GOD CREATED
THE INTEGERS

GOD CREATED THE INTEGERS

THE MATHEMATICAL BREAKTHROUGHS THAT CHANGED HISTORY

EDITED, WITH COMMENTARY, BY

STEPHEN
HAWKING

RUNNING PRESS

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A NOTE ON THE TEXTS

The texts in this book are based on translations of the original, printed editions. We have made no attempt to modernize the authors’ own distinct usage, spelling or punctuation, or to make the texts consistent with each other in this regard.

The Editors

TEXT AND PICTURE CREDITS

Text Permissions:

Selections from Euclid’s Elements, by Thomas L. Heath, courtesy of Dover Publications.

Selections from The Works of Archimedes, by Thomas L. Heath, courtesy of Dover Publications.

Selections from Diophantus of Alexandria, A Study in the History of Greek Algebra, by Thomas L. Heath, reprinted with permission of Cambridge University Press.

The Geometry of Rene Descartes, trans. David E. Smith and Marcia L. Latham courtesy of Dover Publications.

Selections from Isaac Newton’s Principia, notes by David Eugene Smith, courtesy of New York: Daniel Adee, © 1848.

English translation of Leonhard Euler’s On the sums of series of reciprocals (De summis serierum reciprocarum) courtesy of Jordan Bell.

Leonhard Euler’s The Seven Bridges of Konigsberg and Proof that Every Integer is A Sum of Four Squares courtesy of Dover Publications.

Pierre Simon Laplace’s A Philosophical Essay on Probabilities, introductory note by E.T. Bell, courtesy of Dover Publications.

Selection from Jean Baptiste Joseph Fourier’s The Analytical Theory of Heat courtesy of Dover Publications.

Selections from Carl Friedrich Gauss’s Disquisitiones Arithmeticae courtesy of Yale University Press.

Selections from Oeuvres complètes d’Augustin Cauchy reprinted from 1882 version published by Gauthier-Villars, Paris.

Nikolai Ivanovich Lobachevsky’s Geometrical Researches on the Theory of Parallels, trans. Dr. George Bruce Halstead, courtesy of Dover Publications.

János Bolyai’s The Science of Absolute Space, trans. Dr. George Bruce Halstead, courtesy of Dover Publications.

English translation of Evariste Galois’ On the conditions that an equation be soluble by radicals and Of the primitive equations which are soluble by radicals, courtesy of John Anders.

Evariste Galois’ On Groups and Equations and Abelian Integrals, courtesy of Dover publications.

George Boole’s An Investigation of the Laws of Thought courtesy of Dover Publications.

Georg Friedrich Bernhard Riemann’s Ueber die Darstellbarkeit einer Function durch einer trigonometrische Reihe, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, and Ueber die Hypothesen welche der Geometrie zu Grunde liegen, courtesy of Dover Publications.

Karl Weierstrass’ Ausgewählte Kapitel aus der Funktionenlehre: Vorlesung gehalten in Berlin 1886, mit der akademischen Antrittsrede, Berlin 1857, und drei weiteren Originalarbeiten . . . aus den Jahren 1870 bis 1880/86 / K. Weierstrass; herausgegeben, kommentiert und mit einem Anhang versehen von R. Siegmund-Schultze reprinted from R. Siegmund-Schultze (Ed.), Teubner, Leipzig, 1886 (Springer, Berlin), 1998.

Richard Dedekind’s Essays on the Theory of Numbers, trans. Wooster W. Beman, courtesy of Dover Publications.

Selections from Georg Cantor’s Contributions to the Founding of the Theory of Transfinite Numbers, trans. Philip E.B. Jourdain, courtesy of Dover Publications.

Selections from Henri Lebesgue’s Integrale, Longeur, Aire reprinted from Annali di Matematica, Pura ed Applicata, 1902, Ser. 3, vol. 7, pp. 231–359.

Kurt Gödel’s On Formally Undecidable Propositions of Principia Mathematica and Related Systems, trans. B. Meltzer, courtesy of Dover Publications.

Alan Turing’s On computable numbers with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society courtesy of the London Mathematical Society.

Picture Credits:

Euclid: Getty Images.

Archimedes: Getty Images.

Diophantus: Title page of Diophanti Alexandrini Arthimeticorum libri sex. . . ., 1621: Library of Congress, call number QA31.D5, Rare Book/Special Collections Reading Room, (Jefferson LJ239).

Rene Descartes: Getty Images.

Isaac Newton: Time Life Pictures/Getty Images.

Leonhard Euler: Getty Images.

Pierre Simon de Laplace: Getty Images.

Jean Baptiste Joseph Fourier: Science and Society Picture Library, London.

Carl Friedrich Gauss: Getty Images.

Augustin-Louis Cauchy: © Bettmann/CORBIS.

Évariste Galois: © Bettmann/CORBIS.

George Boole: © CORBIS.

Georg Friedrich Bernhard Riemann: © Bettmann/CORBIS.

Karl Weierstrass: © Bettmann/CORBIS.

Richard Dedekind: Frontispiece from Richard Dedekind Gesammelte mathematische Werke. Reprint by Chelsea Publishing Company, Bronx, NY, 1969, of the edition published in Brunswick by F. Vieweg, 1930–32. Reprinted by permission of the Chelsea Publishing Company. Photograph provided by the Library of Congress, call number QA3.D42.

Georg Cantor: © CORBIS.

Henri Lebesgue: Frontispiece from Henri Lebesgue Oeuvres Scientifiques, volume I. Reproduced by permission of L’Enseignement Mathématique, Universite De Geneve, Switzerland. Photograph provided by the Library of Congress, call number QA3.L27, vol. 1, copy 1.

Kurt Gödel: Time Life Pictures/Getty Images.

Alan Turing: Photo provided by King’s College Archive Centre, Cambridge, UK, AMT/K/7/9. Contact the Archive Centre for copyright information.

CONTENTS

Introduction

EUCLID

(C. 325BC–265BC)

His Life and Work

Selections from Euclid’s Elements

Book I: Basic Geometry—Definitions, Postulates, Common Notions; and Proposition 47, (leading up to the Pythagorean Theorem)

Book V: The Eudoxian Theory of Proportion—Definitions & Propositions

Book VII: Elementary Number Theory—Definitions & Propositions

Book IX: Proposition 20: The Infinitude of Prime Numbers

Book IX: Proposition 36: Even Perfect Numbers

Book X: Commensurable and Incommensurable Magnitudes

ARCHIMEDES

(287BC–212BC)

His Life and Work

Selections from The Works of Archimedes

On the Sphere and Cylinder, Books I and II

Measurement of a Circle

The Sand Reckoner

The Methods

DIOPHANTUS

(C. 200–284)

His Life and Work

Selections from Diophantus of Alexandria, A Study in the History of Greek Algebra

Book II Problems 8–35

Book III Problems 5–21

Book V Problems 1–29

RENÉ DESCARTES

(1596–1650)

His Life and Work

The Geometry of Rene Descartes

ISAAC NEWTON

(1642–1727)

His Life and Work

Selections from Principia

On First and Last Ratios of Quantities

LEONHARD EULER

(1707–1783)

His Life and Work

On the sums of series of reciprocals (De summis serierum reciprocarum)

The Seven Bridges of Konigsberg

Proof that Every Integer is A Sum of Four Squares

PIERRE SIMON LAPLACE

(1749–1827)

His Life and Work

A Philosophical Essay on Probabilities

JEAN BAPTISTE JOSEPH FOURIER

(1768–1830)

His Life and Work

Selection from The Analytical Theory of Heat

Chapter III: Propagation of Heat in an Infinite Rectangular Solid (The Fourier series)

CARL FRIEDRICH GAUSS

(1777–1855)

His Life and Work

Selections from Disquisitiones Arithmeticae (Arithmetic Disquisitions)

Section III Residues of Powers

Section IV Congruences of the Second Degree

AUGUSTIN-LOUIS CAUCHY

(1789–1857)

His Life and Work

Selections from Oeuvres complètes d’Augustin Cauchy Résumé des leçons données à l’École Royale Polytechnique sur le calcul infinitésimal (1823), series 2, vol. 4

Lessons 3–4 on differential calculus

Lessons 21–24 on the integral

NIKOLAI IVANOVICH LOBACHEVSKY

(1792–1856)

His Life and Work

Geometrical Researches on the Theory of Parallels

JÁNOS BOLYAI

(1802–1860)

His Life and Work

The Science of Absolute Space

ÉVARISTE GALOIS

(1811–1832)

His Life and Work

On the conditions that an equation be soluble by radicals

Of the primitive equations which are soluble by radicals

On Groups and Equations and Abelian Integrals

GEORGE BOOLE

(1815–1864)

His Life and Work

An Investigation of the Laws of Thought

BERNHARD RIEMANN

(1826–1866)

His Life and Work

On the Representability of a Function by Means of a Trigonometric Series (Ueber die Darstellbarkeit eine Function durch einer trigonometrische Reihe)

On the Hypotheses which lie at the Bases of Geometry (Ueber die Hypothesen, welche der Geometrie zu Grunde liegen)

On the Number of Prime Numbers Less than a Given Quantity (Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse)

KARL WEIERSTRASS

(1815–1897)

His Life and Work

Selected Chapters on the Theory of Functions, Lecture Given in Berlin in 1886, with the Inaugural Academic Speech, Berlin 1857

§ 7 Gleichmässige Stetigkeit (Uniform Continuity)

RICHARD DEDEKIND

(1831–1916)

His Life and Work

Essays on the Theory of Numbers

GEORG CANTOR

(1848–1918)

His Life and Work

Selections from Contributions to the Founding of the Theory of Transfinite Numbers

Articles I and II

HENRI LEBESGUE

(1875–1941)

His Life and Work

Selections from Integrale, Longeur, Aire (Intergral, Length, Area)

Preliminaries and Integral

KURT GÖDEL

(1906–1978)

His Life and Work

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

ALAN TURING

(1912–1954)

His Life and Work

On computable numbers with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society

INTRODUCTION

WE ARE LUCKY TO LIVE IN AN AGE LN WHICH WE ARE STILL MAKING DISCOVERIES. IT IS LIKE THE DISCOVERY OF AMERICA-YOU ONLY DISCOVER IT ONCE. THE AGE IN WHICH WE LIVE IS THE AGE IN WHICH WE ARE DISCOVERING THE FUNDAMENTAL LAWS OF NATURE . . .

—AMERICAN PHYSICIST RICHARD FEYNMAN, SPOKEN IN 1964

Showcasing excerpts from thirty-one of the most important works in the history of mathematics (four of which have been translated into English for the very first time), this book is a celebration of the mathematicians who helped move us forward in our understanding of the world and who paved the way for our current age of science and technology.

Over the centuries, the efforts of these mathematicians have helped the human race to achieve great insight into nature, such as the realization that the earth is round, that the same force that causes an apple to fall here on earth is also responsible for the motions of the heavenly bodies, that space is finite and not eternal, that time and space are intertwined and warped by matter and energy, and that the future can only be determined probabilistically. Such revolutions in the way we perceive the world have always gone hand in hand with revolutions in mathematical thought. Isaac Newton could never have formulated his laws without the analytic geometry of René Descartes and Newton’s own invention of calculus. It is hard to imagine the development of either electrodynamics or quantum theory without the methods of Jean Baptiste Joseph Fourier or the work on calculus and the theory of complex functions pioneered by Carl Friedrich Gauss and Augustin-Louis Cauchy—and it was Henri Lebesgue’s work on the theory of measure that enabled John von Neumann to formulate the rigorous understanding of quantum theory that we have today. Albert Einstein could not have completed his general theory of relativity had it not been for the geometric ideas of Bernhard Riemann. And practically all of modern science would be far less potent (if it existed at all) without the concepts of probability and statistics pioneered by Pierre Simon Laplace.

All through the ages, no intellectual endeavor has been more important to those studying physical science than has the field of mathematics. But mathematics is more than a tool and language for science. It is also an end in itself, and as such, it has, over the centuries, affected our worldview in its own right. Karl Weierstrass provided a new idea of what it means for a function to be continuous, and Georg Cantors work revolutionized peoples idea of infinity. George Boole’s Laws of Thought revealed logic as a system of processes subject to laws identical to the laws of algebra, thus illuminating the very nature of thought and eventually enabling to some degree its mechanization, that is, modern digital computing. Alan Turing illuminated the power and the limits of digital computing, long before sophisticated computations were even possible. Kurt Gödel proved a theorem troubling to many philosophers, as well as anyone else believing in absolute truth: that in any sufficiently complex logical system (such as arithmetic) there must exist statements that can neither be proven nor disproven. And if that weren’t bad enough, he also showed that the question of whether the system itself is logically consistent cannot be proven within the system.

