Reading Legendre's Elements of Geometry

[xwolfhunter]

In discussing proportions (a topic to which I have not been properly exposed) Legendre states that, adding the antecedent of a proportion to the consequent, and comparing the sum to the antecedent, one obtains a proportion equal to the original plus unity. Legendre's book is apparently notoriously obscure, but this statement seems both clear enough and wrong, at first impression, though I suspect it's because I'm misinterpreting what he's referring to with his terms.

The example he gives is:
$$4:6::12:18\\ 6+4:4::18+12:12\\ 10:4::30:12$$

When I was anticipating what would follow when I started reading this part of the text, I thought he would mean that given $4:6$ and obtaining $4:10$ one would have whatever the . . . comparator? . . . was in the first plus one in the second (1.5 in the first, 2.5 in the second, because there is one more antecedent in the consequent), but when he compared the sum to the antecedent ($10:4$) I thought he meant that the "simplified value" of the ratio would be increased from 2/3 to 5/3, which is obviously not the case. The movement of the antecedent to consequent and the sum to the antecedent in both parties of the proportion obviously still results in a proportion, but where the "unity" increase is, is not clear to me, unless he means the former interpretation by inverse (which he did not state).

I suppose I should quote the original text here:

If to the consequent of a ratio we add the antecedent, and compare this sum with the antecedent, this last will be contained once more than it was in the first consequent; the new ratio then will be equal to the primitive ratio increased by unity.

Perhaps it's just that I don't have a clear enough idea of what the value of a ratio is. Any help elucidating the statement would be appreciated.

 

[Samy_A]

What he is saying is that if $$\frac{a}{b}=\frac{c}{d}$$then $$\frac{a+b}{a}=\frac{c+d}{c}$$
and that $$\frac{a+b}{a}=1+\frac{b}{a}$$
(This last is probably what confuses: he is not referring to the original quotient with "primitive ratio".)

 

[xwolfhunter]

What he is saying is that if $$\frac{a}{b}=\frac{c}{d}$$then $$\frac{a+b}{a}=\frac{c+d}{c}$$
and that $$\frac{a+b}{a}=1+\frac{b}{a}$$
(This last is probably what confuses: he is not referring to the original quotient with "primitive ratio".)

Ahhhh, okay. So by "primitive" he means "irreducible", and that's what should have tipped me off. Right?

Thanks, now I can move on.

Man, I totally should have seen that. Disappointing.