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Book V.

Definitions.

I.

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, then twice 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 multipleof a leſs.

Let A > B, then 2 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, then A > B; or, let 3 A > 3 B, then A > B or, let m A > m B, then A > 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 Red dome Red dome Red dome Red dome Red dome be the ſame multiple of Red dome ,
that Yellow home Yellow home Yellow home Yellow home Yellow home is of Yellow home .
that Blue drop Blue drop Blue drop Blue drop Blue drop is of Blue drop .

Then is evident that
Red dome Red dome Red dome Red dome Red dome Yellow home Yellow home Yellow home Yellow home Yellow home Blue drop Blue drop Blue drop Blue drop Blue drop } is the ſame multiple of { Red dome Yellow home Blue drop
which that Red dome Red dome Red dome Red dome Red dome is of Red dome ;
becauſe there are as many magnitudes
in { Red dome Red dome Red dome Red dome Red dome Yellow home Yellow home Yellow home Yellow home Yellow home Blue drop Blue drop Blue drop Blue drop Blue drop } = { Red dome Yellow home Blue drop
as there are in Red dome Red dome Red dome Red dome Red dome = Red dome .

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 Yellow circle Yellow circle Yellow circle , the firſt, be the ſame multiple of Yellow circle , the ſecond, that Red drop Red drop Red drop , the third, is of Red drop , the fourth; and let Blue circle Blue circle Blue circle Blue circle , the fifth, be the ſame multiple of Yellow circle , the ſecond, that Black drop Black drop Black drop Black drop , the ſixth, is of Red drop , the fourth.

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

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 { Yellow square Yellow square Yellow square Yellow square } be the ſame multiple of Red square
which { Black diamond Black diamond Black diamond Black diamond } is of Blue diamond ;
take { Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square } the ſame multiple of { Yellow square Yellow square Yellow square Yellow square ,
which { Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond } is of { Black diamond Black diamond Black diamond Black diamond .

Then it is evident,
that { Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square } is the ſame multiple of Red square
which { Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond } is of Blue diamond ;
becauſe { Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square Red square } contains { Yellow square Yellow square Yellow square Yellow square } contains Red square
as many times as
Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond } contains { Black diamond Black diamond Black diamond Black diamond } contains Blue diamond .

The ſame reaſoning is applicable in all caſes.

If the firſt four, &c.

Definition V.

Four magnitudes Red circle , Yellow square , Blue diamond , Black home , 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

Trans square Trans square Trans square Trans square Trans square Red circle Red circle Trans square Trans square Trans square Trans square Red circle Red circle Red circle Trans square Trans square Trans square Red circle Red circle Red circle Red circle Trans square Trans square Red circle Red circle Red circle Red circle Red circle Trans square Red circle Red circle Red circle Red circle Red circle Red circle

&c.

of the ſecond

Trans square Trans square Trans square Trans square Trans square Yellow square Yellow square Trans square Trans square Trans square Trans square Yellow square Yellow square Yellow square Trans square Trans square Trans square Yellow square Yellow square Yellow square Yellow square Trans square Trans square Yellow square Yellow square Yellow square Yellow square Yellow square Trans square Yellow square Yellow square Yellow square Yellow square Yellow square Yellow square

&c.

of the third

Trans diamond Trans diamond Trans diamond Trans diamond Trans diamond Blue diamond Blue diamond Trans diamond Trans diamond Trans diamond Trans diamond Blue diamond Blue diamond Blue diamond Trans diamond Trans diamond Trans diamond Blue diamond Blue diamond Blue diamond Blue diamond Trans diamond Trans diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Trans diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond

&c.

of the fourth

Trans square Trans square Trans square Trans square Trans square Black home Black home Trans square Trans square Trans square Trans square Black home Black home Black home Trans square Trans square Trans square Black home Black home Black home Black home Trans square Trans square Black home Black home Black home Black home Black home Trans square Black home Black home Black home Black home Black home Black home

&c.

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

If { Red circle Red circle >, = or < Yellow square Yellow square Red circle Red circle >, = or < Yellow square Yellow square Yellow square Red circle Red circle >, = or < Yellow square Yellow square Yellow square Yellow square Red circle Red circle >, = or < Yellow square Yellow square Yellow square Yellow square Yellow square Red circle Red circle >, = or < Yellow square Yellow square Yellow square Yellow square Yellow square Yellow square

then will { Blue diamond Blue diamond >, = or < Black home Black home Blue diamond Blue diamond >, = or < Black home Black home Black home Blue diamond Blue diamond >, = or < Black home Black home Black home Black home Blue diamond Blue diamond >, = or < Black home Black home Black home Black home Black home Blue diamond Blue diamond >, = or < Black home Black home Black home Black home Black home Black home

