Commentary on Euclid I, Proposition 3
Two unequal right lines being given, from the greater to cut off a part equal to the less.
This third problem, likewise, has a variety of cases. For the given unequal right lines are either mutually distant from each other, as with the institutor of the elements, or they are united according to one extreme; or the one cuts the other according to one of its extremities, and this in a two-fold manner. For either the greater cuts the less, or the less the greater. But if they are united according to one extreme, the demonstration is manifest. For employing the common extremity as a centre, and the lesser of the lines for an interval, you will describe a circle, and cut off from the greater, a part equal to the less; since as much as the circle intercepts within itself, will be equal to the less. But if the one cuts the other according to its extreme, either the greater will cut the greater, or the contrary. And if they mutually cut each other, they will either be mutually cut into equal parts, or into unequal; or the one will be cut into equal, and the other into unequal parts, and this in a two-fold respect. For all these present us with an admirable variety of exercise, some of which, out of a many, we shall exhibit. Let there be given the unequal right lines a b, c d, the greater of which is c d, and let it cut a b in one of its extremities c; then with the centre a, but interval a b, let a circle b f be described, and let an equilateral triangle a e c be constructed upon a c, and produce e a, e c. Again, with the centre e, but interval e f, let the circle g f be described; and with the centre c, and interval c g, the circle g l. Because therefore, e f is equal to e g (for the centre is e) of which e a is equal to e c, the remainder a f, shall be equal to the remainder c g. But a f is likewise equal to a b; for the centre is a. Hence, c g will be equal to a b, and this is equal to c l, for the centre is the point c: a b, therefore, is equal to c l, which was required to be done.
But let c d be less than a b, and let it cut a b according to its extremity c; either, therefore, it will cut it in the middle, or not in the middle. Let it in the first place cut it in the middle; c d, therefore, is either the half of a b, and a c is equal to c d, or it is less than half. And in this case with the centre c, and interval c d, describe a circle, and you will cut off from a b a part equal to c d: Or it is greater than half; and then at the point a, placing a f, equal to c d, and describing a circle with the centre a, and interval a f, you will cut off from a b a part equal to a f, that is to c d. But if c d does not cut a b in the middle, c d shall either be its half, or greater than the half, or less. If therefore c d is the half, or less than the half of a b, employing c as a centre, and c d as an interval, you will cut off from a b, a part equal to c d, as was required to be done. But if c d is greater than the half, again at the point a placing a f equal to c d, you will accomplish the same. For with the centre a, but interval a f, you will describe a circle, cutting off from a b a line equal to a f, that is, to c d. But if they mutually intersect, as c d, a b, then with the centre b, but interval b a, describe the circle a f, and let b c be extended to the point f. Because therefore, b f, c d, are the two unequal right lines, and c d cuts b f, according to its extremity, it is possible from c d to make a line equal to b f; for this has been shewn in the first case of this problem. It is therefore possible, that a line equal to a b may be cut off from c d; for a b and b f are mutually equal. Having, therefore, received these cases from division, we have endeavoured to exhibit their variety. But the demonstration of the elementary institutor is admirable, since it accords with all the preceding constructions.
And it is possible, in every position, at the extremity of the greater, to place a line equal to the less, and using the same extreme as a centre, and placing the interval to describe a circle, which shall cut off from the greater, a line equal to the less, whether they mutually intersect, or one cuts the other, or they are constituted in a still different position.