This fascinating volume presents all these and other groundbreaking developments, the central ideas in twenty-five centuries of mathematics, employing the original texts to trace the evolution, and sometimes revolution, in mathematical thinking from its beginnings to today.

Though the first work presented here is that of Euclid, C.300 B.C., the Egyptians and Babylonians had developed an impressive ability to perform mathematical calculations as early as 3,500 B.C. The Egyptians employed this skill to build the great pyramids and to accomplish other impressive ends, but their computations lacked one quality considered essential to mathematics ever since: rigor. For example, the ancient Egyptians equated the area of a circle to the area of a square whose sides were 8/9 the diameter of the circle. This method amounts to employing a value of the mathematical constant pi that is equal to 256/81. In one sense this is impressive—it is only about one half of one percent off of the exact answer. But in another sense it is completely wrong. Why worry about an error of one half of one percent? Because the Egyptian approximation overlooks one of the deep and fundamental mathematical properties of the true number π: that it cannot be written as any fraction. That is a matter of principle, unrelated to any issue of mere quantitative accuracy. Though the irrationality of π wasn’t proved until the late eighteenth century, the early Greeks did discover that numbers existed which could not be written as fractions, and this was both puzzling and shocking to them. This was the brilliance of the Greeks: to recognize the importance of principle plura in mathematics, and that in its essence mathematics is a subject in which one begins with a set of concepts and rules and then rigorously works out their precise consequences.

Euclid detailed the Greek understanding of geometry in his Elements, in Alexandria, around 300 B.C. In the ensuing centuries the Greeks made great strides in both algebra and geometry. Archimedes, the greatest mathematician of antiquity, studied the properties of geometric shapes and created ingenious methods of finding areas and volumes and new approximations for π. Another Alexandrian, Diophantus, looking over the clutter of words and numbers in algebraic problems, saw that an abstraction could be a great simplification. And so, Diophantus took the first step toward introducing symbolism into algebra. Over a millennium later, Frenchman René Descartes united the two fields: geometry and algebra, with his creation of analytic geometry. His work paved the way for Isaac Newton to invent calculus, and with it, a new way of doing science. Since Newton’s day, the pace of mathematical innovation has been almost frenetic, as the fundamental mathematical fields of algebra, geometry, and calculus (or function theory) have fed on and in turn nourished one another, yielding insights into applications as diverse as probability, numbers, and the theory of heat. And as mathematics matured, so did the range of questions it addresses: Kurt Gödel and Alan Turing, the last two thinkers represented in this volume, address perhaps the deepest issue—the question of what is knowable. Like those of the past, future developments in mathematics are sure to affect, directly or indirectly, our ways of living and thinking. The wonders of the ancient world were physical, like the pyramids in Egypt. As this volume illustrates, the greatest wonder of the modern world is our own understanding.

Euclid

(c.325B.C.–c.265B.C.)

HIS LIFE AND WORK

With the possible exception of Isaac Newton, Euclid is the best known mathematician of all time. Until sometime in the twentieth century, his one and only surviving work, The Elements, was the second best selling book of all time, surpassed only by the Bible. Yet there is no reason to believe that Euclid was anything other than a compiler of the mathematics known in his times—akin to Noah Webster, the great nineteenth-century lexicographer who gave his name to the leading American dictionary.

There is very little known about Euclid. He taught at an academy in Alexandria, the Hellenic city that Alexander the Great founded at the mouth of the Nile in Egypt. Because of his work as a compiler, Euclid was familiar with all of the Greek mathematics that had preceded him, and especially familiar with the first crisis in mathematics: the crisis of the irrational.

Pythagoras (died c.475 B.C.) is a mysterious figure from early Greek mathematics. If we know little of Euclid, we know even less of Pythagoras. We do, however, know something about the Pythagorean School. The Pythagoreans thought that the whole of the cosmos could be described in terms of the whole numbers: 1, 2, 3, etc. As Aristotle wrote, “The Pythagoreans . . . having been brought up in the study of mathematics, thought that things are numbers . . . and that the whole cosmos is a scale and a number.”

The Pythagorean theorem (presented in this chapter) demonstrated this. Small whole numbers, like 3, 4, and 5, could be found that not only described the lengths of a right triangle but also had the property that when the areas of the squares erected on the two smaller sides were added together they equaled the area of the square erected on the longest side, the hypotenuse, the side opposite the right angle. Notice how the ancient Greeks phrased the Pythagorean theorem in terms of geometric objects and not in terms of numbers!

Then someone asked an interesting question. If there were a square with each side a unit in length, and a second square with double the area of the first square, how would the side of the second square compare to the side of the first square? This was how the question of the square root of 2 was originally proposed.

The ancient Egyptians found a good approximation to the answer. The side of the second square was to the side of the first square almost as 7 is to 5. This should certainly be no surprise to us since we know can also be expressed as 1.4, which is very close to what to we know to be the decimal expansion of . But close wasn’t good enough for the Pythagoreans. After all, the Pythagorean theorem didn’t assert that the areas of the squares were close to being equal. It asserted that they were, in fact, equal.

Then, someone (whose name we don’t know) hit upon a deep insight. Suppose that the square root of 2 can be expressed as the ratio of two whole numbers and that these two whole numbers share no common divisors except 1, their common unit. Call these whole numbers p and q with the property that the square P erected on a side of length p is exactly twice the area of a square Q erected on a side of length q. Now if P contains twice the number of units than Q, then P must contain an even number of units! The Pythagoreans already knew that if a square contains an even number of units, the number of units the square contains must be a multiple of 4 units of area and its sides must contain an even number of whole units of length.

Still, everyone knew how, given one square, you could find another with one fourth the area: Just erect a square on a side equal to half of that side. In this case, erect a square T whose side t is one half of the side p. Because p contains an even number of whole units of length, the side t must still contain some number of whole units of length. But then, if the square T has one fourth the area of the square P, the square Q must contain twice the number of units of area as the square T. So because the square T contains an even number of units of area, then, just like the square P, the square Q must also contain a multiple of 4 units of area. So, its side q had to contain an even number of units of length. So far, this mathematical argument has been like a tennis match—going back and forth between the players.

Eventually, the argument reaches its climax. It began by supposing that the sides p and q had no divisor in common except 1 and ended up with a contradiction: that they shared the divisor 2! Try as they might, the Pythagoreans could not find a flaw with the argument. Knowing that no one had in fact ever found a way to express the square root of 2 as the ratio of two whole numbers, the Pythagoreans faced up to the reality that they had proven that the square root of 2 could not be expressed as the ratio of two whole numbers.

Thus were born the irrational numbers, mathematical objects that could not be expressed by means of whole numbers, at least not for two millennia: the first of what Kronecker called the works of man.

The Pythagoreans carefully guarded this great discovery because it created a crisis that reached to the very roots of their cosmology. When the Pythagoreans learned that one of their members had divulged the secret to someone outside their circle, they quickly made plans to throw the betrayer overboard and drown him while on the high seas. Whoever he was, this man was the first martyr for mathematics!

The crisis of the irrational also instructed the ancient Greeks that they could not look to arithmetic to form the foundation for the rest of mathematics and with it an explanation for the structure of the cosmos. They had to look elsewhere. They turned to geometry.

Euclid’s Elements is most remembered for its geometry, and especially remembered for its treatment of parallel lines in the definition of parallel lines:

Parallel straight lines are straight lines which, being in the same plane and produced indefinitely in both directions, do not meet one another in either direction.

And in the fifth postulate, the parallel postulate:

That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

This is very different from the form in which it is often presented:

Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

This is an equivalent, but different form given by the Scottish mathematician John Playfair in 1795.

During the height of the Newtonian era, philosophers, such as Immanuel Kant, never doubted the truth of Euclid’s parallel postulate. They merely inquired about the nature of its truth. Was the parallel postulate necessarily true of the cosmos or only contingently true? Of course, since the advent of the Einsteinian revolution, we know that the parallel postulate isn’t true at all about the cosmos. The Einsteinian space-time cosmos that we inhabit is curved. Euclidean geometry and Newtonian physics are only approximations.

So, we may ask, what did the Greeks think about the nature of the parallel postulate? I believe that a brief consideration of the ancient Greek conception of the world will explain that they too viewed the parallel postulate as a useful fiction rather than as a true description of the physical world. After all, the Greeks believed that we inhabited what the historian of science Alexandre Koyré has called a “closed world,” a spherical cosmos in which there were no straight lines actually extending to infinity. Below the moon’s orbits, bodies moved in straight lines either towards the center of the earth or away from it. Above the moon’s orbit, bodies orbited in perfect circles around the center of the earth. In this cosmos, there weren’t actually any straight lines at all.

But the Greeks had a problem. They needed to find a foundation for their mathematics. The Pythagoreans had pursued arithmetic as a foundation and reached a crisis. Needing to find an alternative, another school descending from Thales (died c.547 B.C.) sought to base mathematics on geometry. This school found that they could achieve very little without the parallel postulate! They couldn’t prove the Pythagorean theorem, for example. In fact, they couldn’t prove much geometry at all. This should come as no surprise to us moderns who have the benefit of 2,500 years of hindsight and know that the Pythagorean theorem is false in non-Euclidean geometries. I believe that the ancient Greeks knew that the parallel postulate was only a useful, no, let me say a very useful, approximation.

If the proof of the irrationality of the square root gave us the first crisis in mathematics, it also gave us the first example of the form of argument known since ancient times as reductio ad absurdum, reduction to absurdity. A second example of this form of argument can be seen in Euclid’s proof the infinitude of prime numbers, another proof that certainly originated with someone else.

A prime number is a positive integer, such as 3 or 23, whose only positive integer divisors are 1 and itself. Proving the infinitude of prime numbers is stunningly simple. Suppose there is a largest prime number P. Multiply together all of the prime numbers up to an including P. Now add 1. The result is not divisible by P, nor is it divisible by any of the prime numbers less than P, because P and all of the prime numbers less than it evenly divides into their product before 1 is added to it. Supposing that there is a largest prime number leads to a contradiction. Reductio ad absurdum!

The Greeks noticed that many prime numbers come together in pairs: 11 and 13, 17 and 19, 29 and 31, for example. These are called twin primes. Some have belleves that the Greeks speculated that there were not only an infinitude of prime numbers but also an infinitude of twin primes. But they weren’t able to prove that, nor has any mathematician since.

Neither has any mathematician been able to disprove the existence of an odd perfect number. A perfect number: That sounds quite odd indeed! What is a perfect number? A perfect number is the sum of its integer divisors greater than or equal to 1 but less than itself, what are called its proper divisors. The ancient Greeks found all of the even perfect numbers as follows:

Notice that the sum of the powers of 2 from 1, which is 2⁰, to 2n−1 is equal to 2n−1. In a very simple case, for n = 3, 1 + 2 + 4 = 7 = 8 − 1. Now let’s do some simple arithmetic.

7 | = 1 + 2 + 4 | = 2⁰ + 2¹ + 2²
7 | = 8 − 1 | = 2³ − 2⁰
14 | = 16 − 2 | = 2⁴ − 2¹

These columns all add up to 28. Writing that sum in yet another we see that

28 = 1 + 2 + 4 + 7 + 14 = 2² + 2³ + 2⁴ = 2² × (2⁰ + 2¹ + 2²) = 2² × (2³ − 1)

28 is the sum of its divisors. Notice how these divisors are first all the powers of 2 up to a given exponent, then 1 less than the next power of 2, call this the turning point divisor, and then the turning point divisor times all of the powers of 2 up to that given exponent. And notice that if 7 were not a prime number, then 28 would not be equal to the sum of its proper divisors. If the turning divisor had just one prime aliquot divisor that would cause the sum of all the proper divisors to overflow.

With these observations the Greeks had proven that

If (2n − 1) is a prime number then 2n−1 x (2n − 1) is a perfect number and that even perfect numbers must have this form.

More than two millennia later, no one has ever discovered an odd perfect number. No mathematician believes that an odd perfect number exists. But none has been able to prove that no odd perfect number exists!

The Pythagoreans tried and failed to found all of mathematics on arithmetic. Re-founding mathematics on geometry meant founding arithmetic on geometry.

Quick, which is larger: or ? That may be too easy for you. Try this one instead, without a calculator: which is larger: or ? Try doing it just using multiplication and not division. The Eudoxian theory of proportion, presented in Book V of Euclid’s Elements provides the tools, namely, multiplication, for arriving at an answer.

