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Book IV.

And in the same manner as was done in the pentagon, if, through the points of division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilate. ral and cquiangular quindecagon shall be described about it: And likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circum. Icribed about it.

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I.
Lefs magnitude is said to be a part of a greater magni.
tude, 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 a mutual relation of two magnitudes of the fame See N. • 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 e-
quimultiples 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 ;
H4

of,

Book V. or, if the multiple of the first be greater than that of tho

second, the multiple of the third is also greater than that of
the fourth.

VI.
Magnitudes which have the fame ratio are called proportionals,
N. B. - When four magnitudes are proportionals, it is usu-

ally 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 fecond 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.

XI. See X. When four magnitudes are continual proportionals, the first is

said to bave to the fourth the triplicate ratio of that which it
has to the second, and so on, quadruplicate, &c. increasing
the denomination still by unity, in any number of propor.
tionals.

Definition A, to wit, of compound ratio.
When there are any number of magnitudes of the fame kind,

the first is said to have to the last of them the ratio com-
pounded of the ratio which the firt 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 com-
pounded 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 w B, B to C, and C to D:

And

And if A has to B, the fame ratio which E has to F; and B Book V.

to C, the same ratio that G has to H; and C to D, the same
that K has to L; then, by chis definition, A is said to have
to 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 fame
thing is to be understood when it is more briefly expressed, by
saying A has to D the ratio compounded of the ratios of É
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 fake,
Mis 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 homologous to

one another, as also the consequents to one another.
Geometers make use of the following technical words to lig-
• nify certain ways of changing either the order or magni-
'tude of proportionals, so as that they continue still to be
proportionals,

XIII. Permutando, or alternando, by permutation, or alternately ; See N,

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 thewn 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 fe•
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 excefs 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 proportion-

als,

Book V. als, and it is inferred, that the first is to its excefs above the

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

book 5.

XVIII.
Ex aequali (fc. distantia), or ex aequo, from equality of di-

stance ; 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 aequali, from equality; this term is used fimply 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 le-
cond 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 preceeding definition ; whence this is
called ordinate proportion. It is demonstrated in 22d prop.

book 5

XX.
Ex aequali, in proportione perturbata, seu inordinata ; from e-

quality, 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 23. prop. of book 5.

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E

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

II. Those

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