Commentary on Proposition V.25¶
Back to Proposition V.25.
Algebraically, if
$$a : b = c : d\text{,}$$
and $a$ is the greatest of the four magnitudes and $d$ the least,
$$a + d > b + c.$$
Simson is right in inserting a word in the setting-out, “let AB be the greatest of them and <consequently> F the least.” This follows from the particular case, really included in Def. 5, which Simson makes the subject of his proposition A, the case namely where the equimultiples taken are once the several magnitudes.
The proof is as follows.
Since
$$a : b = c : d\text{,}$$
$$a - c : b - d = a : b. \qquad [\text{V. } 19]$$
But $a > b$; therefore $(a - c) > (b - d)$. [V. 16 and 14]
Add to each $(c + d)$; therefore
There is an important particular case of this proposition, which is, however, not mentioned here, viz. the case where $b = c$. The result shows, in this case, that the arithmetic mean between two magnitudes is greater than their geometric mean. The truth of this is proved for straight lines in VI. 27 by “geometrical algebra,” and the theorem forms the διορισμός for equations of the second degree.
Simson adds at the end of Book v. four propositions, F, G, H, K, which, however, do not seem to be of sufficient practical use to justify their inclusion here. But he adds at the end of his notes to the Book the following paragraph which deserves quotation word for word.
“The 5th book being thus corrected, I most readily agree to what the learned Dr Barrow says, ‘that there is nothing in the whole body of the elements of a more subtile invention, nothing more solidly established, and more accurately handled than the doctrine of proportionals.’ And there is some ground to hope that geometers will think that this could not have been said with as good reason, since Theon’s time till the present.”
Simson’s claim herein will readily be admitted by all readers who are competent to form a judgment upon his criticisms and elucidations of Book v.