Following Eudoxus (died c.355 B.C.), Euclid posed the problem as follows: consider 4 magnitudes of length—a, b, c, and d. How can one determine whether the ratio of a to b is greater than, less than, or equal to the ratio of c to d? Eudoxus began by asserting that “magnitudes are said to have a ratio to one another which is capable, when a multiple of either may exceed the other.” He recognized that if the ratio of a to b is greater than the ratio of c to d, all multiples of the ratio of a to b are greater than the same number of multiples of the ratio of c to d. Recognizing this fact, Eudoxus realized that all he had to do was to find one multiple to use and he could solve the problem. The multiple he chose was the product of b and d. Multiplying the ratio of a to b by the product of b and d, gave the ratio of the product of a, b and d to b, or the area of the rectangle with sides of length a and d. Similarly, multiplying the ratio of c to d by the product of b and d gave the ratio of the product of c, b, and d to d, or the area of a rectangle with sides of c and b.

Thus, the area of the rectangle with side a and d is greater than the area of the rectangle with sides of length c and b, if and only if the ratio of a to b is greater than the ratio of c to d. Whereas the Pythagoreans had tried to arithmetize geometry and failed, when Eudoxus tried to geometrize arithmetic he succeeded! By the way, since 19 × 19 is greater than 30 × 12, is greater than !

Euclid is the greatest mathematical encyclopediast of all times. Today, when mathematicians from different specialties have difficulty understanding work at the frontiers of each others’ specialty, no one mathematician could hope to edit a compendium of all known mathematics. Yet it remains an ideal in the mathematical community. In the second half of the twentieth century, the French mathematical community offered a pale imitation of Euclid in the person of Nicholas Bourbaki. In fact, Nicholas Bourbaki wasn’t even a person. He was the fictitious nom de plume of a collective of more than twenty French mathematicians working in diverse branches of mathematics! Euclid remains, to this day, our model for mathematical texts.

SELECTIONS FROM EUCLID’S ELEMENTS

BOOK I

BASIC GEOMETRY—DEFINITIONS, POSTULATES, COMMON NOTIONS AND PROPOSITION 47 (LEADING UP TO THE PYTHAGOREAN THEOREM)

DEFINITIONS

1. A point is that which has no part.

2. A line is breadthless length.

3. The extremities of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The extremities of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

9. And when the lines containing the angle are straight, the angle is called rectilineal.

10. When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than a right angle.

12. An acute angle is an angle less than a right angle.

13. A boundary is that which is an extremity of anything.

14. A figure is that which is contained by any boundary or boundaries.

15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;

16. And the point is called the centre of the circle.

17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.

19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

POSTULATES

Let the following be postulated:

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to one another.

5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

COMMON NOTIONS

1. Things which are equal to the same thing are also equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equal.

4. Things which coincide with one another are equal to one another.

5. The whole is greater than the part.

PROPOSITION 47

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. [1]

Let ABC be a right-angled triangle having the angle BAC right;

I say that the square on BC is equal to the squares on BA, AC.

For let there be described on BC the square BDEC,

and on BA, AC the squares GB, HC;

[i. 46]

through A let AL be drawn parallel to either BD or CE, and let AD, FC be joined.

Then, since each of the angles BAC, BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AC, AG not lying on the same side make the adjacent angles equal to two right angles;

therefore CA is in a straight line with AG.

[i. 14]

For the same reason

BA is also in a straight line with AH.

And, since the angle DBC is equal to the angle FBA: for each is right: let the angle ABC be added to each;

therefore the whole angle DBA is equal to the whole angle FBC.

[C. N. 2]

And, since DB is equal to BC, and FB to BA,

the two sides AB, BD are equal to the two sides FB, BC respectively, [2]

and the angle ABD is equal to the angle FBC;

therefore the base AD is equal to the base FC,

and the triangle ABD is equal to the triangle FBC.

[i. 4]

Now the parallelogram BL is double of the triangle ABD, for they have the same base BD and are in the same parallels BD, AL.

[i. 41]

And the square GB is double of the triangle FBC,

for they again have the same base FB and are in the same parallels FB, GC.

[i. 41]

[But the doubles of equals are equal to one another.] [3]

Therefore the parallelogram BL is also equal to the square GB.

Similarly, if AE, BK be joined,

the parallelogram CL can also be proved equal to the square HC;

therefore the whole square BDEC is equal to the two squares GB, HC.

[C. N. 2]

And the square BDEC is described on BC,

and the squares GB, HC on BA, AC.

Therefore the square on the side BC is equal to the squares on the sides BA, AC.

Therefore etc.

Q.E.D.

“If we listen,” says Proclus, “to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discovery. But for my part, while I admire those who first observed the truth of this theorem, I marvel more at the writer of the Elements, not only because he made it fast () by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth Book. For in that Book he proves generally that, in right-angled triangles, the figure on the side subtending the right angle is equal to the similar and similarly situated figures described on the sides about the right angle.”

In addition, Plutarch, Diogenes Laertius (viii. 12) and Athenaeus (x. 13) agree in attributing this proposition to Pythagoras. It is easy to point out, as does G. Junge (“Wann haben die Griechen das Irrationale entdeckt?” in Novae Symbolae Joachimicae, Halle a. S., 1907), that these are late witnesses, and that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometrical discovery as due to him. Yet the distich of Apollodorus the “calculator,” whose date (though it cannot be fixed) is at least earlier than that of Plutarch and presumably of Cicero, is quite definite as to the existence of one “famous proposition” discovered by Pythagoras, whatever it was. Nor does Cicero, in commenting apparently on the verses (De nat. deor. iii. C. 36, § 88), seem to dispute the fact of the geometrical discovery, but only the story of the sacrifice. Junge naturally emphasises the apparent uncertainty in the statements of Plutarch and Proclus. But, as I read the passages of Plutarch, I see nothing in them inconsistent with the supposition that Plutarch unhesitatingly accepted as discoveries of Pythagoras both the theorem of the square of the hypotenuse and the problem of the application of an area, and the only doubt he felt was as to which of the two discoveries was the more appropriate occasion for the supposed sacrifice. There is also other evidence not without bearing on the question. The theorem is closely connected with the whole of the matter of Eucl. Book II., in which one of the most prominent features is the use of the gnomon. Now the gnomon was a well-understood term with the Pythagoreans. Aristotle also (Physics iii. 4, 203 a 10–15) clearly attributes to the Pythagoreans the placing of odd numbers as gnomons round successive squares beginning with 1, thereby forming new squares, while in another place (Categ. 14, 15 a 30) the word gnomon occurs in the same (obviously familiar) sense: “e.g. a square, when a gnomon is placed round it, is increased in size but is not altered in form.” The inference must therefore be that practically the whole doctrine of Book II. is Pythagorean. Again Heron (First century. A.D.), like Proclus, credits Pythagoras with a general rule for forming right-angled triangles with rational whole numbers for sides. Lastly, the “summary” of Proclus appears to credit Pythagoras with the discovery of the theory, or study, of irrationals (). But it is now more or less agreed that the reading here should be, not , but , or rather (“of proportionals”), and that the author intended to attribute to Pythagoras a theory of proportion, i.e. the (arithmetical) theory of proportion applicable only to commensurable magnitudes, as distinct from the theory of Eucl. Book V., which was due to Eudoxus. It is not however disputed that the Pythagoreans discovered the irrational (cf. the scholium No. 1 to Book X.). Now everything goes to show that this discovery of the irrational was made with reference to , the ratio of the diagonal of a square to its side. It is clear that this presupposes the knowledge that i. 47 is true of an isosceles right-angled triangle; and the fact that some triangles of which it had been discovered to be true were rational right-angled triangles was doubtless what suggested the inquiry whether the ratio between the lengths of the diagonal and the side of a square could also be expressed in whole numbers. On the whole, therefore, I see no sufficient reason to question the tradition that, so far as Greek geometry is concerned (the possible priority of the discovery of the same proposition in India will be considered later), Pythagoras was the first to introduce the theorem of i. 47 and to give a general proof of it.

On this assumption, how was Pythagoras led to this discovery? It has been suggested and commonly assumed that the Egyptians were aware that a triangle with its sides in the ratio 3, 4, 5 was right-angled. Cantor inferred this from the fact that this was precisely the triangle with which Pythagoras began, if we may accept the testimony of Vitruvius (ix. 2) that Pythagoras taught how to make a right angle by means of three lengths measured by the numbers 3, 4, 5. If then he took from the Egyptians the triangle 3, 4, 5, he presumably learnt its property from them also. Now the Egyptians must certainly be credited from a period at least as far back as 2000 B.C. with the knowledge that 4² + 3² = 5². Cantor finds proof of this in a fragment of papyrus belonging to the time of the 12th Dynasty newly discovered at Kahun. In this papyrus we have extractions of square roots: e.g. that of 16 is 4, that of is , that of is , and the following equations can be traced:

It will be seen that 4² + 3² = 5² can be derived from each of these by multiplying, or dividing out, by one and the same factor. We may therefore admit that the Egyptians knew that 3² + 4² = 5². But there seems to be no evidence that they knew that the triangle (3, 4, 5) is right-angled; indeed, according to the latest authority (T. Eric Peet, The Rhind Mathematical Papyrus, 1923), nothing in Egyptian mathematics suggests that the Egyptians were acquainted with this or any special cases of the Pythagorean theorem.

How then did Pythagoras discover the general theorem? Observing that 3, 4, 5 was a right-angled triangle, while 3² + 4² = 5², he was probably led to consider whether a similar relation was true of the sides of right-angled triangles other than the particular one. The simplest case (geometrically) to investigate was that of the isosceles right-angled triangle; and the truth of the theorem in this particular case would easily appear from the mere construction of a figure. Cantor and Allman (Greek Geometry from Thales to Euclid) illustrate by a figure in which the squares are drawn outwards, as in i. 47, and divided by diagonals into equal triangles; but I think that the truth was more likely to be first observed from a figure of the kind suggested by Bürk (Das Āpastamba-Śulba-Sūtra in Zeitschrift der deuts hen morgenländ. Gesellschaft, LV., 1901) to explain how the Indians arrived at the same thing. The two figures are as shown above. When the geometrical consideration of the figure had shown that the isosceles right-angled triangle had the property in question, the investigation of the same fact from the arithmetical point of view would ultimately lead to the other momentous discovery of the irrationality of the length of the diagonal of a square expressed in terms of its side.

The irrational will come up for discussion later; and our next question is: Assuming that Pythagoras had observed the geometrical truth of the theorem in the case of the two particular triangles, and doubtless of other rational right-angled triangles, how did he establish it generally?

There is no positive evidence on this point. Two possible lines are however marked out. (1) Tannery says (La Géométrie grecque) that the geometry of Pythagoras was sufficiently advanced to make it possible for him to prove the theorem by similar triangles. He does not say in what particular manner similar triangles would be used, but their use must apparently have involved the use of proportions, and, in order that the proof should be conclusive, of the theory of proportions in its complete form applicable to incommensurable as well as commensurable magnitudes. Now Eudoxus was the first to make the theory of proportion independent of the hypothesis of commensurability; and as, before Eudoxus’ time, this had not been done, any proof of the general theorem by means of proportions given by Pythagoras must at least have been inconclusive. But this does not constitute any objection to the supposition that the truth of the general theorem may have been discovered in such a manner; on the contrary, the supposition that Pythagoras proved it by means of an imperfect theory of proportions would better than anything else account for the fact that Euclid had to devise an entirely new proof, as Proclus says he did in i. 47. This proof had to be independent of the theory of proportion even in its rigorous form, because the plan of the Elements postponed that theory to Books V. and VI., while the Pythagorean theorem was required as early as Book II. On the other hand, if the Pythagorean proof had been based on the doctrine of Books I. and II. only, it would scarcely have been necessary for Euclid to supply a new proof.

The possible proofs by means of proportion would seem to be practically limited to two.

(a) One method is to prove, from the similarity of the triangles ABC, DBA, that the rectangle CB, BD is equal to the square on BA, and, from the similarity of the triangles ABC, DAC, that the rectangle BC, CD is equal to the square on CA; whence the result follows by addition.

