The following will not be a faithful translation. The reasons for this are the following:

- It would bore me--I am not a servant to whomever.
- The arithmetic ability of the servant, whom Euler dictated to, was probably superior to a that of a modern reader. So I have to make some additions.
- Some of Euler's message is hidden behind the antique language and measurements.
- Annotations will be written in italics.

**A COMPLETE TRAINING IN ALGEBRA**

**by Mr. Leonhard Euler**

**St. Petersburg**

**1770**

**First Part**

**First Section**

**Chapter 1: MATHEMATICS (Paragraphs 1.-7,)**

A quantity is anything, which can be increased or decreased. Money and weight are quantities.

There are a lot of different quantities, which are treated in different parts of mathematics. Mathematics is the science of quantities.

To measure a quantity, we first call a certain part of this quantity a unit of this quantity. Then we use this unit to measure the quantity. For example, if we want to measure a length, we look how often a unit of length, e.g., a foot, is contained in the length.

Numbers are the results of these measurements: The number 3 can indicate that 1 foot is contained 3-times in a given length. But 3 can also tell that 1 dollar is contained 3-times in a given sum of money.

Algebra is the part of mathematics concerned with numbers independently from the quantities, from which they are derived.

**CHAPTER 2: EXPLANATIONS OF THE SIGNS + PLUS AND - MINUS (Paragraphs 8. - 22.)**

*Euler lived at a time when everything went slower. Because it went slower, things actually got done better and, therefore, faster. I will make some effort to help you slow down. Especially in the beginning, with "simple" things like addition, subtraction, and so on, you might tend to say: "I know all this" and speed up and not pay attention. It is the simple things in our lives we often disregard to our disadvantage. We tend to have our heads in the clouds and are not in contact with the things, which is are the level of our feet. The things, which are at the level of our feet in mathematics, are the numbers and the simple operations plus, minus, times, divided by. There is a lot of reasoning, which went into these concepts. At school, these things are mostly treated in a mechanical and mindless way. Rules are given, but not justified. We will put the reasoning back into these things, which are the foundations of mathematics.*

*For seeing the first ten numbers, we use our hands: Each hand has 5 fingers, which together make 10.** 1, 2, 3, 4, 5 fingers on** one hand. *

*6= 5 + 1, 7 = 5 + 2, 8 = 5 + 3, 9 = 5 + 4, 10 = 5 + 5 fingers if we also use the other hand.*

If you want to add a number to another, you indicate that with the sign +, which you write in front of the number to be increased. + is pronounced *plus*.

5 + 3 says that 5 has to be increased by 3 -- or 3 by 5. 5 + 3 = 8,

We also have

12 + 7 = 19

25 + 16 = 25 + 5 + 11 = 30 + 11 = 41

25 + 41 = 20 + 40 + 5 + 1 = 60 + 6 = 66 .

You can connect several numbers by the sign +, such as in

7 + 5 + 9 = 5 + 2 + 5 + 5 + 4 = 15 + 6 = 15 + 5 + 1 = 21

*Here, we have replaced 7 by 5 + 2, 9 by 5 + 4. If we do that, we get 3 fives or 15. The 2 and 4 together give 6 or 5 + 1. 15 and 5 are 20, and the 1 are 21. We do not need to do it this way. However, I told you that I want to slow you down. In addition, things might appear in a new light, which will be helpful with*

8 + 5 + 13 + 11 + 1 + 3 + 10 = 13 + 13 + 12 + 13 = 10 + 10 + 10 +10 + 3 + 3 + 2 + 3 = 40 + 11 = 51.

*8 + 5 = 13, 11 + 1 + 12, 3 + 10 = 13*

If we want to express the fact that the sum of two numbers is the same whichever we take as the one to be increased, we express this as

a + b = b + a, where a and b might be replaced by any two numbers.

f + m + b + x expresses the sum of the numbers, which replace the letters f, m. b, x.

However, 8 - 5 says that 5 has to be taken away or subtracted from 8:

8 - 5 = 5 + 3 - 5 = 3.

We also have:

12 - 7 = 5 + 5 + 2 - 5 - 2 = 5

20 - 14 = 14 + 6 - 14 = 6