Commentary on Euclid I, Proposition 9
To bisect a given rectilineal angle.
Our author mingles theorems with problems, and connects problems with theorems, and through both completes the whole of his elementary institution, comparing as well subjects as the symptoms subsisting about subjects themselves. Since, therefore, he had shewn in the preceding propositions, both in one triangle, from the equality of the sides, the consequent equality of the angles, and the contrary: and in a similar manner in two triangles, with this exception, that the mode of conversion in one and two triangles is different, he now passes to problems, and orders us to bisect a rectilineal angle. And it is manifest, that the angle here is given according to form: for it is called right-lined, and not of any kind whatever. Indeed, we cannot bisect every angle by the elementary institution; since it is doubtful whether every triangle can be bisected. For, perhaps, you may doubt whether it is possible to bisect a cornicular angle. But the ratio of the section is also distinguished in this problem, and this again not in vain. For to divide an angle in any given ratio, transcends the present construction: as, for example, into three, four, or five equal parts. Indeed, to trisect a right angle is possible, by employing a few of the propositions which are afterwards delivered: but this cannot be effected in an acute angle, without passing on to other lines of a mixt species. And this is manifested by the geometricians who propose to trisect a given rectilineal angle. For Nichomedes, indeed, from conchoidal lines, the origin, order, and symptoms of which, he delivers, as he was the inventor of their properties, trisects every right-lined angle. But others effect this from the quadrantal lines of Hippias and Nichomedes, by employing mixt quadrantal lines. Others, again, being incited from the Helices of Archimedes, divide a given rectilineal angle, in a given ratio. But the consideration of these, because difficult to learners, we shall for the present omit; as it will, perhaps, be more convenient to examine this in the third book, where the institutor of the Elements bisects a given circumference. For there the same mode of enquiry presents itself with respect not only to bisection, but also trisection; and the ancients endeavoured, by employing the same lines, to divide every circumference into three equal parts. With great propriety, therefore, he who only mentions a right line and a circumference, alone bisects a right angle and a circumference.
But conceiving that the species composed from these, through mixture, are difficult to explain and enumerate, without a curious examination, he omits all such enquiries as involve mixt lines in their consideration, and proposes to investigate in first and simple forms alone, such things as can either be produced or considered from these. And such, indeed, is the proposition of the present problem, to bisect a given right lined angle. For in the construction of this he uses one petition, and the first and third problem: but in the demonstration he employs the eighth theorem alone. Since problems entirely require demonstration (as we have already observed) and through this they obtain a power of producing science. But perhaps, some may oppose the geometrician, by asserting that an equilateral triangle may be constituted by him, not having its vertex within the two right lines, but either upon, or external to each; and that this may be manifested by the elements. For let there be an angle b a c, which it is required to bisect. Then let b a be taken equal to a c, and let b c be connected, and upon it, let an equilateral triangle b c d be constructed. This point d, therefore, is either within the right lines a b, a c, or upon a b, or a c, or external to both. Now the institutor of the Elements assumes them within; and hence, those who oppose the demonstration, will say the point is either placed on one of the right lines, or external to both. Let the point d then be placed on the line a b, so that the triangle b c d may be equilateral: d b, therefore, is equal to d c, and the angles at the base c b d, b c d, are equal. Hence, the whole, b c e, is greater than the angle c b d. Again, because a b, c a, are equal, the triangle a b c, is isosceles, and the angles under the base b c, will be equal. The angle, therefore, b c e, is equal to the angle c b d. But it was also greater, which is impossible. Hence, the vertex of the equilateral triangle cannot be in the right line a b d. In like manner we may shew that it cannot be in the right line a c e. Let it therefore, if possible be placed externally. Because, then b d is equal to c d, the angles at the base are equal, viz. b c d, and c b d. Hence, the angle b c d, is greater than the angle c b f.
Much more, therefore, is the angle b c e, greater than c b f: but it is also equal, because these angles are under the base b c, of an isosceles triangle a b c, and this is impossible. Hence, the point cannot fall in these parts external to the two right lines; and it may be similarly shewn that this is impossible in other parts. Here too you may again observe, that we destroy objections by using the second part of the fifth proposition, that the angles under the base of an isosceles triangle are equal. And this is what we have previously observed, that many things opposing science, are shewn to be debile, and easy of confutation, by the assistance of this theorem; and that such is the utility it affords to geometry.
But if any one should say that there is no place under the base, and yet that it is requisite to constitute the equilateral triangle at the same parts, in which the lines b a, a c, are situated; it will be necessary that the lines which are constituted should either coincide with b a, a c, if they also are equal to the base c b: or that they should fall external to them, if they are less than the base b c: or within, if b a, a c, are greater than b c. Let them, in the first place, coincide, and let b a c be an equilateral triangle, and let there be taken in the side a b, the point d, and make a e in the side a c, equal to a d, and connect the lines d e, b e, c d, a f. Because, therefore, a b is equal to a c, and a d to a e, the two b a, a e, are equal to the two c a, a d, and they comprehend the same angle. Hence, they are all equal to all, and the angle d b e, is equal to the angle e c d. But d b is also equal to e c, and b e to c d. All, therefore, are equal to all. Hence, the angle d e b, is equal to the angle e d c: for they subtend equal sides. And d f is equal to e f, (by the sixth.) Because, therefore, a e is equal to a d, and a f is common, and the base d f, is equal to the base e f, the angle d a e is bisected, which was required to be done.
But if the sides of the equilateral triangle fall external to the right lines b a, a c, let them be b d, d c, and having connected d a, let it be produced to the point e. Because, therefore b d, d c, are equal, but d a is common, and the bases b a, a c, are equal, the angle, also, b d a, (by the eighth) is equal to the angle c d a. Again, b d, d c, are equal, and d e is common, and they contain equal angles as we have shewn, the base also b e, is equal (by the fourth) to the base e c. Because, therefore, a b is equal to a c, and a e is common, the angle, also b a e, is equal to the angle c a e, which was to be shewn.
But if the sides of the equilateral triangle fall within the right lines a b, a c, as b d, d c, let again a d be connected. Because, therefore, b a, is equal to a c, and a d is common, but the base b d, is equal to the base c d, hence, the angle b a d (by the eighth) is equal to c a d. The angle, therefore, at the point a, is bisected, in whatever manner the equilateral triangle may be constituted. And having thus summarily spoken concerning these, we shall now proceed to the following theorems, only adding, that the given angle may be given in a four-fold respect. In position, as when we say to this right line, and to this point to place an angle: for after this manner it is given. But in form, as when we call the angle right, or acute, obtuse, right-lined, or mixed. And in proportion, as when we call it double, or triple, greater, or less. And lastly, in magnitude, as when we call it the third part of a right angle. But the present angle is only given in form.