Ratios, Proportions, and Their Uses

Ratio, as a basic construct in mathematical language, is used to explore the reason why of the sense of beauty, including shapes and music. Thus, ratio is a human notion of geometry and time, as a part of human perception. The reasoning on ratio is proportion, that two ratios are equivalent but different in size, from which a mathematical theory of ratio was born, and variant mathematical subjects were derived. The concept of ratio did become an important tool in the history of ancient mathematics, especially in differential calculus, accompanied with other mathematical concepts such as divisibility and multiplication. But the modern definition of ratio is mainly arithmetic, that treats a ratio as a value rather than the ratio itself, thus the meaning and possible uses of ratios are lost. This book tries to study ratios from a historical and methodological respect, to recover the possible uses of ratios in the history, thus we may be able to use similar methods and concepts on new things. That is, ratio will be treated as a conceptual tool that can help, rather than merely an arithmetic entity with a value.

The Notion of Ratio
Before the formal discussion of ratio, we explore the occurrence of the notion itself, thus we can know where it can be used and the reason why of related talk.

Comparison of Two Quantities
Let's regard ratio as an innate functionality of human mind. When we say someone is thin, it means the height is relatively larger than the extent of the person, this is where the notion of ratio functions -- the sense of thin is derived from the function of ratio. Here, the word function means an activity of mind that accepts input (height and extent), performs some process (comparison of height and extent and match of the concept of thin) and produces an output (the utterance of thin).

This thin ratio is generally called an aspect ratio which has a strict meaning: "the ratio of a geometric shape in different dimensions", which produces the relative notions of thin, fat, wide, narrow, deep, etc. That is, we have ratio as an useful tool to comprehend the geometric property of a thing, and use this property to analyze similar things. For example, a dendritic spine could be considered thin, which has a biological implication.

Imagining that there are people who lack of the cognitive ability of ratio, or this ability is not well developed, as a result they cannot innately use this notion to comprehend and analyze things; instead, they can only manually follow the definitions of related concepts given by other people like a machine, e.g.: "Spines with head to neck diameter ratio greater than 1.100 μm are considered Thin." For a machine, the word Thin here is only a literal concept without any sense nor implied meaning.

Geometric Organization - Shapes
Technically, ratio is used to describe the geometric organization of a shape. When we say someone's face is beautiful, it means the sizes and positions of the face parts are perfectly arranged in our opinion. There is no comparison of face parts, we just want to measure each part and make a list of them. Someone's eye could be larger than the nose, while someone's nose could be larger than the eye, but they could be both considered beautiful. Here, ratio means the geometric organization of the parts in a whole.

A basic example is the triangle, which three sides is represented by a ratio $$a:b:c$$. In Pythagoreans' viewpoint, the triangle $$3:4:5$$ is considered sacred because it has a particular ratio, shape (right-triangle), and rule (Pythagorean theorem). This $$3:4:5$$ triangle actually represent a shape, or a class of triangles of the same ratio, which are called similar, and they obey the same rule - Pythagorean theorem. Thus, the equation $$3:4:5 = 6:8:10$$ actually means there are two similar objects, and they are not equal; this sense is different from the arithmetic equality of fractions such as $$3/4 = 6/8$$.

Ratio in Euclid's Elements
Ratio as the notion of comparing two quantities is arithmetic. But there is a neutral definition of ratio developed in Euclid's Elements, Book V.

Ratio and Fraction
Ratio and fraction are originally two independent concepts, used in different situations.