A leſs magnitude is ſaid to be an aliquot part or ſubmultiple of a greater magnitude, when the leſs meaſures the greater; that is, when the leſs is contained a certain number of times exactly in the greater.

II.

A greater magnitude is ſaid to be a multiple of a leſs, when the greater is meaſured by the leſs; that is, when the greater contains the leſs a certain number of times exactly.

III.

Ratio is the relation which one quantity bears to another of the ſame kind, with reſpect to magnitude.

IV.

Magnitudes are ſaid to have a ratio to one another, when they are of the ſame kind; and the one which is not the greater can be multiplied ſo as to exceed the other.

The other definitions will be given throughout the book where their aid is firſt required.

Axioms.

I.

Equimultiples or equiſubmultiples of the ſame, or of equal magnitudes, are equal.

If A = B, thentwice A = twice B, that is,2 A = 2 B;3 A = 3 B;4 A = 4 B&c. &c.and
1/2
of A =1/2
of B;
1/3
of A =1/3
of B;
&c. &c.

II.

A multiple of a greater magnitude is greater than the ſame multiple of a leſs.

Let A > B, then2 A > 2 B;3 A > 3 B;4 A > 4 B;&c. &c.

III.

That magnitude, of which a multiple is greater than the ſame multiple of another, is greater than the other.

Let 2 A > 2 B, thenA > B;or, let 3 A > 3 B, thenA > Bor, let m A >m B, thenA > B.&c. &c.

Proposition I. Theorem.

If any number of magnitudes be equimultiples of as many others, each of each: what multiple soever any one of the firſt is of its part, the ſame multiple ſhall of the firſt magnitudes taken together be of all the others taken together.

Let be the ſame multiple of , that is of . that is of .

Then is evident that
}
is the ſame multiple of
{ which that is of ; becauſe there are as many magnitudes
in
{}={ as there are in =.

The ſame demonſtration holds in any number of magnitudes, which has here been applied to three.

∴ If any number of magnitudes, &c.

Proposition II. Theorem.

If the firſt magnitude be the ſame multiple of the ſecond that the third is of the fourth, and the fifth the ſame multiple of the ſecond that the ſixth is of the fourth, then ſhall the firſt, together with the fifth, be the ſame multiple of the ſecond that the third, together with the ſixth, is of the fourth.

Let , the firſt, be the ſame multiple of , the ſecond, that , the third, is of , the fourth; and let , the fifth, be the ſame multiple of , the ſecond, that , the ſixth, is of , the fourth.

Then it is evident, that
{},
the firſt and fifth together, is the ſame multiple of , the ſecond, that
{},
the third and ſixth together, is of the ſame multiple of , the fourth; becauſe there are as many magnitudes in
{}= as there are in
{}=.

∴ If the firſt magnitude, &c.

Proposition III. Theorem.

If the firſt four magnitudes be the ſame multiple of the ſecond that the third is of the fourth, and if any equimultiples whatever of the the firſt and third be taken, thoſe ſhall be equimultiples; one of the ſecond, and the other of the fourth.

Let
{}
be the ſame multiple of which
{}
is of ; take
{}
the ſame multiple of
{, which
{}
is of
{.

Then it is evident,
that
{}
is the ſame multiple of which
{}
is of ; becauſe
{}
contains
{}
contains as many times as
}
contains
{}
contains .

The ſame reaſoning is applicable in all caſes.

∴ If the firſt four, &c.

Definition V.

Four magnitudes ,,,, are ſaid to be proportionals when every equimultiple of the firſt and third be taken, and every equimultiple of the ſecond and fourth, as,

of the firſt

&c.

of the ſecond

&c.

of the third

&c.

of the fourth

&c.

Then taking every pair of equimultiples of the firſt and third, and every pair of equimultiples of the ſecond and fourth,

If
{>, = or <>, = or <>, = or <>, = or <>, = or <

then will
{>, = or <>, = or <>, = or <>, = or <>, = or <

That is, if twice the firſt be greater, equal, or leſs than twice the ſecond, twice the third will be greater, equal, or leſs than twice the fourth; or, if twice the firſt be greater, equal, or leſs than three times the ſecond, twice the third will be greater, equal, or leſs than three times the fourth, and so on, as above expreſſed.

