The Elements covered plane and solid geometry, the theory of numbers, and irrationals. Euclid used the accumulated mathematical theorems of his contemporaries and of his Greek predecessors, including Pythagoras of Samos and Thales of Miletus, and arranged them in a reasonable order, such that each theorem could be proved through the use of theorems proved previously. This is called the axiomatic deductive method. Naturally, this system of organization eventually worked back to something unprovable: If each theorem had to be proved with the help of one already proved, how could one prove theorem No. 1? The solution was to begin with a statement of truths so obvious and acceptable to all as to need no proof. His truths consisted of twenty-three definitions, five postulates (statements about geometrical properties supposed to be self-evident), and five common notions or general axioms. These were the foundations, the “givens,” of his system. He could use them at any time he chose. From these basics, he proved his first proposition. With this behind him, he could then blend his definitions, postulates, common notions, and this first proposition into a proof of his second.(8) And on it went.

 

Euclid’s axiomatic approach has certain requirements. The axioms must be complete, which means that all the theorems could be derived from them. They must be independent, which means that no axiom can be deduced from the other axioms of the set. Finally, the axioms must be consistent, which means that no contradictory theorems could be deduced from the set.(9)

 

Overall, the Elements constructs its geometrical demonstrations with a minimum of simple tools and assumptions. Accordingly, it avoids inaccuracy because the chance of error increases with the number and complexity of tools and with the number of assumptions, which are statements believed to be true but which have no proof.(10) Euclid allowed only a straightedge and a compass for tools.

 

The Elements opens with twenty-three definitions, beginning with “a point is that which has no part” and “a line is breadthless length.” He also includes definitions for a straight line; surface; obtuse and acute angles; the diameter of a circle; equilateral, isosceles, and scalene triangles; a square; an oblong; and a rhombus, to name just over half of them.

 

Next are the five postulates:

1. (It is possible) to draw a straight line from any point to any point.

2. A straight line can be extended indefinitely.

3. A circle with any radius can be drawn around any point.

4. All right angles are equal.

5. Parallel lines do not cross. (Note: This is the famous parallel postulate that led to the discovery of non-Euclidean geometry in the 1800s.)

 

Finally, he stated five common notions or axioms that concern proportional relationships:

1. Things equal to the same thing equal each other.

2. Equals added to equals produce equals.

3. Equals subtracted from equals produce equal remainders.

4. Things that coincide with each other are equal.

5. The whole is greater than a part.

 

From these givens alone, he built an intricate and majestic system of “Euclidean geometry.” Never was so much constructed so well from so little. “Euclid’s reward is that his textbook has remained in use, with but minor modification, for more than 2,000 years.”(11) The high level of logical sophistication and his obvious success at weaving the pieces of his mathematics into a continuous fabric from the basic assumptions to the most sophisticated conclusions served as a model for all subsequent mathematical work.(12) “To this day, in the arcane fields of topology or abstract algebra or functional analysis, mathematicians will first present the axioms and then proceed, step-by-step, to build up their theories. It is the echo of Euclid, 23 centuries after he lived.”(13)

 

Conclusion

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Because geometry was and continues to be one of the core subjects of elementary education, Euclid’s influence was profound and far-reaching. His influence was immediate. Archimedes, the greatest mathematician-scientist of the classical world and a younger contemporary of Euclid, acknowledged Euclid in his own writings. Moreover, Euclid’s works helped form the minds of twentieth-century scientists such as Albert Einstein. Scientists were not the only ones to benefit directly from the Elements. The eighteenth-century German philosopher Immanuel Kant was so impressed with Euclid’s method that he modeled his own reasoning on it. Abraham Lincoln said that he read the Elements to improve his ability to put together persuasive legal and political arguments.

 

There are three significant factors that made Euclid’s Elements so influential. One, it introduced into mathematical reasoning new standards of logical rigor. Two, it established a geometrical perspective in Western mathematics, in contrast to an algebraic one, that lasted into the 1800s (over 2,000 years). Third, our conception of space, our physical world, came to be defined using the contents of Euclid’s Elements. Today the term “Euclidean geometry” is synonymous with the word geometry. “...the axioms of geometry are our basic facts about physical space and vast branches of mathematics and of physical science use the properties of Euclidean geometry...”(14)

 

Logical Rigor

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Euclid provided all thinkers with a method of rational thought and step-by-step construction of proofs of ideas. Accordingly, it not only passed on the heart of Greek mathematics to later times but also taught a rigorous logic that contributed to the rise of the scientific method.(15) This new standard of rigor remained intact in the subsequent history of Greek mathematics, but then a period of logical weakness followed with the European revival of mathematics in the 1500s. By the 1800s, the strict logical standards followed by the Greeks were once again evident in mathematical works produced in Europe.

