In Philosophiae Naturalis Principia Mathematica, Sir Isaac Newton reveals the grand laws that govern the universe with mathematical precision, introducing concepts that forever change how we view the world. Through the three laws of motion and the law of universal gravitation, he shows how all objects, from the smallest stone to the distant planets, are ruled by the same natural forces. His work explains celestial motion, the behavior of physical bodies, and offers a mathematical framework for understanding nature itself. This text feels nothing short of revolutionary—a window into the clockwork of the cosmos, where mathematics becomes the language of the heavens.
Of all the equations known to modern mathematicians, none appear so marvelous, so simple, and so elegant as Euler’s identity: \( e^{i\pi} + 1 = 0 \). In one single stroke, it unites five of the most esteemed numbers: \( e \), \( i \), \( \pi \), \( 1 \), and \( 0 \), rendering them all equal in a most remarkable manner.
I. The Number \( e \)
- The number \( e \), known to us as Euler’s constant, is a quantity approximately equal to 2.718. It is paramount in the study of natural logarithms and arises frequently in the consideration of continuous growth, particularly in matters pertaining to finance and population increase.
We know thus: \( e \approx 2.718 \).
II. The Imaginary Quantity \( i \)
- The quantity \( i \), though termed "imaginary," is not without its use. It is defined by the property \( \sqrt{-1} \), a curious notion indeed, which allows us to solve equations hitherto deemed insoluble, such as \( x^2 = -1 \). The introduction of this number has afforded mathematicians a new dimension in which to work, where rotations and transformations are achieved with ease.
\( i = \sqrt{-1} \), a most curious yet invaluable discovery.
III. The Sacred Circle and \( \pi \)
- \( \pi \), a constant revered since the ancients, is the ratio of a circle's circumference to its diameter, approximately 3.14159. This number reveals itself in all manner of mathematical works related to the geometry of the heavens and the earth alike, serving as a bridge between the abstract and the practical.
\( \pi \approx 3.14159 \), an ancient wonder of geometry.
IV. The Foundational Numbers: \( 1 \) and \( 0 \)
- These two numbers, \( 1 \) and \( 0 \), are the cornerstones of all mathematics. The number \( 1 \) represents the singular and the whole, the basis of counting and measurement. Meanwhile, \( 0 \), representing nothingness, is as vital in its absence as \( 1 \) is in its presence, a paradox of sorts which we must accept and embrace.
Thus, we have the concepts of "something" and "nothing" in \( 1 \) and \( 0 \).
Uniting the Elements
Now, Euler’s great identity reveals itself when these elements are combined. The exponential \( e^{i\pi} \) implies a rotation in the realm of the imaginary. When this rotation completes half a revolution (or \( \pi \) radians), we find ourselves at \( -1 \), on the opposite side of the axis from our starting point.
To explain by a simple analogy, consider the face of a clock. If you begin at the hour of 3 (representing \( +1 \)), and proceed to rotate the hands by exactly 180 degrees, you shall end at 9 o’clock, which represents \( -1 \). Adding \( 1 \) to \( -1 \) restores the clock to 12 o’clock, or \( 0 \), thus completing the identity:
\( e^{i\pi} + 1 = 0 \)
A Brief Allegory of the Circle
Picture yourself treading upon a circular path. At the outset, you stand at the point \( 1 \). Moving forward by \( \pi \) radians—a half-turn upon the circle—you find yourself directly opposite, at \( -1 \). If you add \( 1 \) to \( -1 \), you return to the origin, to \( 0 \), thus closing the circle, in both a literal and symbolic sense.
In this equation, we see the union of many seemingly disparate branches of mathematics: the arithmetic of numbers, the geometry of circles, and the mysteries of the complex plane, all intertwined in one succinct and beautiful expression, which we must view with awe and reverence.