If
{>, = or <>, = or <>, = or <>, = or <>, = or <

then will
{>, = or <>, = or <>, = or <>, = or <>, = or <

In other terms, if three times the firſt be greater, equal, or leſs than twice the ſecond, three times the third will be greater, equal, or leſs than twice the fourth; or, if three times the firſt be greater, equal, or leſs than three times the ſecond, then will three times the third be greater, equal, or leſs than three times the fourth; or if three times the firſt be greater, equal, or leſs than four times the ſecond, then will three times the third be greater, equal, or leſs than four times the fourth, and so on. Again,

If
{>, = or <>, = or <>, = or <>, = or <>, = or <

then will
{>, = or <>, = or <>, = or <>, = or <>, = or <

And so on, with any other equimultiples of the four magnitudes, taken in the ſame manner.

Euclid expreſſes this definition as follows:—

The firſt of four magnitudes is ſaid to have the ſame ratio to the ſecond, which the third has to the fourth, when any equimultiples whatſoever of the firſt and third being taken, and any equimultiples whatſoever of the ſecond and fourth; if the multiple of the firſt be leſs than that of the second, the multiple of the third is alſo leſs than that of the fourth; or, if the multiple of the firſt be equal to that of the ſecond, the multiple of the third is alſo equal to that of the fourth; or, if the multiple of the firſt be greater than that of the ſecond, the multiple of the third is alſo greater than that of the fourth.

In future we ſhall expreſs this definition generally, thus:

If M>, =<m,when M>, =<m,

Then we infer that , the firſt, has the ſame ratio to , the ſecond, which , the third, has to the fourth: expreſſed in the ſucceeding demonſtrations thus:

::::;or thus, :=:;or thus,
/=/: and is read,

“as is to , so is to .”

And if :::: we ſhall infer if
M>, = or <m, then will
M>, = or <m.

That is, if the firſt be to the second, as the third is to the fourth; then if M times the firſt be greater than, equal to, or leſs than m times the ſecond, then ſhall M times the third be greater than, equal to, or leſs than m times the fourth, in which M and m are not to be conſidered particular multiples, but every pair of multiples whatever; nor are ſuch marks as ,,, &c. to be conſidered any more than repreſentatives of geometrical magnitudes.

The ſtudent ſhould thoroughly underſtand this definition before proceeding further.

Proposition IV. Theorem.

If the firſt of four magnitudes have the ſame ratio to the ſecond, which the third has to the fourth, then any equimultiples whatever of the firſt and third shall have the ſame ratio to any equimultiples of the ſecond and fourth; viz., the equimultiple of the firſt ſhall have the ſame ratio to that of the ſecond, which the equimultiple of the third has to that of the fourth.

Let ::::, then 3 : 2 :: 3 : 2 , every equimultiple of 3 and 3 are equimultiples of and , and every equimultiple of 2 and 2 , are equimultiples of and (B. 5. pr. 3.)

That is, M times 3 and M times 3 are equimultiples of and , and m times 2 and m 2 are equimultiples of 2 and 2 ; but :::: (hyp); ∴ if M 3 >, = or <m 2 , then M 3 >, = or <m 2 (def. 5.) and therefore 3 : 2 :: 3 : 2 (def. 5.)

The ſame reaſoning holds good if any other equimultiple of the firſt and third be taken, any other equimultiple of the ſecond and fourth.

∴ If the firſt four magnitudes, &c.

Proposition V. Theorem.

If one magnitude be the ſame multiple of another, which a magnitude taken from the firſt is of a magnitude taken from the other, the remainder ſhall be the ſame multiple of the remainder, that the whole is of the whole.

Let
=M′
and =M′ ,
∴
minus =M′
minus M′ ,
=M′
(
minus ),
and ∴=M′ .

∴ If one magnitude, &c.

Proposition VI. Theorem.

If two magnitudes be equimultiples of two others, and if equimultiples of theſe be taken from the firſt two, the remainders are either equal to theſe others, or equimultiples of them.

Let
=M′ ; and =M′ ;
then
minus m′ =

M′ minus m′= (M′ minus m′) ,
and minus m′=M′ minus m′= (M′ minus m′) .

