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ELEMENTS OF EUCLID.
the greater is measured by the less, that is, « when the
III. • Ratio is a mutual relation of two magnitudes of the same see Note. • kind to one another, in respect of quantity.'
the second, which the third has to the fourth, when any
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.
the fifth definition, the multiple of the first is greater than
to have to the third the duplicate ratio of that which it has
is said to have to the fourth the triplicate ratio of that
Definition A, to wit, of compound ratio.
kind, the first is said to have to the last of them the ratio
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
E to F, G to H, and K to L.
N the same ratio which A has to D: then, for shortness'
to one another, as also the consequents to one another.
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
and it is inferred, that the second is to the first, as the fourth
it is inferred, that the excess of the first above second, is to
XVII. Convertendo, by conversion; when there are four proportion. als, and it is inferred, that the first is to its excess above the
Book V. second, as the third to its excess above the fourth. Prop. E,
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.'
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.
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.
are equal to one another.
* 4. Prop. lib. 2. Archimedis de spliæra et cylindro.
II. Those magnitudes of which the same, or equal magnitudes are equimultiples, are equal to one another.
multiple of another, is greater than that other magnitude.
PROP. I. THEOR.
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;
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
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