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 { Red circle Red circle Red circle >, = or < Trans square Trans square Trans square Trans square Trans square Yellow square Yellow square Red circle Red circle Red circle >, = or < Trans square Trans square Trans square Trans square Yellow square Yellow square Yellow square Red circle Red circle Red circle >, = or < Trans square Trans square Trans square Yellow square Yellow square Yellow square Yellow square Red circle Red circle Red circle >, = or < Trans square Trans square Yellow square Yellow square Yellow square Yellow square Yellow square Red circle Red circle Red circle >, = or < Trans square Yellow square Yellow square Yellow square Yellow square Yellow square Yellow square

then will { Blue diamond Blue diamond Blue diamond >, = or < Trans square Trans square Trans square Trans square Trans square Black home Black home Blue diamond Blue diamond Blue diamond >, = or < Trans square Trans square Trans square Trans square Black home Black home Black home Blue diamond Blue diamond Blue diamond >, = or < Trans square Trans square Trans square Black home Black home Black home Black home Blue diamond Blue diamond Blue diamond >, = or < Trans square Trans square Black home Black home Black home Black home Black home Blue diamond Blue diamond Blue diamond >, = or < Trans square Black home Black home Black home Black home Black home Black home

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 { Red circle Red circle Red circle Red circle >, = or < Yellow square Yellow square Red circle Red circle Red circle Red circle >, = or < Yellow square Yellow square Yellow square Red circle Red circle Red circle Red circle >, = or < Yellow square Yellow square Yellow square Yellow square Red circle Red circle Red circle Red circle >, = or < Yellow square Yellow square Yellow square Yellow square Yellow square Red circle Red circle Red circle Red circle >, = or < Yellow square Yellow square Yellow square Yellow square Yellow square Yellow square

then will { Blue diamond Blue diamond Blue diamond Blue diamond >, = or < Black home Black home Blue diamond Blue diamond Blue diamond Blue diamond >, = or < Black home Black home Black home Blue diamond Blue diamond Blue diamond Blue diamond >, = or < Black home Black home Black home Black home Blue diamond Blue diamond Blue diamond Blue diamond >, = or < Black home Black home Black home Black home Black home Blue diamond Blue diamond Blue diamond Blue diamond >, = or < Black home Black home Black home Black home Black home Black home

And so on, with an 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 Red circle >, = < m Yellow square , when M Blue diamond >, = < m Black home ,

Then we infer that Red circle , the firſt, has the ſame ratio to Yellow square , the ſecond, which Blue diamond , the third, has to Black home the fourth: expreſſed in the ſucceeding emonſtrations thus:

Red circle : Yellow square :: Blue diamond : Black home ; or thus, Red circle : Yellow square = Blue diamond : Black home ; or thus, Red circle / Yellow square = Blue diamond / Black home : and is read,

“as Red circle is to Yellow square , so is Blue diamond to Black home .

And if Red circle : Yellow square :: Blue diamond : Black home we ſhall infer if
M Red circle >, = or < m Yellow square , then will
M Blue diamond >, = or < m Black home .

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 Red circle , Black home , Yellow square , &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 ſeond, 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 Yellow circle : Black square :: Red diamond : Blue home , then 3 Yellow circle : 2 Black square :: 3 Red diamond : 2 Blue home , every equimultiple of 3 Yellow circle and 3 Red diamond are equimultiples of Yellow circle and Red diamond , and every equimultiple of 2 Black square and 2 Blue home , are equimultiples of Black square and Blue home (B. 5. pr. 3.)

That is, M times 3 Yellow circle and M times 3 Red diamond are equimultiples of Yellow circle and Red diamond , and m times 2 Black square and m 2 Blue home are equimultiples of 2 Black square and 2 Blue home ; but Yellow circle : Black square :: Red diamond : Blue home (hyp); if M 3 Yellow circle >, = or < m 2 Black square , then M 3 Red diamond >, = or < m 2 Blue home (def. 5.) and therefore 3 Yellow circle : 2 Black square :: 3 Red diamond : 2 Blue home (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 Blue drop Blue drop Blue drop Yellow dome = M Black triangle Red square
and Yellow dome = M Red square ,
Blue drop Blue drop Blue drop Yellow dome minus Yellow dome = M Black triangle Red square minus M Red square ,
Blue drop Blue drop Blue drop = M′ ( Black triangle Red square minus Red square ),
and Blue drop Blue drop Blue drop = M Black triangle .

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 Yellow drop Yellow drop Yellow drop Yellow drop = M Red square ; and Black dome Black dome = M Blue triangle ;
then Yellow drop Yellow drop Yellow drop Yellow drop minus m Red square =

M′ Red square minus m′ Red square = (M′ minus m′) Red square ,
and Black dome Black dome minus m′ Blue triangle = M′ Blue triangle minus m′ Blue triangle = (M′ minus m′) Blue triangle .