Hence, (M′ minus m′) and (M′ minus m′) are equimultiples of and , and equal to and , when M′ minus m′= 1.

∴ If two magnitudes be equimultiples, &c.

Proposition A. Theorem.

If the firſt of the four magnitudes has the ſame ratio to the ſecond which the third has to the fourth, then if the firſt be greater than the ſecond, the third is alſo greater than the fourth; and if equal, equal; if leſs, leſs.

Let ::::; therefore, by the fifth definition, if >, then will >; but if >, then > and >, and ∴>.

Similarly, if =, or <, then will =, or <.

∴ If the firſt of four, &c.

Definition XIV.

Geometricians make uſe of the technical term “Invertendo,” by inverſion, when there are four proportionals, and it is inferred, that the ſecond is to the firſt as the fourth to the third.

Let A : B :: C : D, then, by “invertendo” it is inferred B : A :: D : C.

Proposition B. Theorem.

If four magnitudes are proportionals, they are proportionals alſo when taken inverſely.

Let M<m, that is, m>M, ∴M<m, or, m>M; ∴ if m>M, then will m>M.

In the ſame manner it may be ſhown,

that if m= or <M, then will m=, or <M; and therefore, by the fifth definition, we infer
that :::.

∴ If four magnitudes, &c.

Proposition C. Theorem.

If the firſt be the ſame multiple of the ſecond, or the ſame part of it, that the third is of the fourth; the firſt is to the ſecond, as the third is to the fourth.

Let , be the firſt, the ſame multiple of , the ſecond,
that , the third, is of , the fourth.

Then :::: take M,m,M,m; becauſe is the ſame multiple of that is of (according to the hypotheſis);
and M is taken the ſame multiple of that M is of , ∴ (according to the third proposition),
M is the ſame multiple of that M is of .

Therefore, if M be of a greater multiple than m is, then M is a greater multiple of than m is; that is, if M be greater than m, then M will be greater than m; in the ſame manner it can be ſhewn, if M be equal m, then
M will be equal m.

And, generally, if M>, = or <m then M will be >, = or <m; ∴ by the fifth definition,
::::.

Next, let be the ſame part of that is of .

In this caſe alſo ::::.

For, becauſe
is the ſame part of that is of , therefore is the ſame multiple of that is of .

Therefore, by the preceding caſe,
::::; and ∴::::, by propoſition B.

∴ If the firſt be the ſame multiple, &c.

Proposition D. Theorem.

If the firſt be to the ſecond as the third to the fourth, and if the firſt be a multiple, or a part of the ſecond; the third is the ſame multiple, or the ſame part of the fourth.

Let ::::; and firſt, let be a multiple ; ſhall be the ſame multiple of .

Take =.

Whatever multiple is of take the ſame multiple of , then, becauſe :::: and of the ſecond and fourth, we have taken equimultiples,
and , therefore (B. 5. pr. 4),
::::, but (conſt.),
=∴ (B. 5. pr. A.) = and is the ſame multiple of that is of .

Next, let ::::, and alſo a part of ; then ſhall be the ſame part of .

Inverſely (B. 5.), ::::, but is a part of ; that is, is a multiple of ; ∴ by the preceding caſe, is the ſame multiple of that is, is the ſame part of that is of .

∴ If the firſt be to the ſecond, &c.

Proposition VII. Theorem.

Equal magnitudes have the ſame ratio to the ſame magnitude, and the ſame has the ſame ratio to equal magnitudes.

Let = and any other magnitude;
then :=: and :=:.

Becauſe =, ∴M=M;

∴ if M>, = or <m, then
M>, = or <m, and ∴:=: (B. 5. def. 5).

From the foregoing reaſoning it is evident that,
if m>, = or <M, then
m>, = or <M ∴:=: (B. 5. def. 5).

∴ Equal magnitudes, &c.

Definition VII.

When of the equimultiples of four magnitudes (taken as in the fifth definition), the multiple of the firſt is greater than that of the ſecond, but the multiple of the third is not greater than the multiple of the fourth; then the firſt is ſaid to have to the ſecond a greater ratio than the third magnitude has to the fourth: and, on the contrary, the third is ſaid to have the fourth a leſs ratio than the firſt has to the ſecond.

