## Preface and Arithmetic and Geometry philosophy of mathematics

Mathematics gives access to the part of philosophy including introduction, arithmetic, and geometry. In other words, mathematics concerns the way people perceive this physical world and hence essential to the physical world. The arithmetic is a philosophical part of mathematics that deal with the property and manipulation of numbers. For instance, in arithmetic, the sequence has the constant difference between the two consecutive terms, while in the geometry the sequence has a ratio between the terms.

## Practice of Mathematics

This includes different complexes of mathematical problems or core problems that a mathematician should understand to acquire a broad competence in mathematics. For example, such problems involve acquiring theorems to make a prove, seeking the peer review, reason quantitatively and abstractly, and applying suitable tools strategically among others.

## The world-out-there

This is a different world that contains strange things and is independent of the human mind.

The mathematical-world-out-there

In mathematics, this is the world that involves abstract objects such as spheres and numbers existing beyond space and time. This world is identified through mystical apprehension, intuition, or cognition.

## Euclid's axiomatic method

This is a system applicable to the Euclidean geometry philosophy in which true statements or basic statement of geometry can be derived from certain fundamental propositions, which are postulates or axioms.

## Euclid's proof of the Pythagorean Theorem

According to Euclid, since angle CAB=BAG = right angle, C, A, G, and B, A, H, are collinear.

Angle CBD = FBA = Right angle, hence angle ABD = angle FBC.

Since A-K-L form a straight line, which is parallel to DB, then the area of rectangle BDLK is twice the area of triangle ABD. This is because they share the same altitude BK and base BD.

## The area of Square BAGF is twice the area of FBC since C is collinear with G and A.

Therefore, rectangle BDLK = BAGF = AB2 and CKLE = ACIH = AC2

AB2 + AC2 = BD x BK + KL x KC

Since KL=BD, BD × BK + KL × KC = BD(KC+BK) = BD × BC

Therefore, BC2 = AC2 + AB2

## Composite numbers

Composite numbers are whole numbers that can be divided evenly by itself, 1, and other numbers. For instance, 8 can be divided evenly by 4, 2, 1, and itself.

## Prime numbers;

Prime numbers are those numbers that cannot be divided evenly by other numbers apart from 1 or itself. For instance, 5 can only be divided by itself or 1.

## Babylonian zodiac

Babylonian Zodiac or MUL.APIN is one of the zodiacs that provides the initial discovery of zodiac in ancient Greek that have 12 constellations as shown below.

Derived http://www.ancient-wisdom.com/zodiac.htm

## Eratosthenes' measurement of the circumference of the Earth (diagram);

Eratosthenes in ancient Egypt measured the circumference of the earth using the diagram above. The sunbeam hits the ground with two rays one at Alexandria and the other at Syene. The angle of gnomons and Sunbeam at Alexandria allowed Eratosthenes to estimate the circumference and radius of the earth.

## Lo-Shu (magic square);

In traditional Chinese, the Lo Shu Square is a magical and unique normal square of order 3. The pattern of the square contains the doted representation of the integers composed of a 3 X 3 grid.

## Kepler's three laws of planetary motion

These laws include the Law of Eclipse, The Law of Equal Areas, and The Law of Harmonies. The Law of Eclipse states that planets move around the sun in an elliptical path. The Law of Equal Areas suggests that an imaginary line connecting planet to the sun occupies equal areas given equal times. The last law states that the ratio of two planets' squares of periods is proportional to the ratio of a cube of the distance from the sun.

## Heliocentric model of the Solar System

In Greek, Helios means the sun. Therefore, the heliocentric model depicts the sun at the center of all planets, which revolve around it through the orbits as shown below.

Newton's law of universal gravitation (f = G m1 m2 /d2),

Isaac Newton described the force of universal gravitational attraction for two objects as the function of their distances and their masses. Mathematically, this can be expressed as f = G m1 m2 /d2. Where F represents Force (Newtons), both m1 and m2 represents the masses of the two objects (kilograms), G represents gravitational constant, and d represents the distance (meters) between the two objects.

## Homework:2

Proportion Filippo Brunelleschi;

Filippo Brunelleschi is a renowned Italian engineer and architect, who was among the early pioneers of the Italian Renaissance architecture. He was known for using mathematical proportions through simple proportional relationships in his designs to create a geometrical harmonious quality.

## Linear perspective

This is a tool for drawing an object that applies vanishing points and lines to determine how the size of an object apparently changes with space.

## Proportion; Euclid's division of a line into mean and extreme ratio

According to Euclid, a straight line that has been divided in extreme mean and extreme ratio is the one that line is to the greater portion, so is the greater to the less.

## According to the definition, the division of a line into mean and extreme ratio is expressed as

X Y = X+YX

## Fibonacci series;

This is a series where a subsequent number is derived from adding the two proceeding numbers in the series. For example, 2,3,5,8

## Phyllotaxis

This is a pattern structure especially in botany where the leaves on the stem are arranged in alternate, whorled, or opposite. For example, the Brabejum Stellatiflium is a tree with whorled phyllotaxis.

## When and why do artists start using the "golden section"?

Golden section can be used structures like the Parthenon. Artists use golden ratio to set the height and width of the image they include in their design. The golden section can also be considered in organic and geometrical constructions; especially those entail harmony and rhythm to produce pleasing proportions.

## Works Cited

Ancient Wisdom. "The Origin of the Zodiac." www.ancient-wisdom.com/zodiac.htm.