This paper introduces a new method for quantifier elimination in real closed fields, which is significantly more efficient than previous methods. The method, discovered by George E. Collins in 1973, was presented at various academic conferences and published in the proceedings of the EUROSAM 74 Conference. The computational complexity of this method is dominated by a polynomial function, making it potentially practical for real-world applications. In contrast, earlier methods, including those by Tarski, Seidenberg, and Cohen, had exponential complexity in the number of variables and polynomials. The key idea behind the method is cylindrical algebraic decomposition, which decomposes the real space into disjoint connected sets (cells) where polynomials in a given set are invariant in sign. This decomposition helps in determining the truth value of quantified formulas by evaluating sample points in each cell. The method's efficiency is further supported by recent results from Fischer and Rabin, which suggest that the best achievable bound for any deterministic method is likely of the form \(2^{k}\) where \(k \leq 8\).This paper introduces a new method for quantifier elimination in real closed fields, which is significantly more efficient than previous methods. The method, discovered by George E. Collins in 1973, was presented at various academic conferences and published in the proceedings of the EUROSAM 74 Conference. The computational complexity of this method is dominated by a polynomial function, making it potentially practical for real-world applications. In contrast, earlier methods, including those by Tarski, Seidenberg, and Cohen, had exponential complexity in the number of variables and polynomials. The key idea behind the method is cylindrical algebraic decomposition, which decomposes the real space into disjoint connected sets (cells) where polynomials in a given set are invariant in sign. This decomposition helps in determining the truth value of quantified formulas by evaluating sample points in each cell. The method's efficiency is further supported by recent results from Fischer and Rabin, which suggest that the best achievable bound for any deterministic method is likely of the form \(2^{k}\) where \(k \leq 8\).