If, among the equimultiples of four magnitudes, compared as the in the fifth definition, we ſhould find >, but = or <, or if we ſhould find any particular multiple M′ of the firſt and third, and a particular multiple m′ of the ſecond and fourth, ſuch, that M′ times the firſt is >m′ times the ſecond, the M′ times the third is not >m′ times the fourth, i.e.= or <m′ times the fourth; then the firſt is ſaid to have to the ſecond a greater ratio than the third has to the fourth; or the third has to the fourth, under ſuch circumſtances, a leſs ratio than the firſt has to the second: although ſeveral other equimultiples may tend to ſhow that the four magnitudes are proportionals.

This definition will in future be expreſſed thus:—

If M′>m′, but M′= or <m′, then :>:.

In the above general expreſſion, M′ and m′ are to be conſidered particular multiples, not like the multiples M and m introduced in the fifth definition, which are in that definition conſidered to be every pair of multiples that can be taken. It muſt alſo be here obſerved, that ,,, and the like ſymbols are to be conſidered merely the representatives of geometrical magnitudes.

In a partial arithmetical way, this may be ſet forth as follows:

Let us take four numbers, 8,7,10, and 9.

Firſt. 8

Second. 7

Third. 10

Fourth. 9

16
24
32
40
48
46
64
72
80
88
96
104
112
&c;

14
21
28
35
42
49
56
63
70
77
84
91
98
&c;

20
30
40
50
60
70
80
90
100
110
120
130
140
&c;

18
27
36
45
54
63
72
81
90
99
108
117
126
&c;

Among the above multiples we find 16>14 and 20>18; that is, twice the firſt is greater than twice the ſecond, and twice the third is greater than twice the fourth; and 16<21 and 20<27; that is, twice the firſt is leſs than three times the ſecond, and twice the third is leſs than three times the fourth; and among the ſame multiples we can find 72>56 and 90>72: that is 9 times the firſt is greater than 8 times the ſecond, and 9 times the third is greater than 8 times the fourth. Many other equimultiples might be selected, which would tend to ſhow that the numbers 8,7,10,9, were proportionals, but they are not, for we can find a multiple of the firſt > a multiple of the ſecond, but the ſame multiple of the third that has been taken of the firſt not > than the ſame multiple of the fourth which has been taken of the ſecond; for inſtance, 9 times the firſt is > 10 times the ſecond, but 9 times the third is not > 10 times the fourth, that is, 72>70, but 90 not >90, or 8 times the firſt we find > 9 times the ſecond, but 8 times the third is not greater than 9 times the fourth, that is 64>63, but 80 is not >81. When any ſuch multiples as theſe can be found, the first (8) is ſaid to have the ſecond (7) a greater ratio than the third (10) has to the fourth (9), and on the contrary the third (10) is ſaid to have the fourth (9) a leſs ratio than the firſt (8) has to the ſecond (7).

Proposition VIII. Theorem.

Of unequal magnitudes the greater has a greater ratio to the ſame than the leſs has: and the ſame magnitude has a greater ratio to the leſs than it has to the greater.

Let and be two unequal magnitudes, and any other.

We ſhall firſt prove that which is the greater of the two unequal magnitudes, has a greater ratio to than , the leſs, has to ;

that is, :>:; take M′,m′,M′, and m′; ſuch, that M′ and M′ ſhall be each >; alſo take m′ the leaſt multiple of , which will make m′>M′=M′; ∴M′ is not >m′, but M′ is >m′, for,
as m′ is the firſt multiple which firſt becomes >M′, than (m′ minus 1) or m′ minus is not >M′, and is not >M′, ∴m′ minus + muſt be <M′+M′; that is, m′ muſt be <M′; ∴M′ is >m′; but it has been ſhown above that
M′ is not >m′, therefore, by the ſeventh definition,
has to a greater ratio than :.

Next we ſhall prove that has a greater ratio to , the leſs than it has to , the greater;
or, :>:.

Take m′,M′,m′, and M′, the ſame as in the firſt caſe, ſuch that
M′ and M′ will be each >, and m′ the leaſt multiple of , which firſt becomes greater than M′=M′.

∴m′ minus is not >M′, and is not >M′; conſequently
m′ minus + is <M′+M′; ∴m′ is <M′, and ∴ by the ſeventh definition,
has to a greater ratio than has to .

