Home » Curriculum » Quadrivium » QRV-211 Intro to Classical Arithmetic » Intro to Classical Arithmetic, Lesson 03. Units of Linear Measure

Intro to Classical Arithmetic, Lesson 03. Units of Linear Measure

To complete this lesson, complete the following tasks:

1. Study the lesson for mastery.
2. Complete the lesson memory work.
3. Complete the lesson exam.


In our last lesson, we learned that to measure any quantity, we must begin with some known quantity, which we call Unity. We discussed how our everyday tasks do not usually need exact measurements, but in many cases exact measurements are needed. For these tasks, men have worked together to set these units so that all men everywhere can use the same units and share measurements. These include inches, pounds, degrees, minutes and so on. In this lesson, we are going to learn the units that men have established for measuring distance or length.

Different Systems of Units

If you live in America, you will use a different set of units for measuring than the rest of the world does. Americans use four basic units for measuring distance or length: the foot, the inch, the yard and the mile. This is called the American System. However, almost all other countries measure with the Metric System. We will learn about the Metric System later in Arithmetic. If you ever hear of people measuring distance or length in meters, millimeters, centimeters or kilometers–they are using the Metric System.

The Foot

The first unit we need to learn for measuring distance or length is the Foot, which is written “ft.” for short.. The plural form of foot is feet. Thus, if you see 1 ft., it means “one foot”; 7 ft. means “seven feet” and 3 ft. means “three feet”. This unit of measurement was at first based on the size of a man’s foot, for which it is named. It is very easy to measure how wide a room is or how long your driveway is by counting how many feet these quantities contain. However, since men’s feet differ in size this is not a good unit of measurement for everyone to use–especially when we need to share measurements with others. In time, men set the foot at the exact length we find on our ruler today: twelve inches.

A foot is written as “ft.” for short, so 6 ft. means “six feet”, and 3 ft. means “three feet”. The single quotation mark ( ‘ ) is also used to represent the foot, with 4′ meaning “four feet” and 8’ meaning “eight feet”.

The Inch

That leads us to the inch. It is hard to tell where this measurement first came from. Some believe it was the distance across a man’s thumb, at the bottom of his nail. However, in Old English the word uncia means “a twelfth part” and since there are twelve inches in a foot, the word inch probably means “one twelfth-part” of a foot. In everyday life, an inch is about the width of a quarter in coins or a big pink eraser. An inch is written as “in.” for short, so 7 in. means “seven inches”, and 2 in. means “two inches”. The double quotation mark ( ” ) is also used to represent the inch, with 3″ meaning “three inches” and 9″ meaning “nine inches”.

The Yard

It is also important to know of the yard. It is important to note that the yard is mainly used for measuring walking distances on the ground and for buying cloth. A yard is equal to the height of most kitchen counters, or the length of a man’s step when walking. It may also be helpful to think of a yard as the distance from the middle of an average man’s chest to his fingertip when his arm is stretched to the side. If we wish to be exact, a yard contains three feet or thirty-six inches. A yard is written as “yd.” for short, so 5 yds. means “five yards”. An American football field is 100 yards long and every small white line–called a “hash mark”–measures one yard.

The Mile

If we need to measure how wide a room is, we measure with a foot. If we need to measure a piece of wood, we measure with an inch. If we need to measure a driveway or a large field, we can measure with a yard. However, if we need to measure very long distances, like how far the drive to church is, or how far the moon is from the earth, we use the mile, which is written “mi.”. A mile is a very long distance, probably too long for you to think of without some help. In ancient Rome, the unit used to measure long distances was one-thousand steps, which in Latin is mille passuum. From the Latin word mille we get the English word mile, even though a mile is not exactly a thousand steps. Today, a mile is set at 5,280 feet or 1,760 yards. The next time you take a ride in the car ask your parent to tell you when you are a mile away from home. Or, if you think about walking, it would take you 20 to 30 minutes to walk one mile. This will give you a good idea of how far a mile is.

Tools for Measuring

Measuring tools are designed to allow us to measure using different units in different situations. The most basic measuring tools are the ruler, the yardstick, the measuring tape, the measuring wheel and the odometer.

For measuring lengths smaller than one foot, we use a ruler. To read the measurement, we simply count how many inches are contained in the length of the object we are measuring. Later in Arithmetic, we will learn about fractions and this will allow us to read the smaller lines on a ruler to give a more exact measurement.

If we are measuring something more than one foot long, but less than one yard long, we might use a yardstick, which is one yard or three feet long. However, for distances or lengths greater than one foot but less than 100 feet, we usually use a measuring tape, which may be made from soft plastic or from sturdy metal. Your mother may have a soft measuring tape and your father may have a metal one. To read a measuring tape we say how many feet the distance is and then how many inches more. So a man may be 6 ft. and 3 in. tall, which we write as 6′ 3″.

For distances longer than 100 feet, but less than one mile, we can use a measuring wheel. This tool uses the size a wheel as unity and then counts how many times the wheel turns as the user rolls it along the ground, giving a distance in feet and inches.

Lastly, for greater distances that need to be measured in miles, we use an odometer, which is simply another kind of measuring wheel. The odometer uses the wheel of the car or bicycle to measure the distance the car travels and gives the distance in miles. You can see an odomoter in your family car and see how many miles your family has driven. The word odomoeter comes from the Greek word hodos, which means “road” and so means “road-measurer”.

In the metric system, the same tools are used for measurement but the units on the tools are different. We will learn about them later in this course, after we learn about decimals.


As we learned in the last lesson, we cannot measure any quantity unless we find some known quantity to measure it with. The “known quantity” we use is called a “unit”, and we may use any quantity we wish as unity. However, so that men can measure more exactly and share their measurements with others, units of measurement have been established for all. For measuring length using the American system, these units include the inch, the foot, the yard and the mile. We also learned that the system of units used in America is not the same as that used throughout the world, which used the Metric system.

Memory Work

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.

15. (Review) What is Unity, or a Unit?
Unity, or a Unit, is a known quantity we refer to as One.

21. What are the two systems of measurement used in the world?
The two systems used are the American system and the Metric system.

22. What is a Linear Measure?
A Linear Measure is a measure used in measuring distance or length.

23. What are the four basic units of linear measure in the American system?
In the American system, the four basic units of linear measure are the foot, the inch, the yard and the mile.

24. How long is the foot and how is it written?
The foot is approximately the length of a man’s foot, or twelve inches. It is written “ft.” or with a single quotation mark (‘) as in 7 feet, 7 ft. and 7’.

25. How long is the inch and how is it written?
The inch is one twelfth-part of a foot. It is written “in.” or with a double quotation mark ( ” ) as in 3 inches, 3 in. and 3″.

26. How long is the yard and how is it written?
The yard is 3 feet or 36 inches long. It is written “yd.” as in 100 yards, 100 yds.

27. How long is the mile and how is it written?
The mile is 5,280 feet or 1,760 yards long. It is written as “mi.” as in 10 miles, 10 mi.