There are five assignments for this lesson:
- Re-take all of your previous exams in Arithmetic for review.
- Study your Lesson.
- Complete your Memory Work.
- Complete your Lesson Exercises.
- Complete the Lesson Examination.
Lesson
Thus far in this course, we have begun to study the basic concepts of Arithmetic. At this point, you should be comfortable with all of your past memory work on Quantity, Multitude and Magnitude, Unity and Units, More and Fewer, Greater and Lesser, Species and Number, Homogeneous and Heterogeneous Multitudes, Integers and Fractions, Aliquot and Aliquant Parts and the first ten Axioms. That’s a lot of information! Therefore it is required that, before continuing in Lesson 07, you stop here and re-take all of your past exams for review. When you have passed each of your old exams, continue below.
Reason and the Three R’s
If you have been in the Classical Liberal Arts Academy for long, you will know that, when God made man, He made him to be greater than any other creature. He did this not by giving man stronger arms or faster feet than other animals, nor better vision or a larger body. God made man great by giving him Reason. The gift of Reason forces us to think. Unless we are crippled by sins and worldly desires, we cannot live like cattle and sheep, walking around all day with our eyes upon the ground looking for something to eat or a place to rest. Our eyes are naturally turned upward, towards the heavens, from where we can hear God’s voice calling us to Himself.
When we look upon objects in the world, our minds are not content to merely know what they are. An animal may only desire to know if a thing may be eaten, but man wants to know much more. Reason forces us to consider how ideas are related to one another and is constantly trying to make connections between different objects, or find ways to separate them when they are not related. This is called “judgment”.
The Latin word for Reason is Ratio. We say that a man is a “rational” creature because he has ratio or reason. We use this word to refer to the reasons why things are the way they are, or why creatures do what they do. Animals never ask “Why did you do that?”–only man does because he seeks reasons for things. Reason also causes us to seek to understand the relationships between things and this idea of relationships is also called ratio in Latin. Lastly, reason moves us to understand whether things follow any certain rules that will allow us to predict their actions in the future. Again, this idea of a rule is named ratio in Latin. Therefore, Reason (ratio) leads men to seek out the reasons, relationships and rules of things around us. We can think of this as Reason and the three R’s.
Ratio in Arithmetic
When our senses perceive a number of different objects, reason quickly goes to work to discover whether any reasons, relationships or rules may be found. Let’s take a few examples. Look at the four letters below:
In the four boxes above we find four different objects. You will notice, though, that your mind quickly finds what is similar about them and also finds what is different. We find that these are all letter A’s and that they differ in size and color. That’s your reason at work.
Look at the next set of objects:
Once again, you will find your reason leading you to discover whether there is a reason, relationship or rule among these objects–and there is! We find a rule that the numbers 2 and 4 repeat in a pattern, and that the colors red and green repeat in a pattern, but that the numbers differ in size. That is reason at work.
However, what does your reason discover with these two numbers?
They are the same size. They are the same color. There is no pattern to discover because there are only two numbers. The first number is more than the first–but reason is not content with that. Reason continues to work to find that there is a relationship between these two numbers. The first number is exactly twice the quantity of the second. Now you will find that reason can begin to relax. If you keep allowing reason to seek for more relationships between objects, you will often find some. This is called meditation.
Let’s look at another example:
Again, they are numbers of the same size and color. There can be no pattern since there are only two numbers. There is again a relationship that can be found: the second number is twice the quantity of the first.
In Arithmetic, a relationship between two numbers is called a Ratio, which shouldn’t surprise us now that we know what the word means in Latin! (In English, it’s pronounced RAY-shee-oh.). Anytime we see two numbers we can express their relationship or ratio by simply saying “A to B”. In the example above, we find a ratio of 8 to 4. We also saw the ratio 3 to 6. We can write ratios by using the colon ( : ) in place of the word “to”, as 8:4 and 3:6.
In a ratio, the number given first is called the Antecedent, which is simply Latin for “that which goes before”. The second number is called the Consequent, which is Latin for, “that which follows”. If you realize that a “consequence” is something that follows a certain action, the idea will be clear. When we have a ratio in which the antecedent is greater than the consequent, it is called a “Ratio of Greater Inequality”. When we have a ratio in which the antecedent is less than the consequent, it is called a “Ratio of Lesser Inequality”.
Reason is Not Satisfied
Before we saw that reason will not rest merely knowing that one number is greater or less than another. Reason wants to search out every reason, relationship and rule that exists–even when they are buried deeper.
