Commentary on Proposition VI.9

Back to Proposition VI.9.

Simson observes that “this is demonstrated in a particular case, viz. that in which the third part of a straight line is required to be cut off; which is not at all like Euclid’s manner. Besides, the author of that demonstration, from four magnitudes being proportionals, concludes that the third of them is the same multiple of the fourth which the first is of the second; now this is nowhere demonstrated in the 5th book, as we now have it; but the editor assumes it from the confused notion which the vulgar have of proportionals.”

The truth of the assumption referred to is proved by Simson in his proposition D given above (p. 128); hence he is able to supply a general and legitimate proof of the present proposition.

“Let AB be the given straight line; it is required to cut off any part from it.

From the point A draw a straight line AC making any angle with AB; in AC take any point D, and take AC the same multiple of AD that AB is of the part which is to be cut off from it;

A E D B C

join BC, and draw DE parallel to it: then AE is the part required to be cut off.

Because ED is parallel to one of the sides of the triangle ABC, viz. to BC,

[VI. 2]as CD is to DA, so is BE to EA,

and, componendo,

[V. 18]CA is to AD, as BA to AE.

But CA is a multiple of AD;

[Prop. D]therefore BA is the same multiple of AE.
Whatever part therefore AD is of AC, AE is the same part of AB; wherefore from the straight line AB the part required is cut off.”

The use of Simson’s Prop. D can be avoided, as noted by Camerer after Baermann, in the following way. We first prove, as above, that

CA is to AD as BA is to AE.

Then we infer that, alternately,

[V. 16]CA is to BA as AD to AE.

But AD is to AE as n · AD to n · AE (where $n$ is the number of times that AD is contained in AC); [V. 15]

[V. 11]whence AC is to AB as n · AD is to n · AE.

In this proportion the first term is equal to the third; therefore [V. 14]

the second is equal to the fourth,

so that AB is equal to $n$ times AE.

Prop. 9 is of course only a particular case of Prop. 10.