Hence, (M′ minus m′) Red square and (M′ minus m′) Blue triangle are equimultiples of Red square and Blue triangle , and equal to Red square and Blue triangle , 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 Red circle : Black square :: Blue home : Yellow diamond ; therefore, by the fifth definition, if Red circle Red circle > Black square Black square , then will Blue home Blue home > Yellow diamond Yellow diamond ;
but if Red circle > Black square , then Red circle Red circle > Black square Black square
and Blue home Blue home > Yellow diamond Yellow diamond ,
and Blue home > Yellow diamond .

Similarly, if Red circle =, or < Black square , then will Blue home =, or < Yellow diamond .

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 Blue home : Black dome :: Red square : Yellow diamond ,
then inverſely, Black dome : Blue home :: Yellow diamond : Red square .

If M Blue home < m Black dome , then M Red square < m Yellow diamond
by the fifth definition.

Let M Blue home < m Black dome , that is, m Black dome > M Blue home ,
M Red square < m Yellow diamond , or, m Yellow diamond > M Red square ;
if m Black dome > M Blue home , then will m Yellow diamond > M Red square .

In the ſame manner it may be ſhown,

that if m Black dome = or < M Blue home ,
then will m Yellow diamond =, or < M Red square ;
and therefore, by the fifth definition, we infer
that Black dome : Blue home : Yellow diamond : Red square .

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 Blue square Blue square Blue square Blue square , be the firſt, the ſame multiple of Black circle , the ſecond,
that Yellow diamond Yellow diamond Yellow diamond Yellow diamond , the third, is of Red home , the fourth.

Then Blue square Blue square Blue square Blue square : Black circle :: Yellow diamond Yellow diamond Yellow diamond Yellow diamond : Red home
take M Blue square Blue square Blue square Blue square , m Black circle , M Yellow diamond Yellow diamond Yellow diamond Yellow diamond , m Red home ;
becauſe Blue square Blue square Blue square Blue square is the ſame multiple of Black circle
that Yellow diamond Yellow diamond Yellow diamond Yellow diamond is of Red home (according to the hypotheſis);
and M Blue square Blue square Blue square Blue square is taken the ſame multiple of Blue square Blue square Blue square Blue square
that M Yellow diamond Yellow diamond Yellow diamond Yellow diamond is of Yellow diamond Yellow diamond Yellow diamond Yellow diamond ,
(according to the third proposition),
M Blue square Blue square Blue square Blue square is the ſame multiple of Black circle
that M Yellow diamond Yellow diamond Yellow diamond Yellow diamond is of Red home .

Therefore, if M Blue square Blue square Blue square Blue square be of Black circle a greater multiple than m Black circle is, then M Yellow diamond Yellow diamond Yellow diamond Yellow diamond is a greater multiple of Red home than m Red home is; that is, if M Blue square Blue square Blue square Blue square be greater than m Black circle , then M Yellow diamond Yellow diamond Yellow diamond Yellow diamond will be greater than m Red home ; in the ſame manner it can be ſhewn, if M Blue square Blue square Blue square Blue square be equal m Black circle , then
M Yellow diamond Yellow diamond Yellow diamond Yellow diamond will be equal m Red home .

And, generally, if M Blue square Blue square Blue square Blue square >, = or < m Black circle
then M Yellow diamond Yellow diamond Yellow diamond Yellow diamond will be >, = or < m Red home ;
by the fifth definition,
Blue square Blue square Blue square Blue square : Black circle :: Yellow diamond Yellow diamond Yellow diamond Yellow diamond : Red home .

Next, let Black circle be the ſame part of Blue square Blue square Blue square Blue square
that Red home is of Yellow diamond Yellow diamond Yellow diamond Yellow diamond .

In this caſe alſo Black circle : Blue square Blue square Blue square Blue square :: Red home : Yellow diamond Yellow diamond Yellow diamond Yellow diamond .

For, becauſe
Black circle is the ſame part of Blue square Blue square Blue square Blue square that Red home is of Yellow diamond Yellow diamond Yellow diamond Yellow diamond ,
therefore Blue square Blue square Blue square Blue square is the ſame multiple of Black circle
that Yellow diamond Yellow diamond Yellow diamond Yellow diamond is of Red home .

Therefore, by the preceding caſe,
Blue square Blue square Blue square Blue square : Black circle :: Yellow diamond Yellow diamond Yellow diamond Yellow diamond : Red home ;
and Black circle : Blue square Blue square Blue square Blue square :: Red home : Yellow diamond Yellow diamond Yellow diamond Yellow diamond ,
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 Yellow circle Yellow circle Yellow circle : Black square :: Red diamond Red diamond Red diamond Red diamond : Blue home ;
and firſt, let Yellow circle Yellow circle Yellow circle be a multiple Black square ;
Red diamond Red diamond Red diamond Red diamond ſhall be the ſame multiple of Blue home .

