Euler and his wife Katharina had thirteen children together, but only five survived to their teens. While it is true that infant mortality rates were high during the eighteenth century when Euler lived, the death of eight young children would have stung him particularly. Euler was an involved and affectionate father, writing that some of his most important mathematical ideas came when he was holding a baby in his lap. Some of these children, it is likely, were in one stage or another of some fatal disease.
Another significant challenge was the loss of sight in one of Euler’s eyes. He carried on in spite of this, but it was a real blow when he lost the use of his other eye as well, two years before his wife died. Because he had a phenomenal memory, a spirit to match, and supportive friends and family who could be scribes for him, he continued to make contributions to mathematics right through his last day on earth.
People who have learned to maintain a positive outlook toward life in spite of deep pain often learn compassion as well, and are unwilling to knowingly inflict suffering, not even in the form of mild retaliation. Even if Euler had been inclined to cruelty, though, he was aware of far too many impressive mathematical curiosities to have resorted to the nonsense equation mentioned in the Euler-Diderot Incident.
An unusual mathematical imagination allowed him to visualize, between two apparently unrelated concepts, relationships that are generally non-intuitive. He had insight into the significance, in terms of mathematical modeling of observable phenomena, of what came to be known as the natural logarithm, and saw connections between it and infinite series. He calculated a numerical approximation for the base, which he named e, of its inverse function. He also related various infinite series to the natural constant pi (the ratio of the circumference of a circle to its diameter) which yielded many pleasingly symmetrical mathematical formulae. Another beautiful curiosity Euler discovered was the link between harmonic series and prime numbers, which he used to give a new proof for Euclid’s assertion that there are an infinite number of primes.
Euler assigned the notation i to the square root of -1. Euler was largely responsible for making the formerly skeptical mathematical world comfortable with this “imaginary” number by describing the essential role it played in the field of analytical algebra. Euler’s facility with algebraic analysis carried into his explorations in the field of geometry. There was a general feeling among geometers in the eighteenth century that results derived from calculations involving algebraic symbols were less elegant than proofs made up primarily of logical constructions. Leave it to Euler, though, to intuit with his mind’s eye the geometric relationship within the elements of triangles that no other geometer, not even Euclid, had seen, and to use analytic geometry to prove this observation that might have defied proof without Cartesian analysis. This result is the discovery that every triangle has what is called the “Euler line” containing the circumcenter, the orthocenter, and the centroid, and the distance from the centroid to the orthocenter is exactly twice the distance from the orthocenter to the circumcenter. Here again is another beautiful and surprising result that delighted the mathematical aesthete while winning him over to the power of the newer computational innovations, in this case, algebraic analysis within the framework of the Cartesian coordinate system.