Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/160

136 And the same thing is to be understood when it is more briefly expressed by saying, A has to D the ratio com- pounded of the ratios of E 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 that A has to D ; then, for the sake of shortness, M is said to have to N the ratio com- pounded of the ratios of E to F, G to H, and K to L.

12. In proportionals, the antecedent terms are said to be homologous to one another ; as also the consequents to one another.

Geometers make use of the following technical words, to signify certain ways of changing either the order or the magnitude of proportionals, so that they continue still to be proportionals.

13. Permutando, or alternando, by permutation or alternately; when there are four proportionals, and it is inferred that the first is to the third, as the second is to the fourth. V. 16.

14. Invertendo, by inversion; when there are four proportionals, and it is inferred, that the second is to the first as the fourth is to the third. V. B.

15. Componendo, by composition ; when there are four proportionals, and it is inferred, that the first together with the second, is to the second, as the third together with the fourth, is to the fourth. V. 18.

16. Dividendo, by division ; when there are four pro- portionals, and it is inferred, that the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth. V. 17.

17. Convertendo, by conversion; when there are four proportionals, and it is inferred, that the first is to its excess above the second, as the third is to its excess above the fourth. V. E.

18. Ex aequali distantia, or ex aequo, from equality of distance ; when there is any number of magnitudes more than two, and as many others, such that they are propor- tionals 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.