and so is) O to P :

ex aequali, K is to 31, as N to P. (v. 22.)

Therefore, if several ratios, &c. Q. E. D.

PROPOSITION H. THEOREM.

If a ratio which is compounded of several ratios be the same to a ratio which

is compounded of several other ratios ; and if one of the first ratios, or the

ratio which is compounded of several of them, be the same to one of the last

ratios, or to the ratio which is compounded of several of them ; then the re-

maining ratio of the first, or, if there he more than one, the ratio compounded

of the remaining ratios, shall he the same to the remaining ratio of the last,

or, if there be more than one, to the ratio compounded of these remaining ratios.

Let the first ratios be those of ^ to B, B to C, Cto D, D to F, and

FtoF;

and let the other ratios be those of G to H, H to K, K to X, and

L to 3I\

also, let the ratio of A to F, which is compounded of the first ratios,

be the same with the ratio of G to 3d, which is compounded of the

other ratios ;

and besides, let the ratio of A to D, which is compounded of the

ratios of A to B, B to C, C to D, be the same with the ratio of G to

K, which is compounded of the ratios of G to H, and Jf to K.

Then the ratio compounded of the remaining first ratios, to wit, of

the ratios of JD to F, and F to F, which compounded ratio is the ratio

BOOK V. PROP. H, K. 2So

of D to F, shall be the same with the ratio of K to 31, which is

pounded of the remaining ratios of K to L, and L to 31 of the

com-

to iHf of the other

ratios.

A.B.C.D.E.F

G.H.K.L.M

Because, hy the h}q)othesis, A is to D, as G to K,

by inversion, jD is to ^, as ^to G; (v. B.)

and as A is to F, so is G to Jf ; (hyp.)

therefore, ex sequali, D is to F, as ^to If. (v. 22.)

If, therefore, a ratio which is, &c. Q. E. d.

PROPOSITION K. THEOREM.

If there be any number of ratios, and any number of other ratios, such, that

the ratio which is compounded of ratios which are the same to the first ratios,

each to each, is the same to the ratio which is compounded of ratios which

are the same, each to each, to the last ratios ; and if one of the first ratios, or the

ratio which is compounded of ratios which are the same to several of the first

ratios, each to each, be the same to one of the last ratios, or to the ratio which

is compounded of ratios which are the same, each to each, to several of the last

ratios; then the remaining ratio of the first, or, if there be more than one,

the ratio which is compounded of ratios which are the same each to each to

the remaining ratios of the first, shall be the same to the remaining ratio of (he

last, or, if there be more than one, to the ratio which is compounded of ratios

which are the same each to each to these remaining ratios.

Let the ratios of Aio B, C to J), F to F, be the first ratios :

and the ratios of G to H, K to L, 31 to N, to P, Q to B, be the

and let A be to JB, as S to T; and C to D, as T to F; and F to F,

as rto X:

therefore, by the definition of compound ratio, the ratio of S to X is

compounded of the ratios of S to 2] T to V, and Fto X, which are

the same to the ratios of ^ to B, C to D, F to F : each to each.

Also, as G to H, so let Y be to Z; and K to L, as Z to a ;

31 to iV, as a to 6 ; O to P, as J to c ; and Q to P, as c to c? :

therefore, by the same definition, the ratio of I^to d is compounded

of the ratios of YtoZ,Z to a, a to h, h to c, and c to d, which are the

same, each to each, to the ratios of G to H, K to L, 31 to N, O to P,

and Q to P :

therefore, by the hypothesis, S is to X, as T'to d.

Also, let the ratio of A to P, that is, the ratio of S to T, which is

one of the first ratios, be the same to the ratio of e to g, which is com-

pounded of the ratios of e to /, and / to g, which, by the hypothesis,

are the same to the ratios of G to JT, and K to Z, two of the other

ratios ;

and let the ratio of ^ to ^ be that which is compounded of the ratios

of h to k, and h to I, which are the same to the remaining first ratios,

viz. of C to D, and E to F;

234

Euclid's elements.

also, let the ratio of m to p, be that which is compounded of the

ratios of m to w, n to o, and o to p, which are the same, each to each,

to the remaining other ratios, viz. of Mto K, O to P, and Q to JR,

Then the ratio of h to / shall be the same to the ratio of m to p; or

h shall be to I, as m to p.

h, k, 1.

A

B

C, D;

E,

F.

s,

T,

V,

X.

G

H;

K,

L;

M, N;

0,

P;

Q,

R.

Y

z,

a,

b,

c,

d.

e,

f, g.

m, n

0,

p.

(hyp-)

Because e is to/, as (G^ to II, that is, as) Yto Z;

and/ is to ff, as ( JT to X, that is, as) Zto a-,

therefore, ex ajquali, e is to ^, as Yto a: (v. 22.)

and by the hypothesis, A is to JB, that is, S to T, as eto g;

wherefore 6' is to T, as Fto a; (v. 11.)

and by inversion, Tis to aS, as a to Y: (v. B.)

but S is to X, as Fto D; (hyp.)

therefore, ex sequali, T is to X, as a to d :

also, because h is to k, as (Cto D, that is, as) Tto V;

and kisto I as {JE to F, that is, as) 7^ to X ;

therefore, ex aequali, A is to /, as Tto X:

in like manner, it may be demonstrated, that m is to j^, as a to c?;

and it has been shewn, tliat Tis to X, as a to d;

therefore h is to I, as ni to p. (v. 11.) Q. e.d.

The propositions G and X are usually, for the sake of brevity, ex-

pressed in the same terms with propositions F and // : and therefore

it was proper to shew the true meaning of them when they are so

expressed ; especially since they are very frequently made use of by

geometers.

NOTES TO BOOK V.

In the first four Books of the Elements are considered, only the

absolute equality and inequality of Geometrical magnitudes. The Fifth

Book contains an exposition of the principles whereby a more definite

comparison may be instituted of the relation of magnitudes, besides their

simple equality or inequality.

The doctrine of Proportion is one of the most important in the whole

course of mathematical truths, and it appears probable that if the subject

were read simultaneously in the Algebraical and Geometrical form, the

investigations of the properties, under both aspects, would mutually

assist each other, and both become equally comprehensible ; also their

distinct characters would be more easily perceived.

Def. T, II. In the first Four Books the word part is used in the same

sense as we find it in the ninth axiom, *' The whole is greater than its

part :" where the word part means any portion whatever of any whole

magnitude : but in the Fifth Book, the word part is restricted to mean

that portion of magnitude which is contained an exact number of times

in the whole. For instance, if any straight line be taken two, three, four,

or any number of times another straight line, by Euc. i. 3 ; the less line

is called a part, or rather a submultiple of the greater line ; and the greater,

a multiple of the less line. The multiple is composed of a repetition of

the same magnitude, and these definitions suppose that the multiple may

be divided into its parts, any one of which is a measure of the multiple.

And it is also obvious that when there are two magnitudes, one of which

is a multiple of the other, the two magnitudes must be of the same kind,

that is, they must be two lines, two angles, two surfaces, or two solids :

thus, a triangle is doubled, trebled, &c., by doubling, trebling, &c. the

base, and completing the figure. The same may be said of a parallelo-

gram. Angles, arcs, and sectors of equal circles may be doubled, trebled,

or any multiples found by Prop, xxvi â€” xxix. Book tii.

Two magnitudes are said to be commensurable when a third magnitude

of the same kind can be found which will measure both of them ; and

this third magnitude is called their common measure : and when it is the

therefore, if from KO, NP there be taken KH, NM, which are

likewise equimultiples of BE, DF,

the remainders KO, MP are either equal to BE, DF, or equi-

multiples of them. (v. 6.)

