Carl Friedrich Gauss (1777–1853), now known as the Prince of Mathematics or the Greatest Mathematician Since Antiquity, was born to a poor family in the German duchy of Braunschweig-Wolfbuttel. His father was the custodian of a one-classroom school and was therefore permitted to allow little Carl, then 6 years old, to attend classes.

At the end of the school year, the teacher gave his students a final test. He asked them to find the sum of all the consecutive numbers from 1 to 20: (1 + 2 + 3 + · · · + 18 + 19 + 20).

The students used their slate boards and began the long and cumbersome calculations, while the teacher sat relaxed in his chair, knowing that the students would be kept busy for a long time.

Five minutes into the test, when the best, most experienced, 10-year-old student was sweating on the sixth addition, little Carl stood up, approached the teacher and humbly said: “Sir, the answer is 210.” The teacher was shocked. A day earlier, he had spent an hour calculating, checking, and reaching a similar answer. He suspected that Carl was cheating. “How did you figure that out so fast?”

“That is simple,” answered 6-year-old Carl. “Please note that the sum of the first and the last numbers is 21 (1 + 20 = 21). And so is the sum of the second number and the one before the last (2 + 19 = 21), and so is 3 + 18 = 21, 4 + 17 = 21, and so on. Therefore, in order to calculate the sum of all the numbers, we simply have to multiply the sum of the first and the last numbers (21) by the number of the pairs. Since we have 20 numbers, there are 10 pairs, and the answer is therefore 21 × 10 = 210.”

The teacher checked this hypothesis. The calculation appeared flawless and took only several seconds to make.

“This is perfect,” he told Carl, “but why then did it take you so long to come up with the answer?” Carl looked at the teacher and answered, “Sir, we have not yet learned how to multiply.”

In 1969, when echocardiography was practiced by very few, Dr. Pravin Shah and others from the University of Rochester published a paper that clearly demonstrated the mechanism of dynamic left ventricular outflow obstruction. Using M-mode echocardiography, phonocardiography, and carotid pulse tracing recordings, Dr. Shah convincingly proved that the systolic anterior motion of the anterior mitral leaflet was responsible for the outflow obstruction, the systolic murmur, and the characteristic bifid carotid pulse. These groundbreaking observations provided new insights into hypertrophic obstructive cardiomyopathy and established the value of echocardiography as the technique of choice in the diagnosis of this complex disorder.

The M-mode tracings in this milestone paper clearly show the mitral valve and the systolic anterior motion. However, the hypertrophied interventricular septum, an important component of the outflow tract and of hypertrophic obstructive cardiomyopathy, is not shown. In the 1960s, it could not yet be visualized by echocardiography. Shah could demonstrate the site of the septum but was unable to assess its thickness.

Gauss invented the formula for the arithmetic series without the knowledge of how to multiply, and Shah described the left ventricular outflow tract obstruction by systolic anterior motion without the knowledge of how to define the septal anatomy by echocardiography.

The two discoveries are, of course, totally unrelated. However, both talented pioneers showed that great observations can be made even when all the tools are not yet available.