Yellow circle Yellow circle Yellow circle Black square Red diamond Red diamond Red diamond Red diamond Blue home

Red dome Red dome Red dome Black drop Black drop Black drop Black drop

Take Red dome Red dome Red dome = Yellow circle Yellow circle Yellow circle .

Whatever multiple Yellow circle Yellow circle Yellow circle is of Black square
take Black drop Black drop Black drop Black drop the ſame multiple of Blue home ,
then, becauſe Yellow circle Yellow circle Yellow circle : Black square :: Red diamond Red diamond Red diamond Red diamond : Blue home
and of the ſecond and fourth, we have taken equimultiples,
Yellow circle Yellow circle Yellow circle and Black drop Black drop Black drop Black drop , therefore (B. 5. pr. 4),
Yellow circle Yellow circle Yellow circle : Red dome Red dome Red dome :: Red diamond Red diamond Red diamond Red diamond : Black drop Black drop Black drop Black drop , but (conſt.),
Yellow circle Yellow circle Yellow circle = Red dome Red dome Red dome (B. 5. pr. A.) Red diamond Red diamond Red diamond Red diamond = Black drop Black drop Black drop Black drop
and Black drop Black drop Black drop Black drop is the ſame multiple of Blue home
that Yellow circle Yellow circle Yellow circle is of Black square .

Next, let Black square : Yellow circle Yellow circle Yellow circle :: Blue home : Red diamond Red diamond Red diamond Red diamond ,
and alſo Black square a part of Yellow circle Yellow circle Yellow circle ;
then Blue home ſhall be the ſame part of Red diamond Red diamond Red diamond Red diamond .

Inverſely (B. 5.), Yellow circle Yellow circle Yellow circle : Black square :: Red diamond Red diamond Red diamond Red diamond : Blue home ,
but Black square is a part of Yellow circle Yellow circle Yellow circle ;
that is, Yellow circle Yellow circle Yellow circle is a multiple of Black square ;
by the preceding caſe, Red diamond Red diamond Red diamond Red diamond is the ſame multiple of Blue home
that is, Blue home is the ſame part of Red diamond Red diamond Red diamond Red diamond
that Black square is of Yellow circle Yellow circle Yellow circle .

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 Red circle = Blue diamond and Yellow square any other magnitude;
then Red circle : Yellow square = Blue diamond : Yellow square and Yellow square : Red circle = Yellow square : Blue diamond .

Becauſe Red circle = Blue diamond ,
M Red circle = M Blue diamond ;

if M Red circle >, = or < m Yellow square , then
M Blue diamond >, = or < m Yellow square ,
and Red circle : Yellow square = Blue diamond : Yellow square (B. 5. def. 5).

From the foregoing reaſoning it is evident that,
if m Yellow square >, = or < M Red circle , then
m Yellow square >, = or < M Blue diamond
Yellow square : Red circle = Yellow square : Blue diamond (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 Red circle Red circle Red circle Red circle Red circle > Yellow square Yellow square Yellow square Yellow square , but Blue diamond Blue diamond Blue diamond Blue diamond Blue diamond = or < Black square Black square Black square Black square , 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′ Red home > m′ Black dome , but M′ Blue square = or < m′ Yellow diamond ,
then Red home : Black dome > Blue square : Yellow diamond .

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 Red home , Black dome , Blue square , 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 Black triangle Red square and Yellow square be two unequal magnitudes, and Blue circle any other.

We ſhall firſt prove that Black triangle Red square which is the greater of the two unequal magnitudes, has a greater ratio to Blue circle than Yellow square , the leſs, has to Blue circle ;

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

Next we ſhall prove that Blue circle has a greater ratio to Yellow square , the leſs than it has to Black triangle Red square , the greater;
or, Blue circle : Yellow square > Blue circle : Black triangle Red square .

Take m′ Blue circle , M′ Yellow square , m′ Blue circle , and M′ Black triangle Red square ,
the ſame as in the firſt caſe, ſuch that
M′ Black triangle and M′ Red square will be each > Blue circle , and m′ Blue circle the leaſt multiple of Blue circle , which firſt becomes greater than M′ Red square = M′ Yellow square .

m′ Blue circle minus Blue circle is not > M′ Red square ,
and Blue circle is not > M′ Black triangle ; conſequently
m′ Blue circle minus Blue circle + Blue circle is < M′ Red square + M′ Black triangle ;
m′ Blue circle is < M′ Black triangle Red square , and by the ſeventh definition,
Blue circle has to Yellow square a greater ratio than Blue circle has to Black triangle Red square .

Of unequal magnitudes, &c.

The contrivance employed in this propoſitino 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 miſchievious 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 Blue diamond : Yellow square :: Red circle : Yellow square , then Blue diamond = Red circle .