First, let HO, 3IP be equal to BE, DF:

then because AE is to EB, as CF to FD, (hyp.)

and that GK, XiVare equimulti^^les of AE, CF;

therefore GK is to EB, as ZiV to FD : (v. 4. Cor.)

but HO is equal to EB, and MP to FD ;

wherefore GK is to HO, as ZJV to 3IP ;

therefore if GK be greater than HO, ZN is greater than MP ; (v. A.)

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

But let HO, MP be equimultiples of EB, FD.

Then because ^^ is to EB, as CFlo FD, (h}^.)

Â» G K H O L N M P

E B C FD

and that of AE, CFare taken equimultiples GK, ZN;

and of EB, FD, the equimultiples HO, MP ;

if GK be greater than HO, ZN is greater than MP ;

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

which was likewise shewn in the preceding case.

But if GH be greater than KO,

taking KH from both, GK is greater than JTO ; (l. ax. 5.)

wherefore also ZN is greater than 3IP ;

and consequently adding NM to both,

Z3I is greater than NP : (l. ax. 4.)

therefore, if GHhe greater than IÂ£0,

ZM is greater than NP.

In like manner it may be shewn, that if GHhe equal to KO,

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

And in the case in which KO is not greater than KH,

it has been shewn that GHis always greater than KO,

and likewise Z3I greater than NP :

but GH, Z3I are any equimultiples whatever of AB, CD, (constr.)

and KO, NP are any whatever of BE, DF;

therefore, as ^-S is to BE, so is CD to DF. (v. def. 5.)

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

BOOK V. PROP. XIX. 225

PROPOSITION XIX. THEOREM.

If a whole magnitude be to a whole, as a magnitude taken from the first

is to a magnitude taken from the other; the remainder shall be to the

remainder as the ivhole to the whole.

Let the whole ^ J5 be to the whole CD, as AH a, magnitude taken

from ^^ is to CF a magnitude taken from CD.

Then the remainder FB shall be to the remainder FD, as the whole

AÂ£ to the whole CD.

A E B

C F D

Because AB is to CD, as AF to CF:

therefore alternately, BA is to AF, as DC to CF: (v. 16.)

and because if magnitudes taken jointly be proportionals, they are

also proportionals, when taken separately; (v. 17.)

therefore, as BF is to FA, so is DF to FC;

and alternately, as JBF is to DF, so is FA to FC:

but, as AF to CF, so, by the hypothesis, is AF to CD;

therefore also FF the remainder is to the remainder DF, as the whole

AB to the whole CD. (v. 11.)

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

Cor. â€” If the whole be to the whole, as a magnitude taken from

the first is to a magnitude taken from the other; the remainder shall

likewise be to the remainder, as the magnitude taken from the first

to that taken from the other. The demonstration is contained in the

preceding.

PROPOSITION E. THEOREM.

If four magnitudes be proportionals, they are also proportionals by con-

version ; that is, the first is to its excess above the second, as the third to its

excess above the fourth.

Let AB be to BF, as CD to DF.

Then BA shall be to AF, as DC to CF.

F I)

Because AB is to BF, as CD to DF,

therefore by division, AF is to FB, as C'-Fto FD-, (v. 17.)

and by inversion, BF is to FA, as Di^is to CF-, (v. B.)

wherefore, by composition, BA is to AF, as DC is to CF. (v. 18.)

If therefore four, &c. Q.E.D.

l5

226 "EucLm's elements.

PROPOSITION XX. THEOREM.

7/ there be three magnitudes, and other three, tvhich^ taken two and two, have

the same ratio ; then if the first be greater than the third, the fourth shall be

greater than the sixth ; and if equal, equal', and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three, which

taken two and two have the same ratio,

viz. as A is to B, so is D to E;

and as B to C, so is E to F.

If ^ be greater than C, D shall be greater than JP;

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

A B C

D E F

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

and that the greater has to the same magnitude a greater ratio than

the less has to it; (v. 8.)

therefore A has to ^ a greater ratio than C has to B :

but as D is to E, so is ^ to J?; (hyp.)

therefore D has to JG" a greater ratio than Cto ^: (v. 13.)

and because B is to C, as E to F,

by inversion, C is to B, as i^ is to ^ : (v. B.)

and D was shewn to have to E 2i greater ratio th^^ Cto B:

therefore D has to ^ a greater ratio than Fto E: (v. 13. Cor.)

but the magnitude which has a greater ratio than another to the same

magnitude, is the greater of the two; (v. 10.)

therefore Z> is greater than jP.

Secondly, let A be equal to C.

Then J) shall be equal to F.

A B C-

D E F-

Because A and Care equal to one another,

A is to B, as Cisto B: (v. 7.)

but A is to J?, as D to ^; (hyp.)

and Cis to ^, as i^to j&; (hyip.)

wherefore E is to E, as Fto E; (v. 11. and v. B.)

and therefore JD is equal to F (v. 9.)

Next, let A be less than C.

Then E shall be less than F,

A B C

D E F-

For C is greater than A ;

and as was shewn in the first case, Cis to B, as Fto E,

and in like manner, ^ is to ^, as J5^ to D ;

therefore JPis greater than D, by the first case;

that is, jD is less tlian F.

Therefore, if there be three, &c. Q. E. D.

I

BOOK V. PROP. XXI. 227

PROPOSITION XXI. THEOREM.

Jf there he three magnitudes^ and other three^ which have the same ratio

taken two and two, but in a cross order ; then if the first inagnitude he

greater than the third, the fourth, shall be greater than the sixth ; and if

equally equal; and if less, less.

Let A, B, Che three magnitudes, and D, E, jP other three, which

have the same ratio, taken two and two, but in a cross order,

yiz. as A i% Xo B so is E to F,

and as B is to C, so is D to E.

If ^ be greater than C, D shall be greater than F\

and if equal, equal j and if less, less.

A B C-

D E F-

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

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

but as E to F, %oh Ato B; (hyp.)

therefore E has to jP a greater ratio than Cto B: (v. 13.)

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

by inversion, C is to B, as E to I):

and E was shewn to have to i^ a greater ratio than C has to B ;

therefore E has to F a greater ratio than E has to E : (V. 13. Cor.)

but the magnitude to which the same has a greater ratio than it has

to another, is the less of the two : (v. 10.)

therefore F is less than E ;

that is, jD is greater than F.

Secondly, Let A be equal to C;

E shall be equal to F,

A B C-

D E F

Because A and C are equal,

A is to B, as CistoB: (v. 7.)

but A is to B, as E to F-, (hyp.)

and C is to B, as E to E;

wherefore E is to F, as EtoE; fv. 11.)

and therefore E is equal to F. (v. 9.)

Next, let A be less than C:

E shall be less than F.

A B C-

D E F-

I

For C is greater than A ;

and as was shewn, C is to B, as E to D,

and in like manner B is to A, as F to E ;

therefore F is greater than E, by case first j

that is, E is less than F.

Therefore, if there be three, &c. q.e.d.

228 Euclid's elements.

PROPOSITION XXII. THEOREM.

If there he any number of magnitudes ^ and as many others ^ which taken

two and two in order, have the same ratio ; the first shall have to the last of

the first magnitudes, the same ratio which the first has to the last of the

others. N.B. This is usually cited by the words " ex aequali," or " ex

aequo."

First, let there be three magnitudes A, JB, C, and as many others

D, E, F, which taken two and two m order, have the same ratio,

that is, such that A is to JB, as D to JE;

and as Â£ is to C, so is JE to F.

Then A shall be to C, as D to F.

G K M

A B C

D E F-

H L â€” N-

Take of A and D any equimultiples whatever G and H;

and of JB and ^ any equimultiples whatever X and L ;

and of Cand jPany whatever 3/ and N:

then because A is to J?, as JD to F,

and that G, Hare equimultiples of A, I>,

and X, L equimultiples of B, F;

therefore as G is to F, so is ^ to X : (v. 4.)

for the same reason, K is to il/as L to iV:

and because there are three magnitudes G, K, 3f, and other three

JI, L, N, which two and two, have the same ratio ,-

therefore if G be greater than M, H is greater than Nj

and if equal, equal ; and if less, less ; (v. 20.)

but G, Hare any equimultiples whatever of ^, JD,

and M, Nave any equimultiples whatever of C, F; (constr.)

