Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/161

Rh Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken, two and two.

19. Ex aequali. This term is used simply by itself, when the first magnitude is to the second of the first rank, as the first is to the second of the other rank; and the second is to the third of the first rank, as the second is to the third of the other ; and so on in order ; and the inference is that mentioned in the preceding definition. V. 22.

20. Ex aequali 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 Ihe first rank, as the last but one is to the last of the second rank ; and the second is to the third of the first rank, as the last but two is to the last but one of the second rank ; and the third is to the fourth of the first rank, as the last but three is to the last but two of the second rank ; and so on in a crogs order ; and the inference is that mentioned in the eighteenth definition. V. 23.

AXIOMS.

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

2. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another.

3. A multiple of a greater magnitude is greater than the same multiple of a less.

4. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude.