Points in Euclidean geometry

Points are most often considered within the framework of Euclidean geometry, where they are one of the fundamental objects. Euclid originally defined the point vaguely, as "that which has no part". In two dimensional Euclidean space, a point is represented by an ordered pair, (х, y), of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three dimensional Euclidean space, where a point is represented by an ordered triplet, (х, y, z) with the additional third number representing depth and often denoted by z.

In addition to defining points and constructs (построения) related to points, Euclid also postulated idea about points; he claimed that any two points can be connected by a straight line. This is easily confirmed under modern developments of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's postulation of points was neither complete nor definitive, as he occasionally assumed facts about points that didn't follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern developments of the system serve to remove these assumptions.

Unit 3

‘There is no royal road to Geometry.’

Euclid

Grammar: The Gerund. Its Forms and Functions.

Forms

Voice Tense	Active	Passive	Expresses the action
Indefinite	writing	being written	Выражает действие, одновременное с действием глагола-сказуемого.
Perfect	having written	having been written	Выражает действие, предшествующее действию глагола-сказуемого.

I am surprised at Меня удивляет то,

a) his solving problems so quickly. как быстро он решает задачи. b) his having solved the problem so quickly. как быстро он решил эту задачу.

Indefinite Forms of Gerund are used after
prepositions:		the following verbs:
On/upon In After +V-ing Before (Gerund) Without		remember excuse +V-ing thank (Gerund) forgive