Ratios
Ratios are numerical comparisons. Most everything can be compared to some other thing. The speed of one auto to another. The length of one line to another. The weights of two objects. Or, maybe elapsed time durations. A ratio is written as 2 : 1, 1 : 7, or maybe 1 : 1... These are spoken as two to one, one to seven, and one to one.
In math a ratio is also a fractional division problem as 2/1, 1/7, or 1/1. By dividing, a decimal fraction is achieved to as 2.0, 0.142857..., and 1.0.
In Trigonometry all the values of sine, cosine, tangent, secant, and etc. are decimal fraction radio comparison of one (1). A number of 1.432 is larger than 1. And, 0.1825 is smaller. The next chapter goes into some Trigonometry. You may find this interesting because most all the strange names you have heard about in Trigonometry, some as just mentioned, are nothing more than straight line length ratio comparisons of a single straight line radius of 1.
A few hundred years ago ratios were more at the forefront of math, but now they lurk in the midst of math. You use them without recognizing them as such. They come as a problem of cross multiply and divide for the answer. And, we just memorize such, without a good understanding. If I went 20 miles in 120 minutes, how far did I go in 60 minutes... [20 : 120] : [X : 60], or 20/120 : X/60. Cross multiplying 20 times 60 equals 1200, and dividing by 120 equals X equals 10 miles . I am not teaching math herein... but, to point out different ways of viewing items.
In Fig.-21 I took from the previous chapter, and some from the next chapter. There are four right triangles A, B, C, & D. These triangles are all similar but of different sizes. Triangle B is twice the distance from point P than triangle A. Thus B to A is 2 : 1 larger. Or A is half that of B, or 1 : 2. Triangle D is twice the distance, (doubling), from point P as triangle C, thus being twice as large as C. (Note the lines of the triangles are doubling... and etc. The area is not being considered... at this moment, since it would be quadrupling.)
Triangle C is set up as you will see typically in Trigonometry. The hypotenuse of triangle C is also the radius of the quarter circle. This has been given a value of one (1). This triangle with the hypotenuse of one is what all and any other sized right triangles are compared against. They stay directly proportional as they get larger or smaller. For example the vertical line called Sine is twice as tall as the same line for triangle B, and the line Sine is half a high as the same line in Triangle D.
If the top radiated line were to move along the circle arc, and the angle shown is changed... the same angle in all the triangles changes the same. And, lines that change in all the triangles all change proportionately. For example Sine may grow taller. But, all the other vertical triangle leg lines grow proportionately taller. With sine longer, cosine becomes shorter, as will all the other triangle typical base lines
This is a way of visually showing something radiating as doubling with distance, having ratios between lines within one triangle and relating to similar proportionate changes in other triangles.
Another synonym for ratio is when something is said to relative to another something. It has been said that the total Universe is in constant total motion with everything relative to all other things. Ratios...
