The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...

PROPOSITION VIII. THEOREM.

THe greatest of Two Magnitudes, hath a greater Reason to the same third, than the lesser; and the same Third hath a lesser Reason to the greater Magnitude than it hath to the lesser.

I suppose that one should compare the Magnitudes AB and C, with the same third EF;

[illustration] diagram of lines AB, C, and EF

and that AB be greater than C. I say that there is a greater Reason of

AB to EF, than of C to EF. Let AD be taken equal to C: and let EF be divided in the middle, and the half again in the middle, and so continuedly untill such time one find GF, an Aliquot part of EF, to be less than DB.

Demonstration. AD, and C are equal; thence there is the same Reason of AD to EF, as there is of C to EF (by the 7th.) and (by the 4th. Definition) AD shall contain as many times GF an Aliquot part of EF, as C containeth the same. Now AB containeth the same once more, seeing DB is greater than GF: whence (by the 5th. Definition.) the Reason of AB to EF, is greater than that of C to the same third EF.

Secondly I say, that EF hath a lesser Reason to AB, than to the Magnitude C. Let there be taken any Aliquot part of C, for Example the one Fourth, as many times as it can be taken in EF. Let us suppose that it be found therein five times, either it will leave something of the Magnitude EF: after having been taken five times. Or will leave nothing, that is to say, it will measure exactly EF, it is evident that five fourths of the Magnitude AB will be a greater Line, than the Fourth of C taken five times:

so that it cannot be found to be five times in EF. And if the fourth of C taken five times, falleth short as in G: either the fourth of AB taken five times, will reach to F; or fall short in I. If it rea∣cheth to F; there will be the same Reason of EF to AB, as of EG to C: by the preceding Argument, EF to C hath a greater Reason to C, than to AB. That if the Quarter of AB taken five times reacheth to I; there will the same Reason of EI to AB, as of EG and C. Now EI, or EF hath a greater Reason than EG to C. Therefore EF to C hath a greater Reason than the same EF to AB.