Chapter XXIII – Of the Method of expressing Logarithms
Summary: Explains decimal fractions as the practical notation for logarithms, introduces the characteristic, and shows how logarithm tables are used to recover products and roots. (source: chapter-1.1.23)
Sources: chapter-1.1.23
Last updated: 2026-04-24
§242–246: Decimal Fractions and Decimal Logarithms
Euler says that values like cannot be expressed exactly by ordinary fractions and are therefore approximated by decimal fractions. (source: chapter-1.1.23)
He explains decimal place value using examples such as , where digits to the right of the point successively represent tenths, hundredths, thousandths, and so on. (source: chapter-1.1.23)
He then gives tabular decimal approximations: (source: chapter-1.1.23)
§247–251: Characteristic and Common Decimal Part
Numbers between and have logarithms between and , numbers between and have logarithms between and , and in general the integral part of the logarithm is one less than the number of digits. (source: chapter-1.1.23)
Euler calls this integral part the characteristic. (source: chapter-1.1.23)
He also observes that numbers formed from the same significant digits but shifted by powers of share the same decimal part in their logarithms. Thus , , , , and differ only in characteristic. (source: chapter-1.1.23)
To avoid negative characteristics in practice, he mentions the tabular convention of adding to them in some settings. (source: chapter-1.1.23)
§252–255: Tables in Practice
Common tables carry logarithms to seven decimal places, while larger tables may go to ten. (source: chapter-1.1.23)
Euler explains that tables usually print only the decimal part, since the characteristic is easy to restore from the size of the number. (source: chapter-1.1.23)
He then gives worked uses of tables: adding logarithms to multiply numbers, and halving a logarithm to extract a square root. His example is found from , so half is , corresponding to approximately . (source: chapter-1.1.23)