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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 etc. etc. and 1 / 2 of A = 1 / 2 of B; 1 / 3 of A = 1 / 3 of B; etc. etc.

II.

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

Let A > B, then 2 A > 2 B; 3 A > 3 B; 4 A > 4 B; etc. etc.

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. etc. etc.

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, etc.

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, etc.

Proposition III. Theorem.

If the first of 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, etc.

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

etc.

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

etc.

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

etc.

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

etc.

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 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 Red circle >, = or < m Yellow square , then M Blue diamond >, = or < 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 demonstrations 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 , etc. 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 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, etc.

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, etc.

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, etc.

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, etc.

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, etc.

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, etc.

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, etc.

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, etc.

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, but 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, etc.

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 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.) Blue diamond : Yellow square :: Red circle : Yellow square ,
therefore, by the first case, Blue diamond = Red circle .

∴ Magnitudes which have the same ratio, etc.

This may be shown otherwise, as follows:—

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 < Red circle , 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, etc.

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, etc.

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 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, etc.

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 because 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, etc.

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, etc.

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 generally applicable, we have

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

∴ Magnitudes have the same ratio, etc.

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 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, etc. 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.

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, etc.

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, etc.

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 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, etc.

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, etc.

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, etc.

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 proportion. 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, etc.

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 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, etc.

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, etc.

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 : Blue drop , 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, etc.

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 have 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, etc.

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, etc.

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.

When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second; and so on, quadruplicate, etc. increasing the denomination still by unity, in any number of proportionals.

For example, let A, B, C, D, be four continued proportionals, that is, A : B :: B : C :: C : D; A said to have to D, the triplicate ratio of A to B;

or A / D = the cube of A / B .

This definition will be better understood and applied to a greater number of magnitudes than four that are continued proportionals, as follows:—

Let ar3, ar2, ar, a, be four magnitudes in continued proportion,
that is, ar3 : ar2 :: ar2 : ar :: ar : a,
then ar3 / a = r3 = the cube of ar3 / ar2 = r.

Or, let ar5, ar4, ar3, ar2, ar, a, be six magnitudes in proportion, that is

ar5 : ar4 :: ar4 : ar3 :: ar3 : ar2 :: ar2 : ar :: ar : a,
then the ratio ar5 / a = r5 = the fifth power of ar5 / ar4 = r.

Or, let a, ar, ar2, ar3, ar4, be five magnitudes in continued proportion; then a / ar4 = 1 / r4 = the fourth power of a / ar = 1 / r .

Definition A.

To know a compound ratio:—

When there are any number of magnitudes of the same kind, the first is said to have to the last of them the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth; and so on, unto the last magnitude.

A B C D
E F G H K L
M N

For example, if A, B, C, D, be four magnitudes of the same kind, the first A is said to have to the last D the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D.

And if A has to B the same ratio which E has to F, and B to C the same ratio that G has to H, and C to D the same that K has to L; then by this definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L. And the same thing is to be understood when it is more briefly expressed by saying, A has to D the ratio compounded of the ratios of E to F, G to H, and K to L.

In like manner, the same things being supposed; if M has to N the same ratio which A has to D, then for shortness sake, M is said to have to N the ratio compounded of the ratios of E to F, G to H, and K to L.

This definition may be better understood from an arithmetical or algebraical illustration; for, in fact, a ratio compounded of several other ratios, is nothing more than a ratio which has four its antecedent the continued product of all the antecedents of the ratios compounded, and for its consequent the continued product of all the consequents of the ratios compounded.

Thus, the ratio compounded of the ratios of
2 : 3, 4 : 7, 6 : 11, 2: 5,
is the ratio of 2 × 4 × 6 × 2 : 3 × 7 × 11 × 5,
or the ratio of 96 : 1155, or 32: 385.

And of the magnitudes A, B, C, D, E, F, of the same kind, A : F is the ratio compounded of the ratios of

A : B, B : C, C : D, D : E, E : F;
for A × B × C × D × E : B × C × D × E × F,
or A × B × C × D × E / B × C × D × E × F = A / F or the ratio of A : F.

