Geometry

Excerpts

• Geometry

  But we do not need to consider who is observing the things, or whether he becomes acquainted with them by sight or touch or hearing. In short, we ignore all particular sensations.

• Geometry

  Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every proposition refers to any things with such and such geometrical properties.

• Geometry

  The answer is that the triangles are in all respects equal, if:---

• Geometry

  Also there is the still simpler correlation between the angles of the triangle, namely, that their sum is equal to two right angles;

• Geometry

  It is as though the great science of Anthropology were named the Study of Noses, owing to the fact that noses are a prominent part of the human body.

• Geometry

  The peculiarity of geometry is the fixity and overwhelming importance of the one particular example which occurs to our minds.

• Geometry

  The abstract logical form of the propositions when fully stated is, “If any collections of things have such and such abstract properties, they also have such and such other abstract properties.”

• Geometry

  The number of the archangels can be counted just because they are things.

Equations

• Geometry

  \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

  In a triangle, the sine of each angle divided by the side opposite it has the same value for all three angles.

• Geometry

  a^{2} = b^{2} + c^{2} - 2bc \cos A

  The square of one side of a triangle equals the sum of the squares of the other two sides minus twice their product times the cosine of the angle between them.

• Geometry

  A + B + C = 180°

  The three angles of a triangle, counted in degrees, add up to 180 degrees (two right angles).

• Geometry

  a < b + c

  Each side of a triangle is shorter than the sum of the other two sides (the same holds for b and c in the book's companion inequalities b < c + a and c < a + b).

Problems

No exercises in this chapter.