THE

ELEMENTS OF EUCLID.

BOOK V.

DEFINITIONS. "

a

1.
A LESS magnitude is said to be a part of a greater mag-
nitude, 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 numbor of times ex-
.actly.

III. • Ratio is a mutual relation of two magnitudes of the same see Note. • kind to one another, in respect of quantity.'

IV.
Magnitudes are said to have a ratio to one another, when the
less can be multiplied so as to exceed the other.

V. .
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 ta-
ken, 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.

.

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Book V.

that of the fourth ; or, if the multiple of the first be greatet than that of the second, the multiple of the third is also greater than that of the fourth.

VI. Magnitudes which have the same ratio are called proportionals. N. B. "When four magnitudes are proportionals,

it' is usually expressed by saying, the first is to the se"cond, as the third to the fourth.

VII.
İVhen 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 fourtlı; then the first is
said to have to the second a greàter ratio than the third mag-
nitude has to the fourth ; and, on the contrary, the third is
said to have to the fourth a less ratio than the first has to
second.

VIII.
“Analogy, or proportion, is the similitude of ratios.”

IX.
Proportion consists in three terñs at least:

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.

XI.
When fous 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, &e.
increasing the denomination still by unity, in any number
of proportionals.

Definition A, to wit, of 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. 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 Á 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 have to B the same ratio which I has to l'; and B Book

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.
În likc manner, the same things being supposed, if I have 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 tó F, G to H, and K to L.

XII.
In proportionals, the antecedent terms are called homologous

to one another, as also the consequents to one another.
Geometers make use of the following technical words to sig.

nify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals.'

XIII. Permutando, or alternando, by permutation, or alternately; this See Note

word 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 to the fourth; as is shown in the 16th prop.
of this 5th book.

XIV.
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. Prop. B, book 5.

XV.
Componendo, by composition; when there are four proportion-
als, and it is inferred, that the first, together with the se-
cond, is to the second, as the third together with the fourth,
is to the fourth. 18th Prop. book 5.

XVI.
Dividendo, by division; when there are four proportionals, and

it is inferred, that the excess of the first above second, is to
the second, as the excess of the third above the fourth, is to
the fourth. 17th. prop. book 5.

XVII. Convertendo, by conversion; when there are four proportion. als, and it is inferred, that the first is to its excess above the

Q

Book V. second, as the third to its excess above the fourth. Prop. E,

book 5.

XVIII.

Ex æquali (sc. distantia), or ex æquo, from equality of dis

tance; when there is any number of magnitudes more than two, and as many others, so 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.'

XIX,

Ex æquali, from cquality; 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 the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in 23d prop. book 5.

XX.

Ex æquali, in proportione perturbata, seu inordinata ; 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 a cross order: and the inference is as in the 18th definition. It is demonstrated in tlie 23d

prop. of book 5.

AXIOMS.

I.
EQUIMULTIPLES of the saine, or of equal magnitudes,

are equal to one another.

* 4. Prop. lib. 2. Archimedis de spliæra et cylindro.

Bool V.

II. Those magnitudes of which the same, or equal magnitudes are equimultiples, are equal to one another.

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

IV.
That magnitude of which a multiple is greater than the same

multiple of another, is greater than that other magnitude.

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54

PROP. I. THEOR.

IF

any number of magnitudes be equimultiples of as many, each of each; what multiple soever any one. of them is of its part, the same multiple shall all the first magnitudes be of all the other.

Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each; whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together.

Because AB is the same multiple of Ethat CD is of F, as many magnitudes as are in AB equal to E, sd many are there in CD equal to F.. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into A CH, HD equal each of them to F: the numbértherefore of the magnitudes CH,HD shall be equal to the number of the others AG,GB;

G

E and because AG equal to E, and CH to F, therefore AG and CH together are equal

B to a É and F together; for the same reason,

& Ax, 2. 5. because GB is equal to E, and HD to F; GB and HD together are equal to E and F to- с gether. Wherefore, as many magnitudes as are in AB equal to E, so many are there in

H
AB, CD together equal to E and F together.
Therefore, whatsoever multiple AB is of E,
the same multiple is AB and CD together of D
E and F together.

Therefore, if any magnitudes, how many soever, be equi. multiples of as many, each of each, whatsoever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other: "For the same demonstration