For, if not, let Blue diamond > Red circle , then will
Blue diamond : Yellow square > Red circle : Yellow square (B. 5. pr. 8),
which is abſurd according to the hypotheſis.
Blue diamond is not > Red circle .

In the ſame manner it may be ſhown, that
Red circle is not > Blue diamond ,
Blue diamond = Red circle .

Again, let Yellow square : Blue diamond :: Yellow square : Red circle , then will Blue diamond = Red circle .

For (invert.) : Yellow square :: Red circle : Yellow square ,
therefore, by the ſirſt caſe, Blue diamond = Red circle .

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 Blue home : Yellow square > Red circle : Yellow square , then Blue home > Red circle .

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

Blue home is not = or < r, and
Blue home muſt be > Red circle .

Again, let Yellow square : Red circle > Yellow square : Blue home ,
then, Red circle < Blue home .

For if not, Red circle muſt be > or = Blue home ,
then Yellow square : Red circle < Yellow square : Blue home (B. 5. pr. 8) and (invert.);
or Yellow square : Red circle = Yellow square : Blue home (B. 5. pr. 7), which is abſurd (hyp.);
Red circle is not > or = Blue home ,
and Red circle muſt be < Blue home .

That magnitude which has, &c.

Proposition XI. Theorem.

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

Let Blue diamond : Blue square = Red circle : Yellow home and Red circle : Yellow home = Black triangle : Black circle ,
then will Blue diamond : Blue square = Black triangle : Black circle .

For if M Blue diamond >, =, or < m Blue square ,
then M Red circle >, =, or < m Yellow home ,
and if M Red circle >, =, or < m Yellow home ,
then M Black triangle >, =, or < m Black circle , (B. 5. def. 5);
if M Blue diamond >, =, or < m Blue square , M Black triangle >, =, or < m Black circle ,
and (B. 5. def. 5) Blue diamond : Blue square = Black triangle : Black circle .

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 Red square : Red circle = Black dome : Black drop = Yellow diamond : Yellow home = Blue circle : Blue triangle = Black triangle : Black circle ;
then will Red square : Red circle =
Red square + Black dome + Yellow diamond + Blue circle + Black triangle : Red circle + Black drop + Yellow home + Blue triangle + Black circle .

For if M Red square > m Red circle , then M Black dome > m Black drop ,
and M Yellow diamond > m Yellow home M Blue circle > m Blue triangle ,
alſo M Black triangle > m Black circle . (B. 5. def. 5.)

Therefore, if M Red square + M Black dome + M Yellow diamond + M Blue circle + M Black triangle ,
or M ( Red square + Black dome + Yellow diamond + Blue circle + Black triangle ) be greater
than m Red circle + m Black drop + m Yellow home + m Blue triangle + m Black circle ,
or m ( Red circle + Black drop + Yellow home + Blue triangle + Black circle ).

In the ſame way it may be ſhown, if M times one f 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 Blue home : Blue dome = Red square : Yellow diamond , but Red square : Yellow diamond > Black drop : Black circle ,
then Blue home : Blue dome > Black drop : Black circle .

For becaſe Red square : Yellow diamond > Black drop : Black circle , there are ſome multiples (M′ and m′) of Red square and Black drop , and of Yellow diamond and Black circle , ſuch that M′ Red square > m′ Yellow diamond ,
but M′ Black drop not > m′ Black circle , by the ſeventh definition.

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

(B. 5. def. 5.) if M′ Blue home >, =, or < m′ Blue dome ;
then will M′ Red square >, =, < m′ Yellow diamond ,
but M′ Red square > m′ Yellow diamond (conſtruction);

M′ Blue home > m′ Blue dome ,
but M′ Black drop is not > m′ Black circle (conſtruction);
and therefore by the ſeventh definition,
Blue home : Blue dome > Black drop : Black circle .

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.

Let Red home : Black dome :: Yellow square : Blue diamond , and firſt ſuppoſe
Red home > Yellow square , then will Black dome > Blue diamond .

For Red home : Black dome > Yellow square : Black dome (B. 5. pr. 8), and by the
hypotheſis, Red home : Black dome = Yellow square : Blue diamond ;
Yellow square : Blue diamond > Yellow square : Black dome (B. 5. pr. 13),
Blue diamond < Black dome (B. 5. pr. 10.), or Black dome > Blue diamond .

Secondly, let Red home = Yellow square , then will Black dome = Blue diamond .

For Red home : Black dome = Yellow square : Black dome (B. 5. pr. 7),
and Red home : Black dome = Yellow square : Blue diamond (hyp.);
Yellow square : Black dome = Yellow square : Blue diamond (B. 5. pr. 11),
and Black dome = Blue diamond (B. 5, pr. 9).

