At first the prevailing idea was that all the studies and sciences needing exactitude of formulæ and result should be neglected by the community. On consideration it was felt that some of the most valuable stepping-stones to the loftier ideals of the future would be sacrificed if this were done. The other alternative was chosen. The inventors who had made the calculating machines were set on to find instruments which would accomplish what the mathematicians had had to do for the community. And, one after the other, the years had produced them. Even differential and integral calculus had been superseded by a series of machines that with little guidance worked out all the applications of their intricate formulæ to the sciences. As we advanced from department to department we watched these machines at work confirming the imaginative results of the physicists, the chemists, and the astronomers. The mathematical families were relieved of their duties and distributed, and every member of the scientific families was taught to use all these formulating instruments. Their brain-energy was not monopolised by calculations; the use of the machines was but a routine detail in their wider intellectual life, and absorbed so little of their energy that it seemed to have no effect on their faculties. I was not many days in mastering the details of the formula-machines; for I had paid some attention to mathematics in my buried life and the memory of the subject rapidly revived. I soon came to see the wisdom of the Limanorans in eliminating the study from their scheme of education. It would have been the height of extravagance to waste long periods of their lives in studying and doing what a machine could do better. It was exactly the kind of work best done by a machine, for it had to do with a world rid of all conditions and, mathematically speaking, perfect. The inventors were still busy making new and simpler machines for the use of the scientists; and, though they had to know the new mathematical formulæ needed, they busied their brains rather with their practical application and with the machinery that would use them. It was imagination in the practice of mechanics rather than the mechanical use of methods and formulæ that they were engaged on. Hence it was that they avoided the old unpracticality of the mathematical families, and stood in no danger of thinking themselves infallible and the only treasuries of absolute truth.