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

Next, let there be four magnitudes A, B, C, JD,

and other four F, F, G, H, which two and two have the same ratio,

viz. as A is to B, so is F to F;

and as B to C, so i^ to 6^ ;

and as C to D, so G to JEC.

Then A shall be to D, as F to IT.

A.B.C.D

E.F.G.H

Because A, B, Care three magnitudes, and F, F, G other three,

which taken two and two, have the same ratio ;

therefore by the foregoing case, A is to C, as ^ to G i

but C is to D, as G is to H;

wherefore again, by the first case A is to D, as F to H:

and so on, whatever be the number of magnitudes.

Therefore, if there be any number, &c. Q. E. d.

BOOK V. PROP. XXIII. 229

PROPOSITION XXIII. THEOREM.

If there be any number of magnitudes, and as many others, which

taken two and two in a cross ordey, have the same ratio ; the first shall have

to the last of the first magnittides the same ratio which the first has to the

last of the others. N.B. This is usually cited by the words *' ex sequali

in proportione perturbata ;" or " ex aequo perturbato."

First, let there be three magnitudes A, B, C, and other three D,

E, F, which taken two and two in a cross order have the same ratio,

that is, such that A is to B, as JE to F;

and as B is to C, so is D to F.

Then A shall be to C, as D to F.

G H L

A B C

D E F

K M N-

Take of A, B, D any equimultiples whatever G, H, K;

and of C, F, F any equimultiples whatever Z, 31, N:

and because G, If are equimultiples of A, B,

and that magnitudes have the same ratio which their equimultiples

have; (v. 15.)

therefore as A Is to B, so is G to IT:

and for the same reason, as F is to F, so is 31 to N:

but as A is to B, so is F to F; (hyp.)

therefore as G is to //, so is 31 to 3^: (v. 11.)

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

and that II, K are equimultiples of B, D, and L, 31 oi C, F;

therefore as His to L, so is K to 3f: (v. 4.)

and it has been shewn that G is to II, as 31 to JV:

therefore, because there are three magnitudes G, II, L, and other

three K, 31, N, which have the same ratio taken two and two in a

cross order;

if G be greater than L, K is greater than Ni

and if equal, equal ; and if less, less : (V. 21.)

but G, K are any equimultiples whatever of ^, 7); (constr.)

and L, N any whatever of C, F;

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

Next, let there be four magnitudes A, B, C, D, and other four F,

F, G, H, which taken two and two in a cross order have the same

ratio,

viz. A to B, as G to H)

Bto C, asjPto G;

and Cto D, as F to F.

Then A shall be to D, as F to H,

A.B.C.D

E.F.G.H

Because A, B, C are three magnitudes, and F, G, H other three,

which taken two and two in a cross order, have the same ratio ;

230 Euclid's elements.

by the jfirst case, A is to C, as J^ to ^;

but C is to D, as JE is to F;

wherefore again, by the first case, A is to D, as F to H ;

and so on, whatever be the number of magnitudes.

Therefore, if there be any number, &c. Q. E. D.

PROPOSITION XXIV. THEOREM.

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

and the fifth to the second the same ratio which the sixth has to the fourth; the

first and fifth together shall have to the second, the same ratio which the third

and sixth together have to the fourth.

Let AB the first have to Cthe second the same ratio which DS

the third has to F the fourth ;

and let BG the fifth have to C the second the same ratio which

FH the sixth has to F the fourth.

Then A G, the first and fifth together, shall have to C the second,

the same ratio which DH, the third and sixth together, has to F the

fourth.