Thirdly, if Red home < Yellow square , then will Black dome < Blue diamond ;
becauſe Yellow square > Red home and Yellow square : Blue diamond = Red home : Black dome ;
Blue diamond > Black dome , by the firſt caſe,
that is, Black dome < Blue diamond .

If the firſt has the ſame ratio, &c.

Proposition XV. Theorem.

Magnitudes have the ſame ratio to one another which their equimultiples have.

Let Red circle and Yellow square be two magnitudes;
then Red circle : Yellow square :: M′ Red circle : M′ Yellow square .

For Red circle : Yellow square = Red circle : Yellow square = Red circle : Yellow square = Red circle : Yellow square

Red circle : Yellow square :: 4 Red circle : 4 Yellow square . (B. 5. pr. 12).

An the ſame reaſoning is genrally applicable, we have

Red circle : Yellow square :: M′ Red circle : M′ Yellow square .

Magnitudes have the ſame ratio, &c.

Definition XIII.

The technical term permutando or alternando, by permutation or laternately , 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 Yellow circle : Black diamond :: Red home : Blue square ,
by “permutando” or “alternando” it is
inferred Yellow circle : Red home :: Black diamond : Blue square .

It may be neceſſary here to remark that the magnitudes Yellow circle , Black diamond , Red home , Blue square , 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 altrnately.

Let Red home : Black dome :: Yellow square : Blue diamond , then Red home : Yellow square :: Black dome : Blue diamond .

For M Red home : M Black dome :: Red home : Black dome (B. 5. pr. 15),
and M Red home : M Black dome :: Yellow square : Blue diamond (hyp.) and (B. 5. pr. 11);
alſo m Yellow square : m Blue diamond :: Yellow square : Blue diamond (B. 5. pr. 15);
M Red home : M Black dome :: m Yellow square : m Blue diamond (B. 5. pr. 14),
and if M Red home >, =, or < m Yellow square ,
then will M Black dome >, =, or < m Blue diamond (B. 5. pr. 14);
therefore by the fifth definition,
Red home : Yellow square :: Black dome : Blue diamond .

If four magnitudes of the ſame kind, &c.

Definition XVI.

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 Red home + Black dome : Black dome :: Yellow square + Blue diamond : Blue diamond ,
then will Red home : Black dome :: Yellow square : Blue diamond .

Take M Red home > m Black dome to each add M Black dome ,
then we have M Red home + M Black dome > m Black dome + M Black dome ,
or M ( Red home + Black dome ) > (m + M) Black dome :
but becauſe Red home + Black dome : Black dome :: Yellow square + Blue diamond : Blue diamond (hyp.),
and M ( Red home + Black dome ) > (m + M) Black dome ;
M ( Yellow square + Blue diamond ) > (m + M) Blue diamond (B. 5. def. 5);
M Yellow square + M Blue diamond > m Blue diamond + M Blue diamond ;
M Yellow square > m Blue diamond , by taking M Blue diamond from both ſides:
that is, when M Red home > m Black dome , then M Yellow square > m Blue diamond .

In the ſame manner it may be proved, that if
M Red home = or < m Black dome , then will M Yellow square = or < m Blue diamond ;
and Red home : Black dome :: Yellow square : Blue diamond (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 “invertionB 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 Red home : Black dome :: Yellow square : Blue diamond ,
then Red home + Black dome : Black dome :: Yellow square + Blue diamond : Blue diamond ;
for if not, let Red home + Black dome : Black dome :: Yellow square + Black circle : Black circle ,
ſuppoſing Black circle not = Blue diamond ;
Red home : Black dome :: Yellow square : Black circle (B. 5. pr. 17);
but Red home : Black dome :: Yellow square : Blue diamond (hyp.);
Yellow square : Black circle :: Yellow square : Blue diamond (B. 5. pr. 11);
Black circle = Blue diamond (B. 5. pr. 9),
which is contrary to the ſuppoſition;
Black circle is not unequal to Blue diamond ;
that is Black circle = Blue diamond ;
Red home + Black dome : Black dome :: Yellow square + Blue diamond : Blue diamond .

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.

Let Red home + Black dome : Blue square + Yellow diamond :: Red home : Blue square ,
then will Black dome : Yellow diamond :: Red home + Black dome : Blue square + Yellow diamond ,

For Red home + Black dome : Red home :: Blue square + Yellow diamond : Blue square (alter.),

Black dome : Red home :: Yellow diamond : Blue square (divid.),
again Black dome : Yellow diamond :: Red home : Blue square (alter.),
but Red home + Black dome : Blue square + Yellow diamond :: Red home : Blue square (hyp.);
therefore Black dome : Yellow diamond :: Red home + Black dome : Blue square + Yellow diamond
(B. 5. pr. 11).

If a whole magnitude be to a whole, &c.

Definition XVII.

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.