Proposition F. Theorem.

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

Let A : B :: F : G, B : C :: G : H, C : D :: H : K, and D : E :: K : L.
A B C D E
F G H K L

Then, the ratio which is compounded of the ratios of A : B, B : C, C : D, D : E, or the ratio of A : E, is the same as the ratio compounded of the ratios of F : G, G : H, H : K, K : L, or the ratio of F : L.

For A / B = F / G , B / C = G / H , C / D = H / K , D / E = K / L ;

∴ A × B × C × D / B × C × D × E = F × G × H × K / G × H × K × L

and ∴ A / E = F / L ,
or the ratio of A : E is the same as the ratio of F : L.

The same may be demonstrated of any number of ratios so circumstanced.

Next, let A : B :: K : L, B : C :: H : K, C : D :: G : H, D : E :: F : G.

Then the ratio which is compounded of the ratios of A : B, B : C, C : D, D : E, or the ratio of A : E, is the same as the ratio compounded of the ratios of K : L, H : K, G : H, F : G, or the ratio of F : L.

For A / B = K / L , B / C = H / K , C / D = G / H , and D / E = F / G ;

∴ A × B × C × D / B × C × D × E = K × H × G × F / L × K × H × G

and ∴ A / E = F / L ,
or the ratio of A : E is the same as the ratio of F : L.

∴ Ratios which are compounded, etc.

Proposition G. Theorem.

If several ratios be the same to several ratios, each to each, the ratio which is compounded of ratios which are the same to the first ratios, each to each, shall be the same to the ratio compounded of ratios which are the same to the other ratios, each to each.

A B C D E F G H

a b c d e f g h

P Q R S T

V W X Y Z

If A : B :: a : b C : D :: c : d E : F :: e : f and G : H :: g : h
and A : B :: P : Q C : D :: Q : R E : F :: R : S G : H :: S : T
a : b :: V : W c : d :: W : X e : f :: X : Y g : h :: Y : Z

then P : T = V : Z.

For P/Q = A/B = a/b = V/W , Q/R = C/D = c/d = W/X , R/S = E/F = e/f = X/Y , S/T = G/H = g/h = Y/Z ;

and ∴ P × Q × R × S / Q × R × S × T = V × W × X × Y / W × X × Y × Z ,

and ∴ P / T = V / Z ,

or P : T = V : Z.

∴ If several ratios, etc.

Proposition H. Theorem.

If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios; and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is compounded of several of them; then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall be the same to the remaining ratio of the last, or if there be more than one, to the ratio compounded of these remaining ratios.

A B C D E F G H
P Q R S T X

Let A : B, B : C, C : D, D : E, E : F, F : G, G : H, be the first ratios, and P : Q, Q : R, R : S, S : T, T : X, the other ratios; also, let A : H, which is compounded of the first ratios, be the same as the ratio of P : X, which is the ratio compounded of the other ratios; and let the ratio of A : E, which is compounded of the ratios of A : B, B : C, C : D, D : E, be the same as the ratio of P : R, which is compounded of the ratios P : Q, Q : R.

Then the ratio which is compounded of the remaining first ratios, that is, the ratio compounded of the ratios E : F, F : G, G : H, that is the ratio of E : H, shall be the same as the ratio of R : X, which is compounded of the ratios of R : S, S : T, T : X, the remaining other ratios.

Because A × B × C × D × E × F × G / B × C × D × E × F × G × H = P × Q × R × S × T / Q × R × S × T × X ,

or A × B × C × D / B × C × D × E × E × F × G / F × G × H = P × Q / Q × R × R × S × T / S × T × X ,

and A × B × C × D / B × C × D × E = P × Q / Q × R ,

∴ E × F × G / F × G × H = R × S × T / S × T × X ,

∴ E / H = R / X ,

∴E : H = R : X.

∴ If a ratio which, etc.

Proposition K. Theorem.