∴ Of unequal magnitudes, &c.

The contrivance employed in this propoſition for finding among multiples taken, as in the fifth definition, a multiple of the firſt greater than the multiple of the ſecond, but the ſame multiple of the third which has been taken of the firſt, not greater than the ſame multiple of the fourth which has been taken of the ſecond, may be illuſtrated numerically as follows:—

The number 9 has a greater ratio to 7 than 8 has to 7: that is, 9 : 7>8 : 7; or, 8+1 : 7>8 : 7.

The multiple of 1, which firſt becomes greater than 7, is 8 times, therefore we may multiply the firſt and third by 8, 9, 10, or any other greater number; in this caſe, let us multiply the firſt and third by 8, and we have 64+ 8 and 64: again, the firſt multiple of 7 which becomes greater than 64 is 10 times; then, by multiplying the ſecond and fourth by 10, we ſhall have 70 and 70; then, arranging theſe multiples, we have—

64+ 8

70

64

70

Conſequently, 64+ 8, or 72, is greater than 70, but 64 is not greater than 70, ∴ by the ſevenfth definition, 9 has a greater ratio to 7 than 8 has to 7.

The above is merely illuſtrative of the foregoing demonſtration, for this property could be ſhown of theſe or other numbers very readily in the following manner; becauſe if an antecedent contains it conſequent a greater number of times than another antecedent contains its conſequent, or when a fraction is formed of an antecedent for the numerator, and its conſequent for the denominator be greater than another fraction which is formed of another antecedent for the numerator and its conſequent for the denominator, the ratio of the firſt antecedent to its conſequent is greater than the ratio of the laſt antecedent to its conſequent.

Thus, the number 9 has a greater ratio to 7, than 8 has to 7, for
9/7
is greater than
8/7.

Again, 17 : 19 is a greater ratio than 13 : 15, becauſe
17/19=17 × 15/19 × 15=255/285,
and
13/15=13 × 19/15 × 19=247/285,
hence it is evident that
255/285
is greater than
247/285,
∴17/19
is greater than
13/15,
and, according to what has been above ſhown, 17 has to 19 a greater ratio than 13 has to 15.

So that the general terms upon which a greater, equal, or leſs ratio exiſts are as follows:—

If
A/B
be greater than
C/D,
A is ſaid to have to B a greater ratio than C has to D; if
A/B
be equal to
C/D,
then A has to B the ſame ratio which C has to D; and if
A/B
be leſs than
C/D,
A is ſaid to have to B a leſs ratio than C has to D.

The ſtudent ſhould underſtand all up to this propoſition perfectly before proceeding further, in order to fully comprehend the following propositions in of this book. We therefore ſtrongly recommend the learner to commence again, and read up to this ſlowly, and carefully reaſon at each ſtep, as he proceeds, particularly guarding againſt the miſchievous ſyſtem of depending wholly on the memory. By following theſe inſtructions, he will find that the parts which uſually preſent conſiderable difficulties will preſent no difficulties whatever, in proſecuting the ſtudy of this important book.

Proposition IX. Theorem.

Magnitudes which have the ſame ratio to the ſame magnitude are equal to one another; and thoſe to which the ſame magnitude has the ſame ratio are equal to one another.

Let ::::, then =.

For, if not, let >, then will
:>: (B. 5. pr. 8),
which is abſurd according to the hypotheſis.
∴ is not >.

In the ſame manner it may be ſhown, that
is not >, ∴=.

Again, let ::::, then will =.

For (invert.) ::::, therefore, by the ſirſt caſe, =.

∴ Magnitudes which have the ſame ratio, &c.

Let A : B=A : C, then B=C, for as the fraction
A/B= the fraction
A/C,
and the numerator of one equal to the numerator of the other, therefore the denominator of theſe fractions are equal, that is B=C.

Again, if B : A=C : A, B=C. For, as
B/A=C/A,
B muſt =C.

Proposition X. Theorem.

That magnitude which 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 :>:, then >.

For if not, let = or <; then, :=: (B. 5. pr. 7) or
::<: (B. 5. pr. 8) and (invert.),
which is abſurd according to the hypotheſis.

∴ is not = or <, and
∴ muſt be >.

