Today virtually every large liquid rocket that flies into space takes advantage of the performance-enhancement techniques we pioneered in conjunction with the Apollo moon flights. NASA’s reusable space shuttle, for example, employs modern versions of optimal fuel biasing and postflight trajectory reconstruction. However, more of the critical steps are accomplished automatically by the computer.

Russia’s huge tripropellant rocket, which was designed to burn kerosene-oxygen early in its flight, the switch to hydrogen-oxygen for the last part, yields important performance gains for precisely the same reason the Programmed Mixture Ratio scheme did. In short, the fundamental ideas we pioneered are still providing a rich legacy for today’s mathematicians and rocket scientists most of whom have no idea how it all crystallized more that 40 years ago.

*Illustration 1.* below summarizes the performance gains and a sampling of the mathematical procedures we used in figuring out how to send 4700 extra pounds of payload to the moon on each of the manned Apollo missions. We achieved these performance gains by using a number of advanced mathematical techniques, nine of which are listed on the chart. No costly hardware changes were necessary. We did it all with pure mathematics!

In those days each pound of payload was estimated to be worth five times its weight in 24-karat gold. As the calculations in the box in the lower right-hand corner of Illustration 1. indicate, the total saving per mission amounted to $280 million, measured in 2009 dollars. And, since we flew nine manned missions from the earth to the moon, the total savings amounted to $2.5 billion in today’s purchasing power!

We achieved these savings by using advanced calculus, partial differential equations, numerical analysis, Newtonian mechanics, probability and statistics, the calculus of variations, non linear least squares hunting procedures, and matrix algebra. These were the same branches of mathematics that had confused us, separately and together, only a few years earlier at Eastern Kentucky University, the University of Kentucky, UCLA, and USC.

*Illustration 1. Over a period of two years or so a small team of rocket scientists and mathematics used at least nine branches of advanced mathematics to increase the performance capabilities of the Saturn V moon rocket by more than 4700 pounds of translunar payload. As the calculations in the lower right-hand corner of this figure indicate, the net overall savings associated with the nine manned missions we flew to the moon totaled $2,500,000,000 in today’s purchasing power. These impressive performance gains were achieved with pure mathematical manipulations. No hardware modifications at all were required. *

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