THE

ELEMENTS OF EUCLID.

BOOK V.

DEFINITIONS.

I.

A less magnitude is said to be a part 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 6 of times exactly.

III.

« Ratio is a mutual relation of two magnitudes of the See N.

“ same 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 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.

VI.
Magnitudes which have the same ratio are called pro-
portionals. • N.B. When four magnitudes are pro-

portionals, it is usually expressed by saying, the
• first is to the second, as the third to the fourth.'

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 to the fourth
a less ratio than the first has to the second.

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

IX,
Proportion consists in three terms 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.

See N.

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, qua-
druplicate, &c. 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 nas 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 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.

XII.

In proportionals, the antecedent terms are called homo

logous to one another, as also the consequents to one

another. • Geometers make use of the following technical words, 'to signify certain ways of changing either the order

or magnitude of proportionals, so that they continue 6 still to be proportionals.'

XIII.

Permutando, or alternando, by permutation or alter- See N.

nately. This 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 shewn in the 16th Prop. of this fifth Book.

XIV.

Invertendo, by inversion; when there are four propor

tionals, 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

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. 18th Prop. Book 5.

XVI. Dividendo, by division; when there are four propor

tionals, 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. 17th Prop. Book 5.

XVII.

Convertendo, by conversion; when there are four pro

portionals, and it is inferred, that the first is to its excess above the second, as the third to its excess above the fourth. Prop. E. Book 5.

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.'

XIX.

Ex æquali, from equality. This term is used simply by so on in order: and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in the 22d Prop. Book 5.

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 a cross order : and the inference is as in the 18th definition. It is demonstrated in the 23rd Prop. of Book 5.

AXIOMS.

I.

EQUIMULTIPLES of the same, or of equal magnitudes, , are equal to one another.

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.

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 equimul

4 Prop. lib. 2. Archimedis de sphærâ et cylindro.