It will be observed that this proof is in substance identical with that of Euclid, the only difference being that the equality of the two smaller squares to the respective rectangles is inferred by the method of Book VI. instead of from the relation between the areas of parallelograms and triangles on the same base and between the same parallels established in Book I. It occurred to me whether, if Pythagoras’ proof had come, even in substance, so near to Euclid’s, Proclus would have emphasised so much as he does the originality of Euclid’s, or would have gone so far as to say that he marvelled more at that proof than at the original discovery of the theorem. But on the whole I see no difficulty; for there can be little doubt that the proof by proportion is what suggested to Euclid the method of i. 47, and the transformation of the method of proportions into one based on Book I. only, effected by a construction and proof so extraordinarily ingenious, is a veritable tour de force which compels admiration, notwithstanding the ignorant strictures of Schopenhauer, who wanted something as obvious as the second figure in the case of the isosceles right-angled triangle, and accordingly (Sämmtliche. Werke, iii. § 39 and i. § 15) calls Euclid’s proof “a mouse-trap proof” and “a proof walking on stilts, nay, a mean, underhand, proof” (“Des Eukleides stelzbeiniger, ja, hinterlistiger Beweis”).

(b) The other possible method is this. As it would be seen that the triangles into which the original triangle is divided by the perpendicular from the right angle on the hypotenuse are similar to one another and to the whole triangle, while in these three triangles the two sides about the right angle in the original triangle, and the hypotenuse of the original triangle, are corresponding sides, and that the sum of the two former similar triangles is identically equal to the similar triangle on the hypotenuse, it might be inferred that the same would also be true of squares described on the corresponding three sides respectively, because squares as well as similar triangles are to one another in the duplicate ratio of corresponding sides. But the same thing is equally true of any similar rectilineal figures, so that this proof would practically establish the extended theorem of Eucl. vi. 31, which theorem, however, Proclus appears to regard as being entirely Euclid’s discovery.

On the whole, the most probable supposition seems to me to be that Pythagoras used the first method (a) of proof by means of the theory of proportion as he knew it, i.e. in the defective form which was in use up to the date of Eudoxus.

(2) I have pointed out the difficulty in the way of the supposition that Pythagoras’ proof depended upon the principles of Eucl. Books I. and II. only. Were it not for this difficulty, the conjecture of Bretschneider, followed by Hankel, would be the most tempting hypothesis. According to this suggestion, we are to suppose a figure like that of Eucl. II. 4 in which a, b are the sides of the two inner squares respectively, and a + b is the side of the complete square. Then, if the two complements, which are equal, are divided by their two diagonals into four equal triangles of sides a, b, c, we can place these triangles round another square of the same size as the whole square, in the manner shown in the second figure, so that the sides a, b of successive triangles make up one of the sides of the square and are arranged in cyclic order. It readily follows that the remainder of the square when the four triangles are deducted is, in the one case, a square whose side is c, and in the other the sum of two squares whose sides are a, b respectively. Therefore the square on c is equal to the sum of the squares on a, b. All that can be said against this conjectural proof is that it has no specifically Greek colouring but rather recalls the Indian method. Thus Bhāskara (born 1114 A.D.) simply draws four right-angled triangles equal to the original one inwards, one on each side of the square on the hypotenuse, and says “see!”, without even adding that inspection shows that

Though, for the reason given, there is difficulty in supposing that Pythagoras used a general proof of this kind, which applies of course to right-angled triangles with sides incommensurable as well as commensurable, there is no objection, I think, to supposing that the truth of the proposition in the case of the first rational right-angled triangles discovered, e.g. 3, 4, 5, was proved by a method of this sort. Where the sides are commensurable in this way, the squares can be divided up into small (unit) squares, which would much facilitate the comparison between them. That this subdivision was in fact resorted to in adding and subtracting squares is made probable by Aristotle’s allusion to odd numbers as gnomons placed round unity to form successive squares in Physics III. 4; this must mean that the squares were represented by dots arranged in the form of a square and a gnomon formed of dots put round, or that (if the given square was drawn in the usual way) the gnomon was divided up into unit squares. Zeuthen has shown (“Théorème de Pythagore,” Origine de la Géométrie scientifique in Comptes rendus du IIme Congrès international de Philosophie, Genève, 1904), how easily the proposition could be proved by a method of this kind for the triangle 3, 4, 5. To admit of the two smaller squares being shown side by side, take a square on a line containing 7 units of length (4 + 3), and divide it up into 49 small squares. It would be obvious that the whole square could be exhibited as containing four rectangles of sides 4, 3 cyclically arranged round the figure with one unit square in the middle. (This same figure is given by Cantor to illustrate the method given in the Chinese “Chóu-peï”.) It would be seen that

(I) the whole square (7²) is made up of two squares 3² and 4², and two rectangles 3, 4;

(II) the same square is made up of the square EFGH and the halves of four of the same rectangles 3, 4, whence the square EFGH, being equal to the sum of the squares 3² and 4², must contain 25 unit squares and its side, or the diagonal of one of the rectangles, must contain 5 units of length.

Or the result might equally be seen by observing that

(I) the square EFGH on the diagonal of one of the rectangles is made up of the halves of four rectangles and the unit square in the middle, while

(II) the squares 3² and 4² placed at adjacent corners of the large square make up two rectangles 3, 4 with the unit square in the middle.

The procedure would be equally easy for any rational right-angled triangle, and would be a natural method of trying to prove the property when it had once been empirically observed that triangles like 3, 4, 5 did in fact contain a right angle.

Zeuthen has, in the same paper, shown in a most ingenious way how the property of the triangle 3, 4, 5 could be verified by a sort of combination of the second possible method by similar triangles, (b) on p. 13 above, with subdivision of rectangles into similar small rectangles. I give the method on account of its interest, although it is no doubt too advanced to have been used by those who first proved the property of the particular triangle.

Let ABC be a triangle right-angled at A, and such that the lengths of the sides AB, AC are 4 and 3 units respectively.

Draw the perpendicular AD, divide up AB, AC into unit lengths, complete the rectangle on BC as base and with AD as altitude, and subdivide this rectangle into small rectangles by drawing parallels to BC, AD through the points of division of AB, AC.

Now, since the diagonals of the small rectangles are all equal, each being of unit length, it follows by similar triangles that the small rectangles are all equal. And the rectangle with AB for diagonal contains 16 of the small rectangles, while the rectangle with diagonal AC contains 9 of them.

But the sum of the triangles ABD, ADC is equal to the triangle ABC.

Hence the rectangle with BC as diagonal contains 9 + 16 or 25 of the small rectangles; and therefore BC = 5.

Rational right-angled triangles from the arithmetical standpoint.

Pythagoras investigated the arithmetical problem of finding rational numbers which could be made the sides of right-angled triangles, or of finding square numbers which are the sum of two squares; and herein we find the beginning of the indeterminate analysis which reached so high a stage of development in Diophantus. Fortunately Proclus has preserved Pythagoras’ method of solution in the following passage. “Certain methods for the discovery of triangles of this kind are handed down, one of which they refer to Plato, and another to Pythagoras. [The latter] starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity, and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse. For example, taking 3, squaring it, and subtracting unity from the 9, the method takes half of the 8, namely 4; then, adding unity to it again, it makes 5, and a right-angled triangle has been found with one side 3, another 4 and another 5. But the method of Plato argues from even numbers. For it takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds’ unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. For example, taking 4, the method squares half of this, or 2, and so makes 4; then, subtracting unity; it produces 3, and adding unity it produces 5. Thus it has formed the same triangle as that which was obtained by the other method.”

The formula of Pythagoras amounts, if m be an odd number, to

the sides of the right-angled triangle being m, . Cantor, taking up an idea of Röth (Geschichte der abendländischen Philosophie, II. 527), gives the following as a possible explanation of the way in which Pythagoras arrived at his formula. If c² = a² + b², it follows that

a² = c² − b² = (c + b) (c − b).

Numbers can be found satisfying the first equation if (1) c + b and c − b are either both even or both odd, and if further (2) c + b and c − b are such numbers as, when multiplied together, produce a square number. The first condition is necessary because, in order that c and b may both be whole numbers, the sum and difference of c + b and c − b must both be even. The second condition is satisfied if c + b and c − b are what were called similar numbers (); and that such numbers were most probably known in the time before Plato may be inferred from their appearing in Theon of Smyrna (Expositio rerum mathematicarum ad legendum Platonem utilium, ed. Hiller), who says that similar plane numbers are, first, all square numbers and, secondly, such oblong numbers as have the sides which contain them proportional. Thus 6 is an oblong number with length 3 and breadth 2; 24 is another with length 6 and breadth 4. Since therefore 6 is to 3 as 4 is to 2, the numbers 6 and 24 are similar.

Now the simplest case of two similar numbers is that of 1 and a², and, since 1 is odd, the condition (1) requires that a², and therefore a, is also odd. That is, we may take 1 and (2n + 1)² and equate them respectively to c − b and c + b, whence we have

while

As Cantor remarks, the form in which c and b appear correspond sufficiently closely to the description in the text of Proclus.

Another obvious possibility would be, instead of equating c − b to unity, to put c − b = 2, in which case the similar number c + b must be equated to double of some square, i.e. to a number of the form 2n², or to the half of an even square number, say . This would give

a = 2n,

b = n² − 1,

c = n² + 1,

which is Plato’s solution, as given by Proclus.

The two solutions supplement each other. It is interesting to observe that the method suggested by Röth and Cantor is very like that of Eucl. x. (Lemma 1 following Prop. 28). We shall come to this later, but it may be mentioned here that the problem is to find two square numbers such that their sum is also a square. Euclid there uses the property of ii. 6 to the effect that, if AB is bisected at C and produced to D,

In order that uv may be a square, Euclid points out that u and ν must be similar numbers, and further that u and ν must be either both odd or both even in order that b may be a whole number. We may then put for the similar numbers, say, αβ² and αγ², whence (if αβ², αγ² are either both odd or both even) we obtain the solution

But I think a serious, and even fatal, objection to the conjecture of Cantor and Röth is the very fact that the method enables both the Pythagorean and the Platonic series of triangles to be deduced with equal ease. If this had been the case with the method used by Pythagoras, it would not, I think, have been left to Plato to discover the second series of such triangles. It seems to me therefore that Pythagoras must have used some method which would produce his rule only; and further it would be some less recondite method, suggested by direct observation rather than by argument from general principles.

One solution satisfying these conditions is that of Bretschneider, who suggests the following simple method. Pythagoras was certainly aware that the successive odd numbers are gnomons, or the differences between successive square numbers. It was then a simple matter to write down in three rows (a) the natural numbers, (b) their squares, (c) the successive odd numbers constituting the differences between the successive squares in (b), thus:

Pythagoras had then only to pick out the numbers in the third row which are squares, and his rule would be obtained by finding the formula connecting the square in the third line with the two adjacent squares in the second line. But even this would require some little argument; and I think a still better suggestion, because making pure observation play a greater part, is that of P. Treutlein (Zeitschrift für Mathematik und Physik, XXVIII., 1883, Hist.-litt. Abtheilung).

We have the best evidence (e.g. in Theon of Smyrna) of the practice of representing square numbers and other figured numbers, e.g. oblong, triangular, hexagonal, by dots or signs arranged in the shape of the particular figure. (Cf. Aristotle, Metaph. 1092 b 12). Thus, says Treutlein, it would be easily seen that any square number can be turned into the next higher square by putting a single row of dots round two adjacent sides, in the form of a gnomon (see figures on next page).

If a is the side of a particular square, the gnomon round it is shown by simple inspection to contain 2a + 1 dots or units. Now, in order that 2a + 1 may itself be a square, let us suppose

In order that a and a + 1 may be integral, n must be odd, and we have at once the Pythagorean formula

I think Treutlein’s hypothesis is shown to be the correct one by the passage in Aristotle’s Physics already quoted, where the reference is undoubtedly to the Pythagoreans, and odd numbers are clearly identified with gnomons “placed round 1.” But the ancient commentaries on the passage make the matter clearer still. Philoponus says: “As a proof. . . the Pythagoreans refer to what happens with the addition of numbers; for when the odd numbers are successively added to a square number they keep it square and equilateral.. . . Odd numbers are accordingly called gnomons because, when added to what are already squares, they preserve the square form.. . . Alexander has excellently said in explanation that the phrase ‘when gnomons are placed round’ means making a figure with the odd numbers () . . . for it is the practice with the Pythagoreans to represent things in figures ().”

The next question is: assuming this explanation of the Pythagorean formula, what are we to say of the origin of Plato’s? It could of course be obtained as a particular case of the general formula of Eucl. x. already referred to; but there are two simple alternative explanations in this case also. (1) Bretschneider observes that, to obtain Plato’s formula, we have only to double the sides of the squares in the Pythagorean formula,

where however n is not necessarily odd.

