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.

## Lesson

We learned in our first lesson that Mathematics is the study of Quantity. We defined Quantity as anything that may be increased or diminished. When a farmer sells a piece of his land to another, the quantity of his land is diminishing, while the quantity of his money is increasing. In an hourglass, the quantity of sand in the top is a diminishing quantity and the quantity of sand in the bottom is an increasing quantity. Such studies belong to Mathematics and this work of measuring quantity is called Computation.

We cannot measure any quantity unless a quantity of the same kind is already known to us. If we were asked to measure the time needed to drive to church, we would take a certain known quantity–a minute–and then show how many times a minute was contained in the time needed to arrive at church. On a clock, minutes are marked out as units of time. If we were asked to measure the weight of a wagon full of stones, we would take a certain known quantity–a pound–and then show how many times the pound is contained in the full weight of the stones. On a scale, pounds are marked out as units of weight. If we were asked to measure the length of a piece of wood, we would take a certain known quantity–an inch–and then show how many times the the inch is contained in the length of the wood. On a ruler, inches are marked out as units of length.

In computation, then, we select a known quantity and consider it as Unity or a unit–and it can be any quantity we wish. We can measure the length of a piece of wood using an inch as unity, or a hand, or an arm, or a pencil. Once we have the known unit in mind, we then consider how many times the unit is contained in the quantity we wish to measure. Maybe the length of wood contains twenty inches, or five hands, or one arm or four pencils. This number of times is called a proportion. We express this proportion with a number. Therefore, a number is nothing but a proportion, or number of times a known quantity is contained in a quantity we wish to measure. Unity is written by the number one or 1. If we set unity at a foot, then we might say a length of rope measures six feet. The number six or 6 simply shows us how many times the foot is contained in the length of rope. Computation, then is the action of the mind whereby things are referred to unity.

We can use any quantity we wish as unity when measuring a quantity for ourselves and throughout history measurements weren’t all that important. For example, a fishermen who wishes to measure the depth of the water he’s in, may use a “fathom” as a unit, which is simply the length of rope he can stretch across his body–about six feet. It doesn’t matter if he doesn’t have an exact measurement–he only needs a rough idea of how deep the water is. In the same way, a farmer might measure a field by “walking it out”, using a full step as unity and simply counting how many times the field contains the length of his step. A carpenter who is making just one piece of furniture at a time can take a rough measurement then cut, file and shape every piece to fit just right. For most of our everyday needs, we do not need exact measurements and worrying too much about measuring everything perfectly would actually make our work more difficult! We call this kind of measurement Estimation.

However, there are times when we do need exact measurements. In a factory where cars are made, machines make hundreds of cars every day. The cars are not made by people, but by machines. Engineers design the machines so that every bolt, screw and drop of glue goes in exactly the right place. If the measurements are off by the tiniest bit, a car may not run well. Again, your parents have probably bought a piece of furniture at the store and needed to put it together at home. To put it together, they followed the instructions that came with the furniture. If the pieces in the kit were not measured perfectly, it might be impossible to put them together!

Since we sometimes need exact measurements men have agreed on units for every different kind of measurement. This is why we have rulers, thermometers, clocks, measuring spoons, scales, etc.. These instruments help us to always measure quantities with same units and allow us to share our measurements with others. We’ll learn about these units in our next lessons.

However, God has made us for much more than measuring concrete things which we see, hear, smell, taste and touch like animals. As human beings, we pray, communicate with language, compose music and reason. God has given us the gift of Reason, to allows us to think of abstract things–eternal and unchanging things that only men, angels and God know of. We do not need to see five things, or hold five things to understand the quantity five. We can reason of the number five alone without reference to anything else. We do not need to see or feel a concrete thing to think of unity but, by Reason, we may think of unity alone in its perfect form, which is called Absolute Unity. It is this freedom that allows us to fulfill God’s will and rule over the earth. By thinking of numbers and shapes in their perfect forms, we can discover the rules that apply to all quantities–the rules which begin in God Himself and belong to the Wisdom by which He designed and made all things. When we reason in Mathematics we are using the powers that make us like God and set us apart from all other creatures.

This is what has always interested wise and holy men in Mathematics–not money or material things–but God.

## Memory Work

15. What is Unity, or a Unit?

**Unity, or a Unit, is a known quantity we refer to as One.**

16. What is Computation?

**Computation is the action of the mind whereby a quantity is measured by Unity or a Unit.**

17. What are Same Units?

**Same Units are units that are understood under the same notion, such as a pound of stones and a pound of feathers, or an inch of string and an inch of wood.**

18. What are Different Units?

**Different Units are units that are not understood under the same notion, such a pound of stones and a ton of stones, or an inch of string and a foot of string.**

19. What is a Homogeneous Multitude?

**A Homogeneous Multitude is a multitude that is made up of things of the same kind, such as a pile of pennies.**

Note: The word *homogeneous* simply means “of the same kind”. The Greek word *homoios* means “the same” and *genos* means “a kind”.

20. What is a Heterogeneous Multitude?

**A Heterogeneous Multitude is a multitude that is made up of things of different kinds, such as a pile of mixed coins.**

Note: The word *heterogeneous* simply means “of different kinds”. The Greek word *heteros* means “different” and *genos* means “a kind”.