A B G D E H

Because BGh to C, as EH to F;

by inversion, Cis to BG, as i^to FH: (v. B.)

and because, as AB is to C, so is DF to F; (hyp.)

and as C to BG, so is F to FH;

ex Â«quali, AB is to BG, as FF to FH: (v. 22.)

and because these magnitudes are proportionals when taken separately,

they are likewise proportionals when taken jointly; (v. 18.)

therefore as ^6^ is to GB, so is FH to HF:

but as GB to C, so is HF to F: {hjip.)

therefore, ex sequali, as ^ G^ is to C, so is FH to F. (v. 22.)

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

Cor. 1. â€” If the same hypothesis be made as in the proposition, the

excess of the first and fifth shall be to the second, as the excess of the

third and sixth to the fourth. The demonstration of this is the same

with that of the proposition, if division be used instead of composition.

Cor. 2. â€” The proposition holds true of two ranks of magnitudes,

whatever be their number, of which each of the first rank has to the

second magnitude the same ratio that the corresponding one of the

second rank has to a fourth magnitude : as is manifest.

PROPOSITION XXY. THEOREM.

If four magnitudes of the same kind are proportionals, the greatest and

least of them together are greater than the other two together.

Let the four magnitudes AB, CD, F, Fhe proportionals,

viz. AB to CD, as F to F-,

and let AB be the greatest of them, and consequently F the least,

(v. 14. and A.)

BOOK V. PROP. XXV, r. 231

Then AB together with i^ shall be greater than CD together with .Â£'.

A G B c H D

Take A G equal to E, and CII equal to F.

Then because as ^^ is to CD, so is E to F,

and that AG \^ equal to E, and CH equal to F,

therefore AB is to CD, as ^(r to CH: (v. 11, and 7.)

and because AB the whole, is to the whole CD, as -4 6^ is to CII,

likewise the remainder GB is to the remainder II D, as the whole AB

is to the whole CD : (v. 19.)

but AB is greater than CD; (hyp.)

therefore GB is greater than IID ; (v. A.)

and because AG is equal to E, and CII to F;

A G and F together are equal to CH and E together : (l. ax. 2.)

therefore if to the unequal magnitudes GB, HD, of which GB is

the greater, there be added equal magnitudes, viz. to GB the two AG

and F, and CH and E to HD ;

AB and i^ together are greater than CD and E. (1. ax. 4.)

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

PROPOSITION P. THEOREM.

Ratios which are compounded of the same ratios, are the same to one another.

Let A be to B, as D to E-, and B to C, as .E to F.

Then the ratio which is compounded of the ratios of A to B, and B

to C,

which, by the definition of compound ratio, is the ratio of A to C,

shall be the same with the ratio of D to F, which, by the same

definition, is compounded of the ratios of D to E, and E to F.

A.B.C

D.E.F

Because there are three magnitudes A, B, C, and three others Z), E, F,

which, taken two and two, in order, have the same ratio ;

ex sequali, A is to C, as D to F. (v. 22.)

Next, let A be to B, as E to F, and B to Q as Dio E:

A.B.C

D.E.F

therefore, ex aquali in proportione perturhata, (v. 23.)

A is to C, as D to F;

that is, the ratio of A to C, which is compounded of the ratios of

A to B, and B to C, is the same with the ratio of D to F, which is

compounded of the ratios of D to E, and E to F.

And in like manner the proposition may be demonstrated, what-

ever be the number of ratios in either case.

Euclid's elements.

PROPOSITION G. THEOREM.

If several ratios be the same to several ratios, each to each ; the ratio

which is compounded of ratios which are the same to the first ratios, each

to each, shall be the same to the ratio compounded of ratios which are the

same to the other ratios, each to each.

Let A be to J9, as ^ to i?'; and (7 to D, as (? to iT:

and let A be to ^, as jK" to X ; and C to D, as L to M.

Then the ratio of K to 31,

by the definition of compound ratio, is compounded of the ratios

of K to L, and L to 31, which are the same with the ratios of ^ to -B

and C to D.

Again, as JE to F, so let iV be to O ; and as G to //, so let O be to P.