Beyond knowing the ratio of two numbers, reason also wishes to know if there is any relationship between the numbers in the ratio! In the examples above, reason will discover that not only can the ratio be expressed as 8:4, but it also discovers that the antecedent is exactly twice the consequent. Likewise, in the ratio 3:6, the antecedent is contained exactly twice in the consequent. Now, if you’re paying attention in the course, this language should sound familiar to you!
We learned that an aliquot part is a part which, being repeated a number of times, becomes equal to the whole. Thus, in the ratio 3 to 6, 3 is an aliquot part of 6. On the other hand, an aliquant part is a part which, being repeated a number of times, always exceeds or falls short of the whole. If we took the ratio 3:5 or 3:7, we would find that 3 was an aliquant part of 5 and 7.
The reason why our knowledge of aliquot and aliquant parts is important is because knowing the Measures of numbers is very important to understanding more about the relationships between numbers. Any number that is an aliquot part of another is said to be a measure of that number. When our reason discovers this it gets excited because a whole new level of relationships is found! Let’s look at an example.
When our reason goes to work on these numbers, it finds that they are all green and equal in size, but that’s easy. It notices that 4 is a part of 8 and that 4 is a part of 10. It also notices that 8 is a part of 10. However, if reason is given time it will notice that the number 2 measures 4, measures 8 and measures 10. Aha! A relationship! That makes reason happy. Reason then tells us that the number 2 is a Common Measure of 4, 8 and 10, and that the numbers 4, 8 and 10 are all Commensurate Numbers, because they are all measured by some number other than 1 (unity).
On and on reason goes!
Commensurate Numbers
Let us study the concept of “common measure” a bit more in depth. We saw above that a smaller number measures a greater number if it is an aliquot part of it. Given two or more numbers, we are sometimes interested in finding a “common measure,” i.e., a number that measures both of them. For example, as we saw above, 2 is a common measure of 4, 8, and 10. Sometimes, we can find several common measures. For example, given the numbers 12 and 24, we see that 2 is a common measure, 3 is a common measure, 4 is a common measure, 6 is a common measure, and 1 is also a common measure. We are also going to consider 12 as a common measure. Even if 12 is not an aliquot part of 12 (because it is not a part of it, but the whole!), we still say that 12 measures 12, because it goes into it one time.
Note that, given two or more numbers, 1 is always a common measure. Indeed, the unity 1 measures every number. And it makes sense… in some sense, that is why every system of measurement chooses units to measure quantities!
However, if we overlook 1 for a second, there will be numbers which will have other common measures, and numbers which will not have any other common measure other than unity. If two or more numbers have another common measure other than unity, we call them Commensurate Numbers. If they do not have any other common measure, they are not commensurate numbers.
So, for example, 10 and 12 are commensurate numbers, because they are measure by 2, but 5 and 7 are not commensurate numbers, because they only number that measures both of them is 1.
Why This All Matters
We are only getting started in Arithmetic and most of the lessons we learn in this course are not desired for use in this course alone. We are not studying Arithmetic for its own sake! These ideas are desired for use in other areas of life and study.
We will learn later that some ratios are very special. We will learn about one ratio, called “the Golden Ratio” which was used by architects in history to build buildings, by artists to paint pictures, by musicians to create music and by God in the creation of the world!
We will also learn that by learning the measures of a number, our reason will be able to dig deeper and find ever more relationships between numbers–and between the measures of numbers!
That might sound confusing, but it will get clearer in our next lesson. For now, simply work to understand all that this lesson teaches about ratio and measure.
Memory Work
Directions: The following questions help you to memorize the most important points of this lesson. Commit them perfectly to memory and have a parent or praeceptor quiz you to test your mastery before taking your lesson exam.
- What is the Ratio of two numbers?
The Ratio of two numbers is the Comparison between any two homogeneous numbers (the one being taken as part or parts of the other). - What do we call the terms of a Ratio?
In any ratio, the first number is called the Antecedent and the second number the Consequent. - What is a Ratio of Greater Inequality?
A Ratio of Greater Inequality is a ratio of a greater number to a lesser, as 8 to 2. - What is a Ratio of Lesser Inequality?
A Ratio of Lesser Inequality is a ratio of a lesser number to a greater, as 2 to 8. - When does a lesser number Measure a greater number?
A lesser number measures a greater number when it is an Aliquot part of that greater number. - What is a Common Measure?
A Common Measure is that which measures each of two or more numbers, as 3 is a common measure of 3, 6, 9, 12, 15, etc.. - What are Commensurate Numbers?
Commensurate Numbers are number that have some common measure besides Unity, as 3, 6, 9, 12, 15, etc.
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