(2) Treutlein would explain by means of an extension of the gnomon idea. As, he says, the Pythagorean formula was obtained by placing a gnomon consisting of a single row of dots round two adjacent sides of a square, it would be natural to try whether another solution could not be found by placing round the square a gnomon consisting of a double row of dots. Such a gnomon would equally turn the square into a larger square; and the question would be whether the double-row gnomon itself could be a square. If the side of the original square was a, it would easily be seen that the number of units in the double-row gnomon would be 4a + 4, and we have only to put

and we have the Platonic formula

(2n)² + (n² − 1)² = (n² + 1)².

I think this is, in substance, the right explanation, but, in form, not quite correct. The Greeks would not, I think, have treated the double row as a gnomon. Their comparison would have been between (1) a certain square plus a single-row gnomon and (2) the same square minus a single-row gnomon. As the application of Eucl. ii. 4 to the case where the segments of the side of the square are a, 1 enables the Pythagorean formula to be obtained as Treutlein obtains it, so I think that Eucl. ii. 8 confirms the idea that the Platonic formula was obtained by comparing a square plus a gnomon with the same square minus a gnomon. For ii. 8 proves that

4ab + (a − b)² = (a + b)²,

whence, substituting 1 for b, we have

4a + (a − 1)² = (a + 1)²,

and we have only to put a = n² to obtain Plato’s formula.

The “theorem of Pythagoras” in India.

This question has been discussed anew in the last few years as the result of the publication of two important papers by Albert Bürk on Das Āpastamba-Sulba-Sūtra in the Zeitschrift der deutschen morgenländischen Gesellschaft (lv, 1901, and lvi., 1902). The first of the two papers contains the introduction and the text, the second the translation with notes. A selection of the most important parts of the material was made and issued by G. Thibaut in the Journal of the Asiatic Society of Bengal, XLIV., 1875, Part I. (reprinted also at Calcutta, 1875, as The Śulvasūtras, by G. Thibaut). Thibaut in this work gave a most valuable comparison of extracts from the three Āulvasūtras by Bāudhāyana, Āpastamba and Kātyāyana respectively, with a running commentary and an estimate of the date and originality of the geometry of the Indians. Bürk has however done good service by making the Āpastamba-Ś.-S. accessible in its entirety and investigating the whole subject afresh. With the natural enthusiasm of an editor for the work he is editing, he roundly maintains, not only that the Pythagorean theorem was known and proved in all its generality by the Indians long before the date of Pythagoras (about 580—500 B.C.), but that they had also discovered the irrational; and further that, so far from Indian geometry being indebted to the Greek, the much-travelled Pythagoras probably obtained his theory from India. Three important notices and criticisms of Bürk’s work have followed, by H. G. Zeuthen (“Théorème de Pythagore,” Origine de la Géométrie scientifique, 1904, already quoted), by Moritz Cantor (Über die älteste indische Mathematik in the Archiv der Mathematik und Physik, VIII., 1905) and by Heinrich Vogt (Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt? in the Bibliotheca Mathematica, VII₃, 1906. See also Cantor’s Geschichte der Mathematik, I₃.

The general effect of the criticisms is, I think, to show the necessity for the greatest caution, to say the least, in accepting Bürks conclusions.

I proceed to give a short summary of the portions of the contents of the Āpastamba-Ś.-S. which are important in the present connexion. It may be premised that the general object of the book is to show how to construct altars of certain shapes, and to vary the dimensions of altars without altering the form. It is a collection of rules for carrying out certain constructions. There are no proofs, the nearest approach to a proof being in the rule for obtaining the area of an isosceles trapezium, which is done by drawing a perpendicular from one extremity of the smaller of the two parallel sides to the greater, and then taking away the triangle so cut off and placing it, the other side up, adjacent to the other equal side of the trapezium, thereby transforming the trapezium into a rectangle. It should also be observed that Āpastamba does not speak of right-angled triangles, but of two adjacent sides and the diagonal of a rectangle. For brevity, I shall use the expression “rational rectangle” to denote a rectangle the two sides and the diagonal of which can be expressed in terms of rational numbers. The references in brackets are to the chapters and numbers of Āpastamba’s work.

(1) Constructions of right angles by means of cords of the following relative lengths respectively:

(2) A general enunciation of the Pythagorean theorem thus: “The diagonal of a rectangle produces [i.e. the square on the diagonal is equal to] the sum of what the longer and shorter sides separately produce [i.e. the squares on the two sides].”

(i. 4)

(3) The application of the Pythagorean theorem to a square instead of a rectangle [i.e. to an isosceles right-angled triangle]: “The diagonal of a square produces an area double [of the original square].”

(i. 5)

(4) An approximation to the value of ; the diagonal of a square is times the side.

(i. 6)

(5) Application of this approximate value to the construction of a square with side of any length.

(ii. 1)

(6) The construction of a , by means of the Pythagorean theorem, as the diagonal of a rectangle with sides a and a

(ii. 2)

(7) Remarks equivalent to the following:

(c) Generally, the square on any length contains as many rows (of small, unit, squares) as the length contains units.

(iii. 7)

(8) Constructions, by means of the Pythagorean theorem, of

(a) the sum of two squares as one square,

(ii. 4)

(b) the difference of two squares as one square.

(ii. 5)

(9) A transformation of a rectangle into a square.

(ii. 7)

[This is not directly done as by Euclid in II. 14, but the rectangle is first transformed into a gnomon, i.e. into the difference between two squares, which difference is then transformed into one square by the preceding rule. If ABCD be the given rectangle of which BC is the longer side, cut off the square ABEF, bisect the rectangle DE left over by HG parallel to FE, move the upper half DG and place it on AF as base in the position AK. Then the rectangle ABCD is equal to the gnomon which is the difference between the square LB and the square LF. In other words, Āpastamba transforms the rectangle ab into the difference between the squares and .]

(10) An attempt at a transformation of a square (a²) into a rectangle which shall have one side of given length (b).

(iii. 1)

[This shows no sign of such a procedure as that of Eucl. I. 44, and indeed does no more than say that we must subtract ab from a² and then adapt the remainder a² − ab so that it may “fit on” to the rectangle ab. The problem is therefore only reduced to another of the same kind, and presumably it was only solved arithmetically in the case where a, b are given numerically. The Indian was therefore far from the general, geometrical, solution.]

(11) Increase of a given square into a larger square.

(iii. 9)

[This amounts to saying that you must add two rectangles (a, b) and another square (b²) in order to transform a square a² into a square (a + b)². The formula is therefore that of Eucl. ii. 4, a² + 2ab + b² = (a + b)².]

The first important question in relation to the above is that of date. Bürk assigns to the Āpastamba-Śulba-Sūtra a date at least as early as the 5th or 4th century B.C. He observes however (what is likely enough) that the matter of it must have been much older than the book itself. Further, as regards one of the constructions for right angles, that by means of cords of lengths 15, 36, 39, he shows that it was known at the time of the Tāittirīya-Samhitā and the Satapatha-Brāhmana, still older works belonging to the 8th century B.C. at latest. It may be that (as Bürk maintains) the discovery that triangles with sides (a, b, c) in rational numbers such that a² + b² = c² are right-angled was nowhere made so early as in India. We find however in two ancient Chinese treatises (1) a statement that the diagonal of the rectangle (3, 4) is 5 and (2) a rule for finding the hypotenuse of a “right triangle” from the sides, while tradition connects both works with the name of Chou Kung who died 1105 B.C. (D. E. Smith, History of Mathematics).

As regards the various “rational rectangles” used by Āpastamba, it is to be observed that two of the seven, viz. 8, 15, 17 and 12, 35, 37, do not belong to the Pythagorean series, the others consist of two which belong to it, viz. 3, 4, 5 and 5, 12, 13, and multiples of these. It is true, as remarked by Zeuthen, that the rules of ii. 7 and iii. 9, numbered (9) and (11) above respectively, would furnish the means of finding any number of “rational rectangles.” But it would not appear that the Indians had been able to formulate any general rule; otherwise their list of such rectangles would hardly have been so meagre. Āpastamba mentions seven only, really reducible to four (though one other, 7, 24, 25, appears in the Bāudhāyana-Ś.-S., supposed to be older than Āpastamba). These are all that Āpastamba knew of, for he adds (v. 6): “So many recognisable (erkennbare) constructions are there,” implying that he knew of no other “rational rectangles” that could be employed. But the words also imply that the theorem of the square on the diagonal is also true of other rectangles not of the “recognisable” kind, i.e. rectangles in which the sides and the diagonal are not in the ratio of integers; this is indeed implied by the constructions for , etc. up to (cf. ii. 2, viii. 5). This is all that can be said. The theorem is, it is true, enunciated as a general proposition, but there is no sign of anything like a general proof; there is nothing to show that the assumption of its universal truth was founded on anything better than an imperfect induction from a certain number of cases, discovered empirically, of triangles with sides in the ratio of whole numbers in which the property (1) that the square on the longest side is equal to the sum of the squares on the other two sides was found to be always accompanied by the property (2) that the latter two sides include a right angle.

It remains to consider Bürk’s claim that the Indians had discovered the irrational. This is based upon the approximate value of given by Āpastamba in his rule i. 6 numbered (4) above. There is nothing to show how this was arrived at, but Thibaut’s suggestion certainly seems the best and most natural. The Indians may have observed that 17² = 289 is nearly double of 12² = 144. If so, the next question which would naturally occur to them would be, by how much the side 17 must be diminished in order that the square on it may be 288 exactly. If, in accordance with the Indian fashion, a gnomon with unit area were to be subtracted from a square with 17 as side, this would approximately be secured by giving the gnomon the breadth , for . The side of the smaller square thus arrived at would be , whence, dividing out by 12, we have

But it is a far cry from this calculation of an approximate value to the discovery of the irrational. First, we ask, is there any sign that this value was known to be inexact? It comes directly after the statement (i. 6) that the square on the diagonal of a square is double of that square, and the rule is quite boldly stated without any qualification: “lengthen the unit by one-third and the latter by one-quarter of itself less one-thirty-fourth of this part.” Further, the approximate value is actually used for the purpose of constructing a square when the side is given (ii. 1). So familiar was the formula that it was apparently made the basis of a sub-division of measures of length. Thibaut observes (Journal of the Asiatic Society of Bengal, XLIX.) that, according to Bāudhāyana, the unit of length was divided into 12 fingerbreadths, and that one of two divisions of the fingerbreadth was into 34 sesame-corns, and he adds that he has no doubt that this division, which he has not elsewhere met, owes its origin to the formula for . The result of using this subdivision would be that, in a square with side equal to 12 fingerbreadths, the diagonal would be 17 fingerbreadths less 1 sesame-corn. Is it conceivable that a sub-division of a measure of length would be based on an evaluation known to be inexact? No doubt the first discoverer would be aware that the area of a gnomon with breadth and outer side 17 is not exactly equal to 1 but less than it by the square of or by , and therefore that, in taking that gnomon as the proper area to be subtracted from 17², he was leaving out of account the small fraction ; as, however, the object of the whole proceeding was purely practical, he would, without hesitation, ignore this as being of no practical importance, and, thereafter, the formula would be handed down and taken as a matter of course without arousing suspicion as to its accuracy. This supposition is confirmed by reference to the sort of rules which the Indians allowed themselves to regard as accurate. Thus Āpastamba himself gives a construction for a circle equal in area to a given square, which is equivalent to taking π = 3.09, and yet observes that it gives the required circle “exactly” (iii. 2), while his construction of a square equal to a circle, which he equally calls “exact,” makes the side of the square equal to ths of the diameter of the circle (iii. 3), and is equivalent to taking π = 3.004. But, even if some who used the approximation for were conscious that it was not quite accurate (of which there is no evidence), there is an immeasurable difference between arrival at this consciousness and the discovery of the irrational. As Vogt says, three stages had to be passed through before the irrationality of the diagonal of a square was discovered in any real sense. (1) All values found by direct measurement or calculations based thereon have to be recognised as being inaccurate. Next (2) must supervene the conviction that it is impossible to arrive at an accurate arithmetical expression of the value. And lastly (3) the impossibility must be proved. Now there is no real evidence that the Indians, at the date in question, had even reached the first stage, still less the second or third.

The net results then of Bürks papers and of the criticisms to which they have given rise appear to be these. (1) It must be admitted that Indian geometry had reached the stage at which we find it in Āpastamba quite independently of Greek influence. But (2) the old Indian geometry was purely empirical and practical, far removed from abstractions such as the irrational. The Indians had indeed, by-trial in particular cases, persuaded themselves of the truth of the Pythagorean theorem and enunciated it in all its generality; but they had not established it by scientific proof.

Alternative proofs.