Then the ratio of iV to P is compounded of the ratios of H to O,

and to P, which are the same with the ratios of JE to F, and G to

Hi

and it is to be shewn that the ratio of K to 31, is the seme with

the ratio of iV to P ;

or that K is to 31, as N to P.

A.B.C.D. K.L.xM

E.P.G.H. N.O.P

Because K is to Z, as (A to B, that is, as E to F, that is, as) iV to O ;

and as L to 31, so is (C to D, and so is G to H, and so is) O to P :

ex aequali, K is to 31, as N to P. (v. 22.)

Therefore, if several ratios, &c. Q. E. D.

PROPOSITION H. THEOREM.

If a ratio which is compounded of several ratios be the same to a ratio which

is compounded of several other ratios ; and if one of the first ratios, or the

ratio which is compounded of several of them, be the same to one of the last

ratios, or to the ratio which is compounded of several of them ; then the re-

maining ratio of the first, or, if there he more than one, the ratio compounded

of the remaining ratios, shall he the same to the remaining ratio of the last,

or, if there be more than one, to the ratio compounded of these remaining ratios.

Let the first ratios be those of ^ to B, B to C, Cto D, D to F, and

FtoF;

and let the other ratios be those of G to H, H to K, K to X, and

L to 3I\

also, let the ratio of A to F, which is compounded of the first ratios,

be the same with the ratio of G to 3d, which is compounded of the

other ratios ;

and besides, let the ratio of A to D, which is compounded of the

ratios of A to B, B to C, C to D, be the same with the ratio of G to

K, which is compounded of the ratios of G to H, and Jf to K.

Then the ratio compounded of the remaining first ratios, to wit, of

the ratios of JD to F, and F to F, which compounded ratio is the ratio

BOOK V. PROP. H, K. 2So

of D to F, shall be the same with the ratio of K to 31, which is

pounded of the remaining ratios of K to L, and L to 31 of the

com-

to iHf of the other

ratios.

A.B.C.D.E.F

G.H.K.L.M

Because, hy the h}q)othesis, A is to D, as G to K,

by inversion, jD is to ^, as ^to G; (v. B.)

and as A is to F, so is G to Jf ; (hyp.)

therefore, ex sequali, D is to F, as ^to If. (v. 22.)

If, therefore, a ratio which is, &c. Q. E. d.

PROPOSITION K. THEOREM.

If there be any number of ratios, and any number of other ratios, such, that

the ratio which is compounded of ratios which are the same to the first ratios,

each to each, is the same to the ratio which is compounded of ratios which

are the same, each to each, to the last ratios ; and if one of the first ratios, or the

ratio which is compounded of ratios which are the same to several of the first

ratios, each to each, be the same to one of the last ratios, or to the ratio which

is compounded of ratios which are the same, each to each, to several of the last

ratios; then the remaining ratio of the first, or, if there be more than one,

the ratio which is compounded of ratios which are the same each to each to

the remaining ratios of the first, shall be the same to the remaining ratio of (he

last, or, if there be more than one, to the ratio which is compounded of ratios

which are the same each to each to these remaining ratios.

Let the ratios of Aio B, C to J), F to F, be the first ratios :

and the ratios of G to H, K to L, 31 to N, to P, Q to B, be the

and let A be to JB, as S to T; and C to D, as T to F; and F to F,

as rto X:

therefore, by the definition of compound ratio, the ratio of S to X is

compounded of the ratios of S to 2] T to V, and Fto X, which are

the same to the ratios of ^ to B, C to D, F to F : each to each.

Also, as G to H, so let Y be to Z; and K to L, as Z to a ;

31 to iV, as a to 6 ; O to P, as J to c ; and Q to P, as c to c? :

therefore, by the same definition, the ratio of I^to d is compounded

of the ratios of YtoZ,Z to a, a to h, h to c, and c to d, which are the

same, each to each, to the ratios of G to H, K to L, 31 to N, O to P,

and Q to P :

therefore, by the hypothesis, S is to X, as T'to d.