 

Geometrical Perspective

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By leafing through the Elements, one sees few numbers but pages filled with words describing diagrams that show relationships between line segments, and shapes like triangles, squares, and rectangles. As an example, the Elements does not use the language of algebra to prove the Pythagorean theorem, i.e., a² + b² = c² (where a and b represent the two shorter sides of a right triangle and c represents the longest side), but a geometrical one wherein relations between literal squares constructed upon the three sides of a right triangle are used (see stamp illustration for Pythagoras). It was only toward the end of the nineteenth century that the spell of Euclidean geometry began to weaken and that a desire for the “arithmetization of mathematics,” i.e., the use of numbers in algebraic expressions, began to manifest itself.(16) By the 1920s, with the development of quantum mechanics, the physical sciences had returned to the Pythagorean view that number (through algebra) was the secret of all things.(17)

 

Euclid’s geometrical conception of mathematics is reflected in two of the supreme works in the history of thought, Isaac Newton’s Mathematical Principles of Natural Philosophy (written in 1687) and German philosopher Immanuel Kant’s Critique of Pure Reason (written in 1781). Note that both works were written about two thousand years after Euclid wrote the Elements. Add to this that Newton’s masterpiece, Mathematical Principles of Natural Philosophy, inspired later physicists nearly as much as Euclid’s inspired mathematicians. Newton’s work is cast in the form of geometrical proofs that Euclid had made the rule, even though Newton had discovered the calculus and could have written his book using an algebraic perspective. This would have served him better and made him more easily understood by subsequent generations.(18) Kant’s belief in the universal validity of Euclidean geometry led him to a universal characteristic of man that governs all his speculations on knowledge and perception.(19)

 

In contrast to Newton’s and Kant’s works, the great eighteenth century mathematician Joseph-Louis Lagrange wrote his masterpiece, Analytical Mechanics (1788), using algebraic calculus without any diagrams. Lagrange had reduced mechanics (the branch of applied mathematics dealing with motion and tendencies to motion) to general formulas from which specific equations could be derived. These general formulas still dominate mechanics, and bear his name. Thus Euclid’s geometrical style was eventually replaced by an algebraic one as the explanatory model in mathematics. Nonetheless, the world remained geometrical according to Euclid; what had changed was the use of algebraic language (numbers and equations) instead of geometrical language (words and diagrams).

 

Non-Euclidean Geometry

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Perhaps the greatest testimony to Euclid’s influence on mathematicians is that all geometries that differ from his by adopting a special perspective, or point of view, are called non-Euclidean. In the nineteenth century, some refinements in Euclidean geometry were discovered to accurately describe space on the scale of the solar system. This new perspective on our world was called “non-Euclidean geometry”; it supplemented but did not replace Euclid’s geometry. Such innovation began with the novel postulates of Nikolay Ivanovich Lobachevsky, Carl Friedrich Gauss, John Bolyai, and Bernhard Riemann. Their departures from the postulates of the Elements shocked contemporaries, although in each case, the innovators were trying to supplement, rather than replace, Euclid.(20)

 

From the work of Lobachevsky, Gauss, Bolyai, and Riemann two new non-Euclidean geometries were discovered and each was found to be just as valid and consistent as Euclidean geometry. Also, it soon became clear that it is impossible to tell which, if any, of the three geometries is the most accurate as a mathematical representation of the real world. Thus, mathematicians were forced to abandon the cherished concept of a single correct geometry and to replace is with the concept of equally consistent and valid alternative geometries. They were also forced to realize that mathematical systems are not merely natural phenomena waiting to be discovered; instead, mathematicians create such systems by selecting consistent axioms and postulates and studying the theorems that can be derived from them. It is this change of perspective that may prove to be the most important and far-reaching part of Euclid’s intellectual legacy.(21) (See under Riemann bio an extensive discussion of non-Euclidean geometry).

 

 

Footnotes:

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(1) Donald R. Franceschetti (editor), Biographical Encyclopedia of Mathematicians (New York, 1999), p. 174.

(2) Ibid., p. 173.

(3) Robert Maynard Hutchins (Editor in Chief), Great Books of the Western World, Vol. 11 – Works by Euclid; Archimedes; Apollonius of Perga; Nicomachus of Gerasa (Chicago, 1952), p. ix.

(4) William Dunham, Journey Through Genius – The Great Theorems of Mathematics (New York, 1990), p. 66.

(5) Ibid., p. 68.

(6) Ibid., p. 75.

(7) Ibid., p. 80.

(8) Ibid., p. 31.

(9) Donald R. Franceschetti (editor), Biographical Encyclopedia of Mathematicians (New York, 1999), p. 277.

(10) Ibid., p. 175.

(11) Isaac Asimov, The Intelligent Man's Guide to Science, Vol. 1 – The Physical Sciences (New York, 1960), p. 11.

(12) Dunham, p. 32.

(13) Ibid., p. 32.

(14) Morris Kline, Mathematical Thought From Ancient to Modern Times (New York, 1972), p. 880.

(15) Franceschetti, p. 174.

(16) C.C. Gillispie (Editor), Concise Dictionary of Scientific Biography (Chicago, 1980), p. 232.

(17)Ibid., p. 232.

(18) Ibid., p. 232.

(19) Ibid., p. 232.

(20) Franceschetti, p. 174.

(21) Encyclopaedia Britannica, Micropaedia, Volume 4, 1993, 15th Edition, p. 590.

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