Let Blue circle Black drop : Black drop :: Red square Yellow diamond : Yellow diamond ,
then ſhall Blue circle Black drop : Blue circle :: Red square Yellow diamond : Red square ,

Becauſe Blue circle Black drop : Black drop : Red square Yellow diamond : Yellow diamond ;
therefore Blue circle : Black drop :: Red square : Yellow diamond (divid.),

Black drop : Blue circle :: Yellow diamond : Red square (inver.),

Blue circle Black drop : Blue circle :: Red square Yellow diamond : Red square (compo.).

If four magnitudes, &c.

Definition XVIII.

“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 Blue home , Red dome , Yellow square , be the firſt three magnitudes,
and Blue diamond , Red drop , Yellow circle , be the other three,
ſuch that Blue home : Red dome :: Blue diamond : Red drop , and Red dome : Yellow square :: Red drop : Yellow circle .

Then, if Blue home >, =, or < Yellow square , then will Blue diamond >, =, or < Yellow circle .

From the hypotheſis, by alternando, we have
Blue home : Blue diamond :: Red dome : Red drop ,
and Red dome : Red drop :: Yellow square : Yellow circle ;

Blue home : Blue diamond :: Yellow square : Yellow circle (B. 5. pr. 11);

if Blue home >, =, or < Yellow square , then will Blue diamond >, =, or < Yellow circle (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 magitude be greater than the third, the fourth ſhall be greater than the ſixth; and if equal, equal; and if leſs, leſs.

Let Yellow home , Red home , Blue square , be the firſt three magnitudes,
and Blue diamond , Red drop , Yellow circle , the other three,
ſuch that Yellow home : Red home :: Red drop : Yellow circle , and Red home : Blue square :: Blue diamond : Red drop .

Then, if Yellow home >, =, or < Blue square , then
will Blue diamond >, =, or < Yellow circle .

Firſt, let Yellow home be > Blue square :
then, becauſe Red home is any other magnitude,
Yellow home : Red home > Blue square : Red home (B. 5. pr. 8);
but Red drop : Yellow circle :: Yellow home : Red home (hyp.);
Red drop : Yellow circle > Blue square : Red home (B. 5. pr. 13);
and becauſe Red home : Blue square :: Blue diamond : Red drop (hyp.);
Blue square : Red home :: Red drop : Blue diamond (inv.),
and it was ſhown that Red drop : Yellow circle > Blue square : Red home ,
Red drop : Yellow circle > Red drop : Blue diamond (B. 5. pr. 13);
Yellow circle < Blue diamond ,
that is Blue diamond > Yellow circle .

Secondly, let Yellow home = Blue square ; then ſhall Blue diamond = Yellow circle
For becauſe Yellow home = Blue square ,
Yellow home : Red home = Blue square : Red home (B. 5. pr. 7);
but Yellow home : Red home = Red drop : Yellow circle (hyp.),
and Blue square : Red home = Red drop : Blue diamond (hyp. and inv.),
Red drop : Yellow circle = Red drop : Blue diamond (B. 5. pr. 11),
Blue diamond = Yellow circle (B. 5. pr. 9).

Next, let Yellow home be < Blue square , then Blue diamond ſhall be < Yellow circle ;
for Blue square > Yellow home ,
and it has been ſhown that Blue square : Red home = Red drop : Blue diamond ,
and Red home : Yellow home = Yellow circle : Red drop ;
by the firſt caſe Yellow circle is > Blue diamond ,
that is, Blue diamond < Yellow circle .

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 Red home , Blue diamond , Yellow square ,
and as many others Red diamond , Blue drop , Yellow circle ,
ſuch that
Red home : Blue diamond :: Red diamond : Blue drop ,
and Blue diamond : Yellow square :: Blue drop : Yellow circle ;
then ſhall Red home : Yellow square :: Red diamond : Yellow circle .

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

Red home , Blue diamond , Yellow square , Red diamond , Blue drop , Yellow circle ,
and
M Red home , m Blue diamond , N Yellow square , M Red diamond , m Blue drop , N Yellow circle ,
becauſe Red home : Blue diamond :: Red diamond : Blue drop ;
M Red home : m Blue diamond :: M Red diamond : m Blue drop (B. 5. p. 4).

For the ſame reaſon
m Blue diamond : N Yellow square :: m Blue drop : N Yellow circle ;
and becauſe there are three magnitudes,
M Red home , m Blue diamond , N Yellow square ,
and other three M Red diamond , m Blue drop , N Yellow circle ,
which, taken two and two, have the ſame ratio;

if M Red home >, =, < N Yellow square
then will M Red diamond >, =, < N Yellow circle , by (B. 5. pr. 20);
and Red home : Yellow square :: Red diamond : Yellow circle (def. 5).