I. The well-known proof of i. 47 obtained by putting two squares side by side, with their bases continuous, and cutting off right-angled triangles which can then be put on again in different positions, is attributed by an-Nairīzī to Thābit b. Qurra (826–901 A.D.).

His actual construction proceeds thus.

Let ABC be the given triangle right-angled at A.

Construct on AB the square AD;

produce AC to F so that EF may be equal to AC.

Construct on EF the square EG, and produce DH to K so that DK may be equal to AC.

It is then proved that, in the triangles BAC, CFG, KHG, BDK, the sides BA, CE, KH, BD are all equal, and the sides AC, EG, HG, DK are all equal.

The angles included by the equal sides are all right angles; hence the four triangles are equal in all respects.

[i. 4]

Hence BC, CG, GK, KB are all equal.

Further the angles DBK, ABC are equal; hence, if we add to each the angle DBC, the angle KBC is equal to the angle ABD and is therefore a right angle.

In the same way the angle CGK is right;

therefore BCGK is a square, i.e. the square on BC.

Now the sum of the quadrilateral GCLH and the triangle LDB together with two of the equal triangles make the squares on AB, AC, and together with the other two make the square on BC.

Therefore etc.

II. Another proof is easily arrived at by taking the particular case of Pappus’ more general proposition given below in which the given triangle is right-angled and the parallelograms on the sides containing the right angles are squares. If the figure is drawn, it will be seen that, with no more than one additional line inserted, it contains Thābit’s figure, so that Thābit’s proof may have been practically derived from that of Pappus.

III. The most interesting of the remaining proofs seems to be that shown in the accompanying figure. It is given by J. W. Müller, Systematische Zusammenstellung der wichtigsten bisher bekannten Beweise des Pythag. Lehrsatzes (Nürnberg, 1819), and in the second edition (Mainz, 1821) of Ign. Hoffmann, Der Pythag. Lehrsatz mit 32 theils bekannten theils neuen Beweisen [3 more in second edition]. It appears to come from one of the scientific papers of Lionardo da Vinci (1452–1519).

The triangle HKL is constructed on the base KH with the side KL equal to BC and the side LH equal to AB.

Then the triangle HLK is equal in all respects to the triangle ABC, and to the triangle EBF.

Now DB, BG, which bisect the angles ABE, CBF respectively, are in a straight line. Join BL.

It is easily proved that the four quadrilaterals ADGC, EDGE, ABLK, HLBC are all equal.

Hence the hexagons ADEFGC, ABCHLK are equal.

Subtracting from the former the two triangles ABC, EBF, and from the latter the two equal triangles ABC, HLK, we prove that the square CK is equal to the sum of the squares AE, CF.

Pappus’ extension of i. 47.

In this elegant extension the triangle may be any triangle (not necessarily right-angled), and any parallelograms take the place of squares on two of the sides.

Pappus enunciates the theorem as follows:

If ABC be a triangle, and any parallelograms whatever ABED, BCFG be described on AB, BC, and if DE, FG be produced to H, and HB be joined, the parallelograms ABED, BCFG are equal to the parallelogram contained by AC, HB in an angle which is equal to the sum of the angles BAC, DHB.

Produce HB to K; through A, C draw AL, CM parallel to HK, and join LM.

Then, since ALHB is a parallelogram, AL, HB are equal and parallel.

Similarly MC, HB are equal and parallel.

Therefore AL, MC are equal and parallel;

whence LM, AC are also equal and parallel,

and ALMC is a parallelogram.

Further, the angle LAC of this parallelogram is equal to the sum of the angles BAC, DHB, since the angle DHB is equal to the angle LAB.

Now, since the parallelogram DABE is equal to the parallelogram LABH (for they are on the same base AB and in the same parallels AB, DH),

and likewise LABH is equal to LAKN (for they are on the same base LA and in the same parallels LA, HK),

the parallelogram DABE is equal to the parallelogram LAKN.

For the same reason,

the parallelogram BGFC is equal to the parallelogram NKCM.

Therefore the sum of the parallelograms DABE, BGFC is equal to the parallelogram LACM, that is, to the parallelogram which is contained by AC, HB in an angle LAC which is equal to the sum of the angles BAC, BHD.

“And this is far more general than what is proved in the Elements about squares in the case of right-angled (triangles).”

Heron’s proof that AL, BK, CF in Euclid’s figure meet in a point.

The final words of Proclus’ note on i. 47 are historically interesting. He says: “The demonstration by the writer of the Elements being clear, I consider that it is unnecessary to add anything further, and that we may be satisfied with what has been written, since in fact those who have added anything more, like Pappus and Heron, were obliged to draw upon what is proved in the sixth Book, for no really useful object.” These words cannot of course refer to the extension of i. 47 given by Pappus; but the key to them, so far as Heron is concerned, is to be found in the commentary of an-Nairīzī on i. 47, wherein he gives Heron’s proof that the lines AL, FC, BK in Euclid’s figure meet in a point. Heron proved this by means of three lemmas which would most naturally be proved from the principle of similitude as laid down in Book VI., but which Heron, as a tour de force, proved on the principles of Book I. only. The first lemma is to the following effect.

If, in a triangle ABC, DE be drawn parallel to the base BC, and if AF be drawn from the vertex A to the middle point F of BC, then AF will also bisect DE.

This is proved by drawing HK through A parallel to DE or BC, and HDL, KEM through D, E respectively parallel to AGF, and lastly joining DF, EF.

Then the triangles ABE, AFC are equal (being on equal bases), and the triangles DBF, EEC are also equal (being on equal bases and between the same parallels).

Therefore, by subtraction, the triangles ADF, AEF are equal, and hence the parallelograms AL, AM are equal.

These parallelograms are between the same parallels LM, HK; therefore LF, FM are equal, whence DG, GE are also equal.

The second lemma is an extension of this to the case where DE meets BA, CA produced beyond A.

The third lemma proves the converse of Euclid i. 43, that, If a parallelogram AB is cut into four others ADGE, DF, FGCB, CE, so that DF, CE are equal, the common vertex G will be on the diagonal AB.

Heron produces AG till it meets CF in H. Then, if we join HB, we have to prove that AHB is one straight line. The proof is as follows. Since the areas DF, EC are equal, the triangles DGF, ECG are equal.

If we add to each the triangle GCF,

the triangles ECF, DCF are equal;

therefore ED, CF are parallel.

Now it follows from i. 34, 29 and 26 that the triangles AKE, GKD are equal in all respects;

therefore EK is equal to KD.

Hence, by the second lemma,

CH is equal to HF.

Therefore, in the triangles FHB, CHG,

the two sides BF, FH are equal to the two sides GC, CH,

and the angle BFH is equal to the angle GCH;

hence the triangles are equal in all respects,

and the angle BHF is equal to the angle GHC.

Adding to each the angle GHF, we find that the angles BHF, FHG are equal to the angles CHG, GHF,

and therefore to two right angles.

Therefore AHB is a straight line.

Heron now proceeds to prove the proposition that, in the accompanying figure, if AKL perpendicular to BC meet

EC in M, and if BM, MG be joined,

BM, MG are in one straight line.

Parallelograms are completed as shown in the figure, and the diagonals OA, FH of the parallelogram FH are drawn.

Then the triangles FAH, BAC are clearly equal in all respects;

therefore the angle HFA is equal to the angle ABC, and therefore to the angle CAK (since AK is perpendicular to BC).

But, the diagonals of the rectangle FH cutting one another in Y,

FY is equal to YA,

and the angle HFA is equal to the angle OAF.

Therefore the angles OAF, CAK are equal, and accordingly

OA, AK are in a straight line.

Hence OM is the diagonal of SQ;

therefore AS is equal to AQ,

and, if we add AM to each,

FM is equal to MH.

But, since EC is the diagonal of the parallelogram FN,

FM is equal to MN.

Therefore MH is equal to MN;

and, by the third lemma, BM, MG are in a straight line.

1. the square on, , the word or being understood.

subtending the right angle. Here , “subtending,” is used with the simple accusative () instead of being followed by and the accusative, which seems to be the original and more orthodox construction.

2. the two sides AB, BD.. . . Euclid actually writes “DB, BA” and therefore the equal sides in the two triangles are not mentioned in corresponding order, though he adheres to the words “respectively.” Here DB is equal to BC and BA to FB.

3. [But the doubles of equals are equal to one another.] Heiberg brackets these words as an interpolation, since it quotes a Common Notion which is itself interpolated.

BOOK V

THE EUDOXIAN THEORY OF PROPORTION—DEFINITIONS AND PROPOSITIONS

DEFINITIONS

1. A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

2. The greater is a multiple of the less when it is measured by the less.

3. A ratio is a sort of relation in respect of size between two magnitudes of the same kind.

4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.

5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

6. Let magnitudes which have the same ratio be called proportional.

7. When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to have a greater ratio to the second than the third has to the fourth.

8. A proportion in three terms is the least possible.

9. When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.

10. When four magnitudes are <continuously> proportional, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on continually, whatever be the proportion.

11. The term corresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.

12. Alternate ratio means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.

13. Inverse ratio means taking the consequent as antecedent in relation to the antecedent as consequent.

14. Composition of a ratio means taking the antecedent together with the consequent as one in relation to the consequent by itself.

15. Separation of a ratio means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.

16. Conversion of a ratio means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.

17. A ratio ex aequali arises when, there being several magnitudes and another set equal to them in multitude which taken two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;

Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.

18. A perturbed proportion arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.

Proposition 1

If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.

Let any number of magnitudes whatever AB, CD be respectively equimultiples of any magnitudes E, F equal in multitude;

I say that, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.

For, since AB is the same multiple of E that CD is of F, as many magnitudes as there are in AB equal to E, so many also are there in CD equal to F.

Let AB be divided into the magnitudes AG, GB equal to E,

and CD into CH, HD equal to F;

then the multitude of the magnitudes AG, GB will be equal to the multitude of the magnitudes CH, HD.

Now, since AG is equal to E, and CH to F,

therefore AG is equal to E, and AG, CH to E, F.

For the same reason

GB is equal to E, and GB, HD to E, F;

therefore, as many magnitudes as there are in AB equal to E,

so many also are there in AB, CD equal to E, F;

therefore, whatever multiple AB is of E, that multiple will AB, CD also be of E, F.

Therefore etc.

Q.E.D.

De Morgan remarks of v. 1–6 that they are “simple propositions of concrete arithmetic, covered in language which makes them unintelligible to modern ears. The first, for instance, states no more than that ten acres and ten roods make ten times as much as one acre and one rood.” One aim therefore of notes on these as well as the other propositions of Book V. should be to make their purport clearer to the learner by setting them side by side with the same truths expressed in the much shorter and more familiar modern (algebraical) notation. In doing so, we shall express magnitudes by the first letters of the alphabet, a, b, c etc., adopting small instead of capital letters so as to avoid confusion with Euclid’s lettering; and we shall use the small letters m, n, p etc. to represent integral numbers. Thus ma will always mean m times a or the mth multiple of a (counting 1. a as the first, 2. a as the second multiple, and so on).

Prop. 1 then asserts that, if ma., mb, mc etc. be any equimultiples of a, b, c etc., then

ma + mb + mc + . . . = m(a + b + c + . . .).

Proposition 2

If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth.

Let a first magnitude, AB, be the same multiple of a second, C, that a third, DE, is of a fourth, F, and let a fifth, BG, also be the same multiple of the second, C, that a sixth, EH, is of the fourth F;

I say that the sum of the first and fifth, AG, will be the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.

For, since AB is the same multiple of C that DE is of F, therefore, as many magnitudes as there are in AB equal to C, so many also are there in DE equal to F.

For the same reason also,

as many as there are in BG equal to C, so many are there also in EH equal to F; therefore, as many as there are in the whole AG equal to C, so many also are there in the whole DH equal to F.

Therefore, whatever multiple AG is of C, that multiple also is DH of F.

Therefore the sum of the first and fifth, AG, is the same multiple of the second, C, that the sum of the third and sixth, DH, is of the fourth, F.

Therefore etc.

Q.E.D.

To find the corresponding formula for the result of this proposition, we may suppose a to be the “second” magnitude and b the “fourth.” If now the “first” magnitude is ma, the “third” is, by hypothesis, mb; and, if the “fifth” magnitude is na, the “sixth” is nb. The proposition then asserts that ma + na is the same multiple of a that mb + nb is of b.

