The English works of Thomas Hobbes of Malmesbury, Volume 01 (of 11)
Thomas Hobbes · en
22. If there be four quantities, and the proportion of the first to the
second be duplicate of the proportion of the third to the fourth, the
product of the extremes to the product of the means, will be as the
third to the fourth. Let the four quantities be A, B, C and D; and let
the proportion of A to B be duplicate of the proportion of C to D, I say
A D, that is, the product of A into D is to B C, that is, to the product
of the means, as C to D. For seeing the proportion of A to B is
duplicate of the proportion of C to D, if it be as C to D, so D to
another, E, then A. B :: C. E will be proportionals; for the proportion
of A to B is by supposition duplicate of the proportion of C to D; and C
to E duplicate also of that of C to D by the definition, art. 15.
Wherefore, by the last article, A E or A into E is equal to B C or B
into C; but, by coroll. IV. art. 6, A D is to A E as D to E, that is, as
C to D; and therefore A D is to B C, which as I have shown is equal to A
E, as C to D; which was to be proved.
Moreover, if the proportion of the first A to the second B be triplicate
of the proportion of the third C to the fourth D, the product of the
extremes to the product of the means will be duplicate of the proportion
of the third to the fourth. For if it be as C to D so D to E, and again,
as D to E so E to another, F, then the proportion of C to F will be
triplicate of the proportion of C to D; and consequently, A. B :: C. F
will be proportionals, and A F equal to B C. But as A D to A F, so is D
to F; and therefore, also, as A D to B C, so D to F, that is, so C to E;
but the proportion of C to E is duplicate of the proportion of C to D;
wherefore, also, the proportion of A D to B C is duplicate of that of C
to D, as was propounded.
23. If there be four proportionals, and a mean be interposed betwixt the
first and second, and another betwixt the third and fourth, the first of
these means will be to the second, as the first of the proportionals is
to the third, or as the second of them is to the fourth. For let A. B ::
C. D be proportionals, and let E be a mean betwixt A and B, and F a mean
betwixt C and D; I say A. C :: E. F are proportionals. For the
proportion of A to E is subduplicate of the proportion of A to B, or of
C to D. Also, the proportion of C to F is subduplicate of that of C to
D; and therefore A. E :: C. F are proportionals; and by permutation A. C
:: E. F are also proportionals; which was to be proved.