Again, let :>:, then, <.

For if not, muſt be > or =, then :<: (B. 5. pr. 8) and (invert.);
or :=: (B. 5. pr. 7), which is abſurd (hyp.);
∴ is not > or =, and ∴ muſt be <.

∴ That magnitude which has, &c.

Proposition XI. Theorem.

Ratios that are the ſame to the ſame ratio, are the ſame to each other.

Let :=: and := : ,
then will := : .

For if M>, =, or <m, then M>, =, or <m, and if M>, =, or <m, then M>, =, or <m, (B. 5. def. 5);
∴ if M>, =, or <m,M>, =, or <m,
and ∴ (B. 5. def. 5) := : .

∴ Ratios that are the ſame, &c.

Proposition XII. Theorem.

If any number of magnitudes be proportionals as one of the antecedents is to its conſequent, ſo ſhall all the antecedents taken together be to all the conſequents.

Let :=:=:= : = : ;
then will := ++++ : ++++.

For if M>m, then M>m, and M>mM>m,
alſo M>m. (B. 5. def. 5.)

Therefore, if M+M+M+M+M,
or M (++++) be greater
than m+m+m+m+m,
or m (++++).

In the ſame way it may be ſhown, if M times one of the antecedents be equal or leſs than m times one of the conſequents, M times all the antecedents taken together, will be equal to or leſs than m times all the conſequents taken together. Therefore, by the fifth definition, as one of the antecedents is to its conſequent, ſo are all the antecedents taken together to all the conſequents taken together.

∴ If any number of magnitudes, &c.

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 to the ſixth.

Let :=:, but :>:, then :>:.

For becauſe :>:, there are ſome multiples (M′ and m′) of and , and of and , ſuch that M′>m′, but M′ not >m′, by the ſeventh definition.

Let theſe multiples be taken, and take the ſame multiples of and .

∴ (B. 5. def. 5.) if M′>, =, or <m′; then will M′>, =, <m′, but M′>m′ (conſtruction);

∴M′>m′, but M′ is not >m′ (conſtruction);
and therefore by the ſeventh definition,
:>:.

∴ If the firſt has to the ſecond, &c.

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.

An the ſame reaſoning is generally applicable, we have

:::M′:M′.

∴ Magnitudes have the ſame ratio, &c.

Definition XIII.

The technical term permutando or alternando, by permutation or alternately , is uſed when there are four proportionals, and it is inferred that the firſt has the ſame ratio to the third which the ſecond has to the fourth; or that the firſt is to the third as the ſecond is to the fourth: as it ſhown in the following proposition:—

Let ::::, by “permutando” or “alternando” it is
inferred ::::.

It may be neceſſary here to remark that the magnitudes ,,,, muſt be homogeneous, that is, of the ſame nature or ſimilitude of kind; we muſt therefore, in ſuch caſes, compare lines with lines, ſurfaces with ſurfaces, ſolids with ſolids, &c. Hence the ſtudent will readily perceive that a line and a ſurface, a ſurface and a ſolid, or other heterogenous magnitudes, can never ſtand in the relation of antecedent and conſequent.

Proposition XVI. Theorem.

If four magnitudes of the ſame kind be proportionals, they are alſo proportionals when taken alternately.

Dividendo, by diviſion, when there are four proportionals, and it is inferred, that the exceſs of the firſt above the ſecond is to the ſecond, as the exceſs of the third above the fourth, is to the fourth.

Let A : B :: C : D;
by “dividendo” it is inferred
A minus B : B :: C minus D : D.

According to the above, A is ſuppoſed to be greater than B, and C greater than D; if this be not the caſe, but to have B greater than A, and D greater than C, B and D can be made to ſtand as antecedents, and A and C as conſequents, by “invertion”

B : A :: D : C;
then, by “dividendo,” we infer
B minus A : A :: D minus C : C.

Proposition XVII. Theorem.

If magnitudes, taken jointly, be proportionals, they ſhall alſo be proportionals 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 +:::+:, then will ::::.

Take M>m to each add M, then we have M+M>m+M, or M (+) > (m + M) : but becauſe +:::+: (hyp.),
and M (+) > (m + M) ; ∴M (+) > (m + M) (B. 5. def. 5);
∴M+M>m+M; ∴M>m, by taking M from both ſides:
that is, when M>m, then M>m.