More generally, if pa, qa . . . and pb, qb . . . be any further equimultiples of a, b respectively, ma + na + pa + qa + . . . is the same multiple of a that mb + nb + pb + qb + . . . is of b. This extension is stated in Simson’s corollary to v. 2 thus:

“From this it is plain that, if any number of magnitudes AB, BG, GH be multiples of another C; and as many DE, EK, KL be the same multiples of F, each of each; the whole of the first, viz. AH, is the same multiple of C that the whole of the last, viz. DL, is of F.”

The course of the proof which separates m into its units and also n into its units, practically tells us that the multiple of a arrived at by adding the two multiples is the (m + n)th multiple; or practically we are shown that

ma + na = (m + n)a,

or, more generally, that

ma + na + pa + . . . = (m + n + p + . . .)a.

Proposition 3

If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.

Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples EF, GH be taken of A, C;

I say that EF is the same multiple of ? that GH is of D.

For, since EF is the same multiple of A that GH is of C, therefore, as many magnitudes as there are in EF equal to A, so many also are there in GH equal to C.

Let EF be divided into the magnitudes EK, KF equal to A, and GH into the magnitudes GL, LH equal to C;

then the multitude of the magnitudes EK, KF will be equal to the multitude of the magnitudes GL, LH.

And, since A is the same multiple of B that C is of D, while EK is equal to A, and GL to C,

therefore EK is the same multiple of B that GL is of D.

For the same reason

KF is the same multiple of B that LH is of D.

Since, then, a first magnitude EK is the same multiple of a second B that a third GL is of a fourth D,

and a fifth KF is also the same multiple of the second B that a sixth LH is of the fourth D,

therefore the sum of the first and fifth, EF, is also the same multiple of the second B that the sum of the third and sixth, GH, is of the fourth D.

[v. 2]

Therefore etc.

Q.E.D.

Heiberg remarks of the use of ex aequali in the enunciation of this proposition that, strictly speaking, it has no reference to the definition (17) of a ratio ex aequali. But the uses of the expression here and in the definition are, I think, sufficiently parallel, as may he seen thus. The proposition asserts that, if

then m · na is the same multiple of a that m · nb is of b. Clearly the proposition can he extended by taking further equimultiples of the last equimultiples and so on; and we can prove that

p · q . . . m · na is the same multiple of a that p · q . . . m · nb is of b,

where the series of numbers p · q . . . m · n is exactly the same in both expressions;

and ex aequali () expresses the fact that the equimultiples are at the same distance from a, b in the series na, m · na . . . and nb, m · nb . . . respectively.

Here again the proof breaks m into its units, and then breaks n into its units; and we are practically shown that the multiple of a arrived at, viz. m · na, is the multiple denoted by the product of the numbers m, n, i.e. the (mn)th multiple, or in other words that

m · na = mn · a.

Proposition 4

If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.

For let a first magnitude A have to a second B the same ratio as a third C to a fourth D; and let equimultiples E, F be taken of A, C, and G, H other, chance, equimultiples of B, D;

I say that, as E is to G, so is F to H.

For let equimultiples K, L be taken of E, F, and other, chance, equimultiples M, N of G, H.

Since B is the same multiple of A that F is of C,

and equimultiples K, L of E, F have been taken,

therefore K is the same multiple of A that L is of C.

[v. 3]

For the same reason

M is the same multiple of B that N is of D.

And, since, as A is to B, so is C to D,

and of A, C equimultiples K, L have been taken,

and of B, D other, chance, equimultiples M, N,

therefore, if K is in excess of M, L also is in excess of N,

if it is equal, equal, and if less, less.

[v. Def. 5]

And K, L are equimultiples of E, F,

and M, N other, chance, equimultiples of G, H;

therefore, as E is to G, so is F to H.

[v. Def. 5]

Therefore etc.

Q.E.D.

This proposition shows that, if a, b, c, d are proportionals, then

ma is to nb as mc is to nd;

and the proof is as follows:

Take pma, pmc any equimultiples of ma, mc, and qnb, qnd any equimultiples of nb, nd.

Since a : b = c : d, it follows [v. Def. 5] that,

according as pma > = < qnb, pmc > = < qnd.

But the p- and q-equimultiples are any equimultiples;

therefore [v. Def 5]

ma : nb = mc : nd.

It will be observed that Euclid’s phrase for taking any equimultiples of A, C and any other equimultiples of B, D is “let there be taken equimultiples E, F of A, C, and G, H other, chance, equimultiples of B, D,” E, F being called simply, and G, H . And similarly, when any equimultiples (K, L) of E, F come to be taken, and any other equimultiples (M, N) of G, H. But later on Euclid uses the same phrases about the new equimultiples with reference to the original magnitudes, reciting that “there have been taken, of A, C, equimultiples K, L and of B, D, other, chance, equimultiples M, N”; whereas M, N are not any equimultiples whatever of B, D, but are any equimultiples of the particular multiples (G, H) which have been taken of B, D respectively, though these latter have been taken at random. Simson would, in the first place, add in the passages where any equimultiples E, F are taken of A, C and any equimultiples K, L are taken of E, F, because the words are “wholly necessary” and, in the second place, would leave them out where M, N are called of B, D, because it is not true that of B, D have been taken “any equimultiples whatever (), M, N.” Simson adds: “And it is strange that neither Mr. Briggs, who did right to leave out these words in one place of Prop. 13 of this book, nor Dr. Gregory, who changed them into the word ‘some’ in three places, and left them out in a fourth of that same Prop. 13, did not also leave them out in this place of Prop. 4 and in the second of the two places where they occur in Prop. 17 of this book, in neither of which they can stand consistent with truth: And in none of all these places, even in those which they corrected in their Latin translation, have they cancelled the words in the Greek text, as they ought to have done. The same words α are found in four places of Prop. 11 of this book, in the first and last of which they are necessary, but in the second and third, though they are true, they are quite superfluous; as they likewise are in the second of the two places in which they are found in the 12th prop. and in the like places of Prop. 22, 23 of this book; but are wanting in the last place of Prop. 23, as also in Prop. 25, Book XI.”

As will be seen, Simson’s emendations amount to alterations of the text so considerable as to suggest doubt whether we should be justified in making them in the absence of MS. authority. The phrase “equimultiples of A, C and other, chance, equimultiples of B, D” recurs so constantly as to suggest that it was for Euclid a quasi-stereotyped phrase, and that it is equally genuine wherever it occurs. Is it then absolutely necessary to insert in places where it does not occur, and to leave it out in the places where Simson holds it to be wrong? I think the text can be defended as it stands. In the first place to say “take equimultiples of A, C” is a fair enough way of saying take any equimultiples whatever of A, C. The other difficulty is greater, but may, I think, be only due to the adoption of any whatever as the translation of . As a matter of fact, the words only mean chance equimultiples, equimultiples which are the result of random selection. Is it not justifiable to describe the product of two chance numbers, numbers selected at random, as being a “chance number,” since it is the result of two random selections? I think so, and I have translated accordingly as implying, in the case in question, “other equimultiples whatever they may happen to be.”

To this proposition Theon added the following:

“Since then it was proved that, if K is in excess of M, L is also in excess of N, if it is equal, (the other is) equal, and if less, less,

it is clear also that,

if M is in excess of K, N is also in excess of L, if it is equal, (the other is) equal, and if less, less; and for this reason,

as G is to E, so also is H to F.

PORISM. From this it is manifest that, if four magnitudes be proportional, they will also be proportional inversely.”

Simson rightly pointed out that the demonstration of what Theon intended to prove, viz. that, if E, G, F, H be proportionals, they are proportional inversely, i.e. G is to E as H is to F, does not in the least depend upon this 4th proposition or the proof of it; for, when it is said that, “if K exceeds M, L also exceeds N etc.,” this is not proved from the fact that E, G, F, H are proportionals (which is the conclusion of Prop. 4), but from the fact that A, B, C, D are proportionals.

The proposition that, if A, B, C, D are proportionals, they are also proportionals inversely is not given by Euclid, but Simson supplies the proof in his Prop. B. The fact is really obvious at once from the 5th definition of Book V., and Euclid probably omitted the proposition as unnecessary.

Simson added, in place of Theon’s corollary, the following:

“Likewise, if the first has the same ratio to the second which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth : And, in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth.”

The proof, of course, follows exactly the method of Euclid’s proposition itself, with the only difference that, instead of one of the two pairs of equimultiples, the magnitudes themselves are taken. In other words, the conclusion that

ma is to nb as mc is to nd

is equally true when either m or n is equal to unity.

As De Morgan says, Simson’s corollary is only necessary to those who will not admit M into the list M, 2M, 3M etc.; the exclusion is grammatical and nothing else. The same may be said of Simson’s Prop. A to the effect that, “If the first of four magnitudes has to the second the same ratio which the third has to the fourth: then, if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less.” This is needless to those who believe once A to be a proper component of the list of multiples, in spite of multus signifying many.

Proposition 5

If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.

For let the magnitude AB be the same multiple of the magnitude CD that the part AE subtracted is of the part CF subtracted;

I say that the remainder EB is also the same multiple of the remainder FD that the whole AB is of the whole CD.

For, whatever multiple AE is of CF, let EB be made that multiple of CG. [4]

Then, since AE is the same multiple of CF that EB is of GC,

therefore AE is the same multiple of CF that AB is of GF.

[v. 1]

But, by the assumption, AE is the same multiple of CF that AB is of CD.

Therefore AB is the same multiple of each of the magnitudes GF, CD;

therefore GF is equal to CD.

Let CF be subtracted from each;

therefore the remainder GC is equal to the remainder FD.

And, since AE is the same multiple of CF that EB is of GC,

and GC is equal to DF,

therefore AE is the same multiple of CF that EB is of FD.

But, by hypothesis,

AE is the same multiple of CF that AB is of CD;

therefore EB is the same multiple of FD that AB is of CD.

That is, the remainder EB will be the same multiple of the remainder FD that the whole AB is of the whole CD.

Therefore etc.

Q.E.D.

This proposition corresponds to v. 1, with subtraction taking the place of addition. It proves the formula

ma − mb = m(a − b).

Euclid’s construction assumes that, if AE is any multiple of CF, and EB is any other magnitude, a fourth straight line can he found such that EB is the same multiple of it that AE is of CF, or in other words that, given any magnitude, we can divide it into any number of equal parts. This is however not proved, even of straight lines, much less other magnitudes, until vi. 9. Peletarius had already seen this objection to the construction. The difficulty is not got over by regarding it merely as a hypothetical construction; for hypothetical constructions are not in Euclid’s manner. The remedy is to substitute the alternative construction given by Simson, after Peletarius and Campanus’ translation from the Arabic, which only requires us to add a magnitude to itself a certain number of times. The demonstration follows Euclid’s line exactly.

“Take AG the same multiple of FD that AE is of CF;

therefore AE is the same multiple of CF that EG is of CD.

[v. 1]

But AE, by hypothesis, is the same multiple of CF that AB is of CD; therefore EG is the same multiple of CD that KB, is of CD;

wherefore EG is equal to AB.

Take from them the common magnitude AE; the remainder AG is equal to the remainder EB.

Wherefore, since AE is the same multiple of CF that AG is of FD, and since AG is equal to EB,

therefore AE is the same multiple of CF that EB is of FD.

But AE is the same multiple of CF that AB is of CD;

therefore EB is the same multiple of FD that AB is of CD.”

Q.E.D.

Euclid’s proof amounts to this.

Suppose a magnitude x taken such that

ma − mb = mx, say.

Add mb to each side, whence (by v. 1)

Simson’s proof on the other hand, argues thus.

Take x = m (a − b), the same multiple of (a − b) that mb is of b.

Then, by addition of mb to both sides, we have [v. 1]

Proposition 6

If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.

For let two magnitudes AB, CD be equimultiples of two magnitudes E, F, and let AG, CH subtracted from them be equimultiples of the same two E, F;

I say that the remainders also, GB, HD, are either equal to E, F or equimultiples of them.

For, first, let GB be equal to E; I say that HD is also equal to F.

For let CK be made equal to F.

Since AG is the same multiple of E that CH is of F,

while GB is equal to E and KC to F,

therefore AB is the same multiple of E that KH is of F.

[v. 2]

But, by hypothesis, AB is the same multiple of E that CD is of F;

therefore KH is the same multiple of F that CD is of F.

Since then each of the magnitudes KH, CD is the same multiple of F,

therefore KH is equal to CD.

Let CH be subtracted from each;

therefore the remainder KC is equal to the remainder HD.

But F is equal to KC;

therefore HD is also equal to F.

Hence, if GB is equal to E, HD is also equal to F.

Similarly we can prove that, even if GB be a multiple of E, HD is also the same multiple of F.

Therefore etc.

Q.E.D.