If there be any number of ratios, and any number of other ratios, such that the ratio which is compounded of ratios, which are the same to the first ratios, each to each, is the same to the ratio which is compounded of ratios, which are the same, each to each, to the last ratios—and if one of the first ratios, or the ratio which is compounded of ratios, which are the same to several of the first ratios, each to each, be the same to one of the last ratios, or to the ratio which is compounded of ratios, which are the same, each to each, to several of the last ratios—then the remaining ratio of the first; or, if there be more than one, the ratio which is compounded of ratios, which are the same, each to each, to the remaining ratios of the first, shall be the same to the remaining ratio of the last; or, if there be more than one, to the ratio which is compounded of ratios, which are the same, each to each, to these remaining ratios.

h k m n s

A B, C D, E F, G H, K L, M N,

O P, Q R, S T, V W, X Y,

a b c d e f g

a b c d e f g

h k l m n p

Let A : B, C : D, E : F, G : H, K : L, M : N, be the first ratios, and O : P, Q : R, S : T, V : W, X : Y, the other ratios;

and let A : B = a : b, C : D = b : c, E : F = c : d, G : H = d : e, K : L = e : f, M : N = f : g.

Then, by the definition of a compound ratio, the ratio of a : g is compounded of the ratios of a : b, b : c, c : d, d : e, e : f, f : g, which are the same as the ratio of A : B, C : D, E : F, G : H, K : L, M : N, each to each.

Also, O : P = h : k, Q : R = k : l, S : T = l : m, V : W = m : n, X : Y = n : p.

Then will the ratio of h : p be the ratio compounded of the ratios h : k, k : l, l : m, m : n, n : p, which are the same ratios of O : P, Q : R, S : T, V : W, X : Y, each to each.

∴ by the hypothesis, a : g = h : p.

Also, let the ratio which is compounded of the ratios of A : B, C : D, two of the first ratios (or the ratios of a : c, for A : B = a : b, and C : D = b : c), be the same as the ratio of a : d, which is compounded of the ratios a : b, b : c, c : d, which are the same as the ratios of O : P, Q : R, S : T, three of the other ratios.

And let the ratios of h : s, which is compounded of the ratios h : k, k : m, m : n, n : s, which are the same as the remaining first ratios, namely, E : F, G : H, K : L, M : N; also, let the ratio of e : g, be that which is compounded of the ratios e : f, f : g, which are the same, each to each, to the remaining other ratios, namely, V : W, X : Y. Then the ratio of h : s shall be the same as the ratio of e : g; or h : s = e : g.

For A × C × E × G × K × M / B × D × F × H × L × N = a × b × c × d × e × f / b × c × d × e × f × g ,

and O × Q × S × V × X / P × R × T × W × Y = h × k × l × m × n / k × l × m × n × p ,

by the composition of the ratios;

∴ a × b × c × d × e × f / b × c × d × e × f × g , = h × k × l × m × n / k × l × m × n × p , (hyp.),

or a × b / b × c × c × d × e × f / d × e × f × g = h × k × l / k × l × m × m × n / n × p ,

but a × b / b × c = A × C / B × D = O × Q × S / P × R × T = a × b × c / b × c × d = h × k × l / k × l × m ;

∴ c × d × e × f / d × e × f × g = m × n / n × p .

And c × d × e × f / d × e × f × g = h × k × m × n / k × m × n × s (hyp.),

and m × n / n × p = e × f / f × g (hyp.),

∴ h × k × m × n / k × m × n × s = e f / f g ,

∴ h / s = e / g ,

∴ h : s = e : g.

∴ If there be any number, etc.

Algebraical and Arithmetical expositions of the Fifth Book of Euclid are given in Byrne’s Doctrine of Proportion; Published by Williams and Co. London. 1841.

Books.

  1. Book I. Basic plane geometry
  2. Book II. Geometric algebra
  3. Book III. Circles and angles
  4. Book IV. Regular polygons
  5. Book V. Ratios and proportions
  6. Book VI. Geometric proportions