Next, let there be four magnitudes, Blue home , Black diamond , Yellow square , Red diamond ,
and other four Blue drop , Black circle , Yellow rectangle , Red triangle ,
which, taken two and two, have the ſame ratio,
that is to ſay, Blue home : Black diamond :: Blue drop : Black circle ,
Black diamond : Yellow square :: Black circle : Yellow rectangle ,
and Yellow square : Red diamond :: Yellow rectangle : Red triangle ,
then ſhall Blue home : Red diamond :: Blue drop : Red triangle ;
for, becauſe Blue home , Black diamond , Yellow square , are three magnitudes,
and Blue drop , Black circle , Yellow rectangle , other three,
which, taken two and two, have the ſame ratio;
therefore, by the foregoing caſe, Blue home : Yellow square :: Blue drop : Yellow rectangle ,
but Yellow square : Red diamond :: Yellow rectangle : Red triangle ;
therefore again, by the firſt caſe, Blue home : Red diamond :: Blue drop : Red triangle ;
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 Yellow home , Blue dome , Red square ,
and other three, Yellow diamond , Blue drop , Red circle ,
which, taken two and two in a croſs order, have the ſame ratio; that is, Yellow home : Blue dome :: Blue drop : Red circle , and Blue dome : Red square :: Yellow diamond : Red circle , then ſhall Yellow home : Red square :: Yellow diamond : Red circle .

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

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

Next, let there be four magnitudes,
Yellow home , Blue dome , Red square , Yellow diamond ,
and other four, Blue drop , Red circle , Black rectangle , Black triangle ,
which, when taken two and two in a croſs order, have the ſame ratio; namely, Yellow home : Blue dome :: Black rectangle : Black triangle , Blue dome : Red square :: Red circle : Black rectangle , and Red square : Yellow diamond :: Blue drop : Red circle . then ſhall Yellow home : Yellow diamond :: Blue drop : Black triangle . For, becauſe Yellow home , Blue dome , Red square are three magnitudes,
and Red square , Black rectangle , Black triangle , other three,
which, taken two and two in a croſs order, have the ſame ratio,
therefore, by the firſt caſe, Yellow home : Red square :: Red square : Black triangle ,
but Red square : Yellow diamond :: Blue drop : Red circle ,
therefore again, by the firſt caſe, Yellow home : Yellow diamond :: Blue drop : Black triangle ;
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.

Red home
Black dome
Blue square
Yellow diamond
Red drop
Blue circle

Let Red home : Black dome :: Blue square : Yellow diamond , and Red drop : Black dome :: Blue circle : Yellow diamond , then Red home + Red drop : Black dome :: Blue square + Blue circle : Yellow diamond .

Red drop : Black dome :: Blue circle : Yellow diamond (hyp.),
and Black dome : Red home :: Yellow diamond : Blue square (hyp.) and (invert.),

Red drop : Red home :: Blue circle : Blue square (B. 5. pr. 22);
and, becauſe thſe magnitudes are proportionals, they are proportionals when taken jointly,

Red home + Red drop : Red drop :: Blue circle + Blue square : Blue circle (B. 5. pr. 18),
but Red drop : Black dome :: Blue circle : Yellow diamond (hyp.),

Red home + Red drop : Black dome :: Blue circle + Blue square : Yellow diamond (B. 5. pr. 22).

If the firſt, &c.

Proposition XXV. 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 four magnitudes Red home + Black dome , Blue square + Yellow diamond , Black dome , and Yellow diamond ,
of the ſame kind, be proportionals, that is to ſay,

Red home + Black dome : Blue square + Yellow diamond :: Black dome : Yellow diamond ,
and let Red home + Black dome be the greateſt of the four, and
conſequently by pr. A and 14 of Book 5, Yellow diamond is the leaſt;
then will Red home + Black dome + Yellow diamond be > Blue square + Yellow diamond + Black dome ;
becauſe Red home + Black dome : Blue square + Yellow diamond :: Black dome : Yellow diamond ,

Red home : Blue square :: Red home + Black dome : Blue square + Yellow diamond (B. 5. pr. 19),
but Red home + Black dome > Blue square + Yellow diamond (hyp.),

Red home > Blue square (B. 5. pr. A);
to each of theſe add Black dome + Yellow diamond ,
Red home + Black dome + Yellow diamond > Blue square + Black dome + Yellow diamond .

If four magnitudes, &c.

Definition X.

When three magnitudes are proportionals, the firſt is ſaid to have to the third the duplicate ratio of that which it has to the ſecond.

For example, if A, B, C, be continued proportionals, that is A : B :: B : C, A is ſaid to have to C the duplicate ratio of A : B;

or A / C = the ſquare of A / B .

This property will be more readily ſeen of the quantities

ar2, ar, a, for ar2 : ar :: ar : a;

and ar2 / a = r2 = the ſquare of ar2 / ar = r,

or of a, ar, ar2;

for a / ar2 = 1 / r2 = the ſquare of a / ar = 1 / r .

Definition XI.