Also, let the ratio of A to P, that is, the ratio of S to T, which is

one of the first ratios, be the same to the ratio of e to g, which is com-

pounded of the ratios of e to /, and / to g, which, by the hypothesis,

are the same to the ratios of G to JT, and K to Z, two of the other

ratios ;

and let the ratio of ^ to ^ be that which is compounded of the ratios

of h to k, and h to I, which are the same to the remaining first ratios,

viz. of C to D, and E to F;

234

Euclid's elements.

also, let the ratio of m to p, be that which is compounded of the

ratios of m to w, n to o, and o to p, which are the same, each to each,

to the remaining other ratios, viz. of Mto K, O to P, and Q to JR,

Then the ratio of h to / shall be the same to the ratio of m to p; or

h shall be to I, as m to p.

h, k, 1.

A

B

C, D;

E,

F.

s,

T,

V,

X.

G

H;

K,

L;

M, N;

0,

P;

Q,

R.

Y

z,

a,

b,

c,

d.

e,

f, g.

m, n

0,

p.

(hyp-)

Because e is to/, as (G^ to II, that is, as) Yto Z;

and/ is to ff, as ( JT to X, that is, as) Zto a-,

therefore, ex ajquali, e is to ^, as Yto a: (v. 22.)

and by the hypothesis, A is to JB, that is, S to T, as eto g;

wherefore 6' is to T, as Fto a; (v. 11.)

and by inversion, Tis to aS, as a to Y: (v. B.)

but S is to X, as Fto D; (hyp.)

therefore, ex sequali, T is to X, as a to d :

also, because h is to k, as (Cto D, that is, as) Tto V;

and kisto I as {JE to F, that is, as) 7^ to X ;

therefore, ex aequali, A is to /, as Tto X:

in like manner, it may be demonstrated, that m is to j^, as a to c?;

and it has been shewn, tliat Tis to X, as a to d;

therefore h is to I, as ni to p. (v. 11.) Q. e.d.

The propositions G and X are usually, for the sake of brevity, ex-

pressed in the same terms with propositions F and // : and therefore

it was proper to shew the true meaning of them when they are so

expressed ; especially since they are very frequently made use of by

geometers.

NOTES TO BOOK V.

In the first four Books of the Elements are considered, only the

absolute equality and inequality of Geometrical magnitudes. The Fifth

Book contains an exposition of the principles whereby a more definite

comparison may be instituted of the relation of magnitudes, besides their

simple equality or inequality.

The doctrine of Proportion is one of the most important in the whole

course of mathematical truths, and it appears probable that if the subject

were read simultaneously in the Algebraical and Geometrical form, the

investigations of the properties, under both aspects, would mutually

assist each other, and both become equally comprehensible ; also their

distinct characters would be more easily perceived.

Def. T, II. In the first Four Books the word part is used in the same

sense as we find it in the ninth axiom, *' The whole is greater than its

part :" where the word part means any portion whatever of any whole

magnitude : but in the Fifth Book, the word part is restricted to mean

that portion of magnitude which is contained an exact number of times

in the whole. For instance, if any straight line be taken two, three, four,

or any number of times another straight line, by Euc. i. 3 ; the less line

is called a part, or rather a submultiple of the greater line ; and the greater,

a multiple of the less line. The multiple is composed of a repetition of

the same magnitude, and these definitions suppose that the multiple may

be divided into its parts, any one of which is a measure of the multiple.

And it is also obvious that when there are two magnitudes, one of which

is a multiple of the other, the two magnitudes must be of the same kind,

that is, they must be two lines, two angles, two surfaces, or two solids :

thus, a triangle is doubled, trebled, &c., by doubling, trebling, &c. the

base, and completing the figure. The same may be said of a parallelo-

gram. Angles, arcs, and sectors of equal circles may be doubled, trebled,

or any multiples found by Prop, xxvi â€” xxix. Book tii.

Two magnitudes are said to be commensurable when a third magnitude

of the same kind can be found which will measure both of them ; and

this third magnitude is called their common measure : and when it is the

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