In the ſame manner it may be proved, that if
M= or <m, then will M= or <m; and ∴:::: (B. 5. def. 5).

∴ If magnitudes taken jointly, &c.

Definition XV.

The term componendo, by compoſition, is uſed when there are four proportionals; and it is inferred that the firſt together with the ſecond is to the ſecond as the third together with the fourth is to the fourth.

Let A : B :: C : D;
then, by the term “componendo,” it is inferred that
A+B : B :: C+D : D.

By “invertion” B and D may become the firſt and third, A and C the ſecond and fourth as

B : A :: D : C,
then, by “componendo,” we infer that
B+A : A :: D+C : C.

Proposition XVIII. Theorem.

If 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 is to the fourth, the firſt and ſecond together ſhall be to the ſecond as the third and fourth together is to the fourth.

Let ::::, then +:::+:; for if not, let +:::+:, ſuppoſing not =; ∴:::: (B. 5. pr. 17);
but :::: (hyp.);
∴:::: (B. 5. pr. 11);
∴= (B. 5. pr. 9),
which is contrary to the ſuppoſition;
∴ is not unequal to ; that is =; ∴+:::+:.

∴ If magnitudes, taken ſeparately, &c.

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 whole to the whole.

The term “convertendo,” by converſion, is made uſe of by geometricians, when there are four proportionals, and it is inferred, that the firſt is to its exceſs above the ſecond, as the third is to its exceſs above the fourth. See the following proposition:—

Proposition E. Theorem.

If four magnitudes be proportionals, they are alſo proportionals by converſion: that is, the firſt is to its exceſs above the ſecond, as the third is to its exceſs above the fourth.

“Ex æquali” (ſc. diſtantiâ), or ex æquo from equality of diſtance: when there is any number of magnitudes more than two, and as many others, ſuch that they are proportionals when taken two and two of each rank, and it is inferred that the firſt is to the laſt of the firſt rank of magnitudes, as the firſt is to the laſt of the others: “of this there are the two following kinds, which ariſe from the different order in which the magnitudes are taken, two and two.”

Definition XIX.

“Ex æquali,” from equality. This term is uſed ſimply by itſelf, when the firſt magnitude is to the ſecond of the firſt rank, as the firſt to the ſecond of the other rank; and as the ſecond is to the third of the firſt rank, ſo is the ſecond to the third of the other; and ſo on in order: and in the inference is as mentioned in the preceding definition; whence this is called ordinate proposition. It is demonſtrated in Book 5, pr. 22.

Thus, if there be two ranks of magnitudes,

A, B, C, D, E, F, the firſt rank,
and L, M, N, O, P, Q, the ſecond,
ſuch that A : B :: L : M, B : C :: M : N,
C : D :: N : O, D : E :: O : P, E : F :: P : Q;
we infer by the term “ex æquali” that
A : F :: L : Q.

Definition XX.

“Ex æquali in proportione perturbatâ ſeu inordinatâ,” from equality in perturbate, or diſorderly proportion. This term is uſed when the firſt magnitude is to the ſecond of the firſt rank as the laſt but one is to the laſt of the ſecond rank; and as the ſecond is to the third of the firſt rank, ſo is the laſt but two to the laſt but one of the ſecond rank; and as the third is to the fourth of the firſt rank, ſo is the third from the laſt to the laſt but two of the ſecond rank; and ſo on in croſs order: and the inference is in the 18th definition. It is demonſtrated in B. 5. pr. 23.

Thus, if there be two ranks of magnitudes,

A, B, C, D, E, F, the firſt rank,
and L, M, N, O, P, Q, the ſecond,
ſuch that A : B :: P : Q, B : C :: O : P,
C : D :: N : O, D : E :: M : N, E : F :: L : M;
the term “ex æquali in proportione perturbatâ ſeu inordinatâ” infers that
A : F :: L : Q.

Proposition XX. Theorem.

If there be three magnitudes, and other three, which, 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 ,,, be the firſt three magnitudes,
and ,,, be the other three,
ſuch that ::::, and ::::.

Then, if >, =, or <, then will >, =, or <.

