Question

1a) Compare and contrast the development of geometry briefly from Euclid to current geometry.

1b) In detail, explain how we distinguish geometry from analysis or algebra.

Answer 1

a)

In the merging of algebra with geometry, it became gradually apparent that it was possible to relax someof Euclid's postulates in order to create new and more general geometries. Many new geometries wee soon invented: non-Euclidean geometry (1826), affine geometry (1854), and topological geometry. By the mid-nineteen century, the task of finding order among these many geometries was open.

Felix Klein, with the help o Sophos Lie, showed that all these geometries were systematicaly related to each other within the language of group theory. In 1872, upon accepting a chair at the University of Erlangen, Klein made an inaugural adddress, which has become known as the Elargen program. The Erlanger program describes a geometry by its group of symmetries. The groups are formed from the transformations that leave the basic geometric notions invariant - but this relationship can be inverted, so that, in Klein's words, ''geometrical properties are characterized by their invariance under a group of transformations''. Each type of geometry has its own group, but within the framework of that group, each geometry proceeds along analogous lines.

Klein showed that the Euclidean Geometry could be seen as an evolutionary sequence of more and more complex geometry: topological, projective, similarity group, Euclidean geometry, each one being a sub-group of symmetry transformation as the preceding one.

b)

Geometry is mostly about properties of shapes. Simple examples of the properties of shapes are length, area, volume, angles, and the properties of being larger or smaller, parallel or perpendicular, straight or curved, etc. As this is a branch of math we try to describe these things rigorously.

Algebra is mostly about arithmetic considered abstractly. For example, what happens when we square the sum of two numbers, generally speaking? That question would be easy to answer in a particular example: if you add 2+3 and then square the result, you get 25. But a less obvious way to put it, but useful because it's more general, is that when you square the sum of two numbers you get the square of the first number, plus 2 times the product of the two numbers, plus the square of the second number. That's a complicated idea to take in when it's put in the English language like that, so math people find it more efficient to use “variables,” shorter symbols than the English words “the first number” and “the second number” and so on.

So we might say it not like this:

The square of two numbers is the the square of the first number, plus 2 times the product of the two numbers, plus the square of the second number.

But rather like that:

(x+y)^2 = x^2 + 2xy + y^2

Each area of math (algebra and geometry and so on) develops a lot of jargon and special written symbols, and sometimes students find this confusing. In algebra variables are commonly single letters: “a” has a value, “b” has a value, and “ab” represents the product of “a” and “b.” In geometry a single capital letter is often used for a point in a figure; A has no value (in the algebraic sense), and neither does B, but AB is used as a sort of variable, not the product of A and B but the distance from A to B.

The style of reasoning and communicating differs in other ways too. In geometry it's pretty common to justify a proof. We might show a column of mathematical statements that are supposed to prove some conclusion, and for each of these statements we might give a reason from the fairly short list of established justifications. In algebra this is rare. We show algebraic equations one after another and leave it to the reader to figure out that we have combined like terms, or divided each side of the equation by the coefficient, etc. I guess that this is because of our experience that readers would not be able to follow the geometry without statements of the reasoning, while algebra does a better job of speaking for itself.