CONTENTS Introduction § 1. Euler's method. The basic statements § 2. Non-standard analysis § 3. The derivation of Euler's factorizations using non-standard analysis § 4. Euler's construction in the system of non-standard analysis § 5. Factors and coefficients in Euler's construction § 6. A factorization that Euler did not want to complete § 7. The summation of infinitesimals § 8. Formal infinity as a mathematical object § 9. Representations of the exponential as a series and as a power § 10. Comparing the coefficients of non-standard polynomials Conclusion References