From the hypotheſis, by alternando, we have
::::, and ::::;

∴ if >, =, or <, then will >, =, or < (B. 5. pr. 14).

∴ If there be three magnitudes, &c.

Proposition XXI. Theorem.

If there be three magnitudes, and the other three which have the ſame ratio, taken two and two, but in a croſs order; then if the firſt magnitude be greater than the third, the fourth ſhall be greater than the ſixth; and if equal, equal; and if leſs, leſs.

Let ,,, be the firſt three magnitudes,
and ,,, the other three,
ſuch that ::::, and ::::.

Then, if >, =, or <, then
will >, =, or <.

Firſt, let be >: then, becauſe is any other magnitude,
:>: (B. 5. pr. 8);
but :::: (hyp.);
∴:>: (B. 5. pr. 13);
and becauſe :::: (hyp.);
∴:::: (inv.),
and it was ſhown that :>:, ∴:>: (B. 5. pr. 13);
∴<, that is >.

Next, let be <, then ſhall be <; for >, and it has been ſhown that :=:, and :=:; ∴ by the firſt caſe is >, that is, <.

∴ If there be three, &c.

Proposition XXII. Theorem.

If there be 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 of the others has to the laſt of the ſame.

N.B.—This is uſually cited by the words “ex æquali,” or “ex æquo.”

Firſt, let there be magnitudes ,,, and as many others ,,, ſuch that
::::, and ::::; then ſhall ::::.

Let theſe magnitudes, as well as any equimultiples whatever of the antecedents and conſequents of the ratios, ſtand as follows:—

,,,,,, and
M,m,N,M,m,N, becauſe ::::; ∴M:m::M:m (B. 5. p. 4).

For the ſame reaſon
m:N::m:N; and becauſe there are three magnitudes,
M,m,N, and other three M,m,N, which, taken two and two, have the ſame ratio;

∴ if M>, =, <N then will M>, =, <N, by (B. 5. pr. 20);
and ∴:::: (def. 5).

Next, let there be four magnitudes, ,,,, and other four ,,,, which, taken two and two, have the ſame ratio,
that is to ſay, ::::, ::::, and ::::, then ſhall ::::; for, becauſe ,,, are three magnitudes,
and ,,, other three,
which, taken two and two, have the ſame ratio;
therefore, by the foregoing caſe, ::::, but ::::; therefore again, by the firſt caſe, ::::; and ſo on, whatever the number of magnitudes be.

∴ If there be any number, &c.

Proposition XXIII. Theorem.

If there be any number of magnitudes, and as many others, which, taken two and two in a croſs order, have the ſame ratio; the firſt ſhall have to the laſt of the firſt magnitudes the ſame ratio which the firſt of the others has to the laſt of the ſame.

N.B.—This is uſually cited by the words “ex æquali in proportione perturbatâ;” or “ex æquo perturbato.”

Firſt, let there be three magnitudes ,,, and other three, ,,, which, taken two and two in a croſs order, have the ſame ratio;
that is, ::::,and ::::,then ſhall ::::.

Let theſe magnitudes and their reſpective equimultiples be arranged as follows:—

,,,,,, M,M,m,M,m,m, then :::M:M (B. 5. pr. 15);
and for the ſame reaſon
:::m:m; but :::: (hyp.),
∴M:M::: (B. 5. pr. 11);
and becauſe :::: (hyp.),
∴M:m::M:m (B. 5. pr. 4);
then becauſe there are three magnitudes,
M,M,m, and other three, M,m,m, which, taken two and two in a croſs order, have the ſame ratio;
therefore, if M>, =, or <m, then will M>, =, or <m (B. 5. pr. 21),
and ∴:::: (B. 5. def. 5).

Next, let there be four magnitudes,
,,,, and other four, ,,,, which, when taken two and two in a croſs order, have the ſame ratio; namely,
::::,::::,and ::::.then ſhall ::::.
For, becauſe ,, are three magnitudes,
and ,,, other three,
which, taken two and two in a croſs order, have the ſame ratio,
therefore, by the firſt caſe, ::::, but ::::, therefore again, by the firſt caſe, ::::; and ſo on, whatever be the number of ſuch magnitudes.

∴ If there be any number, &c.

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 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 thave to the fourth.