The Surveyor in Foure Bookes

its three equal sides, is called an equilateral triangle; and in respect of its sharp angles, is termed an oxigonium, or acute-angled triangle. But this name—in my opinion—is needless for this kind of triangle, because the term "equilateral triangle" implies the same thing, seeing it can contain no angles but acute ones. For if in any triangle there were either a right or an obtuse angle, the sides could not all be equal, and therefore, by this definition, it would not be equilateral.

DEFIN. XXII.

Three labeled triangles demonstrating different types of isosceles triangles: A (right-angled), B (obtuse-angled), and C (acute-angled). The vertices are labeled with letters: D, E, F for triangle A; G, H, I for triangle B; K, L, M for triangle C.

Euclid, Book 1, Def. 25.

This is the second sort of triangle, and has only two sides of one length, the third being either longer or shorter. As these three triangles A, B, and C have each of them two equal sides, and are therefore called isosceles. But in respect of their angles, the triangle A is called an orthigonium, or right-angled isosceles, because its angle at the point D is a right angle. Also, the triangle B is called an ambligonium, or an obtuse-angled isosceles, because its angle at the point G is a blunt or obtuse angle. And likewise, the triangle C is called an oxigonium, or acute-angled isosceles, because all its angles are acute or sharp. This triangle is also called an equicrural triangle, in respect of its two equal sides.

DEFIN. XXIII.

Three labeled triangles demonstrating different types of scalene triangles: A (right-angled), B (obtuse-angled), and C (acute-angled). The vertices are labeled with letters: D, E, F for triangle A; G, H, I for triangle B; K, L, M for triangle C.

Euclid, Book 1, Def. 26.

This is the third kind of triangle, and has all its sides of several lengths. As these three triangles A, B, and C have each of them all their sides unequal, and are therefore called scalena. But in respect of their...