This proposition corresponds to v. 2, with subtraction taking the place of addition. It asserts namely that, if n is less than m, ma − na is the same multiple of a that mb − nb is of b. The enunciation distinguishes the cases in which m − n is equal to 1 and greater than 1 respectively.

Simson observes that, while only the first case (the simpler one) is proved in the Greek, both are given in the Latin translation from the Arabic; and he supplies accordingly the proof of the second case, which Euclid leaves to the reader. The fact is that it is exactly the same as the other except that, in the construction, CK is made the same multiple of F that GB is of F, and at the end, when it has been proved that KC is equal to HD, instead of concluding that HD is equal to F, we have to say “Because GB is the same multiple of E that KC is of F, and KC is equal to HD, therefore HD is the same multiple of F that GB is of E.”

Proposition 7

Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.

Let A, B be equal magnitudes and C any other, chance, magnitude;

I say that each of the magnitudes A, B has the same ratio to C, and C has the same ratio to each of the magnitudes A, B.

For let equimultiples D, E of A, B be taken, and of C another, chance, multiple F.

Then, since D is the same multiple of A that E is of B,

while A is equal to B,

therefore D is equal to E.

But F is another, chance, magnitude.

If therefore D is in excess of F, E is also in excess of F, if equal to it, equal; and, if less, less.

And D, E are equimultiples of A, B,

while F is another, chance, multiple of C;

therefore, as A is to C, so is B to C.

[v. Def. 5]

I say next that C also has the same ratio to each of the magnitudes A, B.

For, with the same construction, we can prove similarly that D is equal to E; and F is some other magnitude.

If therefore F is in excess of D, it is also in excess of E, if equal, equal; and, if less, less.

And F is a multiple of C, while D, E are other, chance, equimultiples of A, B;

therefore, as C is to A, so is C to B.

[v. Def. 5]

Therefore etc.

PORISM. From this it is manifest that, if any magnitudes are proportional, they will also be proportional inversely.

Q.E.D.

In this proposition there is a similar use of to that which has been discussed under Prop. 4. Any multiple F of C is taken and then, four lines lower down, we are told that “F is another, chance, magnitude. “It is of course not any magnitude whatever, and Simson leaves out the sentence, but this time without calling attention to it.

Of the Porism to this proposition Heiberg says that it is properly put here in the best MS.; for, as August had already observed, if it was in its right place where Theon put it (at the end of v. 4), the second part of the proof of this proposition would be unnecessary. But the truth is that the Porism is no more in place here. The most that the proposition proves is that, if A, B are equal, and C any other magnitude, then two conclusions are simultaneously established, (1) that A is to C as B is to C and (2) that C is to A as C is to B. The second conclusion is not established from the first conclusion (as it ought to be in order to justify the inference in the Porism), but from a hypothesis on which the first conclusion itself depends; and moreover it is not a proportion in its general form, i. e. between four magnitudes, that is in question, but only the particular case in which the consequents are equal.

Aristotle tacitly assumes inversion (combined with the solution of the problem of Eucl. vi. 11) in Meteorologica iii. 5, 376 a 14–16.

Proposition 8

Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.

Let AB, C be unequal magnitudes, and let AB be greater; let D be another, chance, magnitude;

I say that AB has to D a greater ratio than C has to D, and D has to C a greater ratio than it has to AB.

For, since AB is greater than C, let BE be made equal to C;

then the less of the magnitudes AE, EB, if multiplied, will sometime be greater than D.

[v. Def. 4]

[Case 1.]

First, let AE be less than EB;

let AE be multiplied, and let FG be a multiple of it which is greater than D; then, whatever multiple FG is of AE, let GH be made the same multiple of EB and K of C;

and let L be taken double of D, M triple of it, and successive multiples increasing by one, until what is taken is a multiple of D and the first that is greater than K. Let it be taken, and let it be N which is quadruple of D and the first multiple of it that is greater than K.

Then, since K is less than N first,

therefore K is not less than M.

And, since FG is the same multiple of AE that GH is of EB,

therefore FG is the same multiple of AE that FH is of AB.

[v. 1]

But FG is the same multiple of AE that K is of C;

therefore FH is the same multiple of AB that K is of C;

therefore FH, K are equimultiples of AB, C.

Again, since GH is the same multiple of EB that K is of C, and EB is equal to C,

therefore GH is equal to K.

But K is not less than M;

therefore neither is GH less than M.

And FG is greater than D;

therefore the whole FH is greater than D, M together.

But D, M together are equal to N, inasmuch as M is triple of D, and M, D together are quadruple of D, while N is also quadruple of D; whence M, D together are equal to N.

But FH is greater than M, D;

therefore FH is in excess of N,

while K is not in excess of N.

And FH, K are equimultiples of AB, C, while N is another, chance, multiple of D;

[v. Def. 7]

therefore AB has to D a greater ratio than C has to D.

I say next, that D also has to C a greater ratio than D has to AB.

For, with the same construction, we can prove similarly that N is in excess of K, while N is not in excess of FH.

And N is a multiple of D,

while FH, K are other, chance, equimultiples of AB, C;

therefore D has to C a greater ratio than D has to AB.

[v. Def. 7]

[Case 2.]

Again, let AE be greater than EB.

Then the less, EB, if multiplied, will sometime be greater than D.

[v. Def. 4]

Let it be multiplied, and let GH be a multiple of EB and greater than D;

and, whatever multiple GH is of EB, let FG be made the same multiple of AE, and K of C.

Then we can prove similarly that FH, K are equimultiples of AB, C;

and, similarly, let N be taken a multiple of D but the first that is greater than FG,

so that FG is again not less than M.

But GH is greater than D;

therefore the whole FH is in excess of D, M, that is, of N.

Now K is not in excess of N, inasmuch as FG also, which is greater than GH, that is, than K, is not in excess of N.

And in the same manner, by following the above argument, we complete the demonstration.

Therefore etc.

Q.E.D.

The two separate cases found in the Greek text of the demonstration can practically be compressed into one. Also the expositor of the two cases makes them differ more than they need. It is necessary in each case to select the smaller of the two segments AE, EB of AB with a view to taking a multiple of it which is greater than D; in the first case therefore AE is taken, in the second EB. But, while in the first case successive multiples of D are taken in order to find the first multiple that is greater than GH (or K), in the second case the multiple is taken which is the first that is greater than FG. This difference is not necessary; the first multiple of D that is greater than GH would equally serve in the second case. Lastly, the use of the magnitude K might have been dispensed with in both cases; it is of no practical use and only lengthens the proofs. For these reasons Simson considers that Theon, or some other unskilful editor, has vitiated the proposition. This however seems an unsafe assumption; for, while it was not the habit of the great Greek geometers to discuss separately a number of different cases (e.g. in i. 7 and i. 35 Euclid proves one case and leaves the others to the reader), there are many exceptions to prove the rule, e.g. Eucl. iii. 25 and 33; and we know that many fundamental propositions, afterwards proved generally, were first discovered in relation to particular cases and then generalised, so that Book V., presenting a comparatively new theory, might fairly be expected to exhibit more instances than the earlier books do of unnecessary subdivision. The use of the K is no more conclusive against the genuineness of the proofs.

Nevertheless Simson’s version of the proof is certainly shorter, and moreover it takes account of the case in which AE is equal to EB, and of the case in which AE, EB are both greater than D (though these cases are scarcely worth separate mention).

“If the magnitude which is not the greater of the two AE, EB be (1) not less than D, take FG, GH the doubles of AE, EB.

But if that which is not the greater of the two AE, EB be (2) less than D, this magnitude can be multiplied so as to become greater than D whether it be AE or EB.

Let it be multiplied until it becomes greater than D, and let the other be multiplied as often; let FG be the multiple thus taken of AE and GH the same multiple of EB,

therefore FG and GH are each of them greater than D.

And, in every one of the cases, take L the double of D, M its triple and so on, till the multiple of D be that which first becomes greater than GH.

Let N be that multiple of D which is first greater than GH, and M the multiple of D which is next less than N.

Then, because N is the multiple of D which is the first that becomes greater than GH,

the next preceding multiple is not greater than GH;

that is, GH is not less than M.

And, since FG is the same multiple of AE that GH is of EB,

GH is the same multiple of EB that FH is of AB;

[v. 1]

wherefore FH, GH are equimultiples of AB, EB.

And it was shown that GH was not less than M;

and, by the construction, FG is greater than D;

therefore the whole FH is greater than M, D together.

But M, D together are equal to N;

therefore FH is greater than N.

But GH is not greater than N;

and FH, GH are equimultiples of AB, BE,

and N is a multiple of D;

therefore AB has to D a greater ratio than BE (or C) has to D.

[v. Def. 7]

Also D has to BE a greater ratio than it has to AB.

For, having made the same construction, it may he shown, in like manner, that N is greater than GH but that it is not greater than FH;

and N is a multiple of D,

and GH, FH are equimultiples of EB, AB;

Therefore D has to EB a greater ratio than it has to AB.”

[v. Def. 7]

The proof may perhaps be more readily grasped in the more symbolical form thus.

Take the mth equimultiples of C, and of the excess of AB over C (that is, of AE ), such that each is greater than D;

and, of the multiples of D, let pD be the first that is greater than mC, and nD the next less multiple of D.

Then, since mC is not less than nD,

and, by the construction, m(AE) is greater than D,

the sum of mC and m(AE) is greater than the sum of nD and D.

That is, m(AB) is greater than pD.

And, by the construction, mC is less than pD.

Therefore [v. Def. 7] AB has to D a greater ratio than C has to D.

Again, since pD is less than m(AB),

and pD is greater than mC,

D has to C a greater ratio than D has to AB.

Proposition 9

Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal.

For let each of the magnitudes A, B have the same ratio to C;

I say that A is equal to B.

For, otherwise, each of the magnitudes A, B would not have had the same ratio to C; but it has;

[v. 8]

therefore A is equal to B.

Again, let C have the same ratio to each of the magnitudes A, B;

I say that A is equal to B.

For, otherwise, C would not have had the same ratio to each of the magnitudes A, B;

[v. 8]

but it has;

therefore A is equal to B.

Therefore etc.

Q.E.D.

If A is to C as B is to C,

or if C is to A as C is to B, then A is equal to B.

Simson gives a more explicit proof of this proposition which has the advantage of referring back to the fundamental 5th and 7th definitions, instead of quoting the results of previous propositions, which, as will he seen from the next note, may be, in the circumstances, unsafe.

“Let A, B have each of them the same ratio to C;

A is equal to B.

For, if they are not equal, one of them is greater than the other;

let A he the greater.

Then, by what was shown in the preceding proposition, there are some equimultiples of A and B, and some multiple of C, such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C.

Let such multiples be taken, and let D, E be the equimultiples of A, B, and F the multiple of C, so that D may be greater than F, and E not greater than F.

But, because A is to C as B is to C,

and of A, B are taken equimultiples D, E, and of C is taken a multiple F, and D is greater than F,

E must also be greater than F.

[v. Def. 5]

But E is not greater than F: which is impossible.

Next, let C have the same ratio to each of the magnitudes A and B;

A is equal to B.

For, if not, one of them is greater than the other;

let A be the greater.

Therefore, as was shown in Prop. 8, there is some multiple F of C, and some equimultiples E and D of B and A, such that F is greater than E and not greater than D.

But, because C is to B as C is to A,

and F the multiple of the first is greater than E the multiple of the second,

F the multiple of the third is greater than D the multiple of the fourth.

[v. Def. 5]

But F is not greater than D: which is impossible.

Therefore A is equal to B.”

Proposition 10

Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.

For let A have to C a greater ratio than B has to C;

I say that A is greater than B.

For, if not, A is either equal to B or less.

Now A is not equal to B;

for in that case each of the magnitudes A, B would have had the same ratio to C;

[v. 7]

but they have not;

therefore A is not equal to B.

Nor again is A less than B;

for in that case A would have had to C a less ratio than B has to C;

[v. 8]

but it has not;

therefore A is not less than B.

But it was proved not to be equal either;

therefore A is greater than B.

Again, let C have to B a greater ratio than C has to A;

I say that B is less than A.

For, if not, it is either equal or greater.

Now B is not equal to A;

for in that case C would have had the same ratio to each of the magnitudes A, B;

[v. 7]

but it has not;

therefore A is not equal to B.

Nor again is B greater than A;

for in that case C would have had to B a less ratio than it has to A;

[v. 8]

but it has not;

therefore B is not greater than A.

But it was proved that it is not equal either;

therefore B is less than A.

Therefore etc.

Q.E.D.

No better example can, I think, be found of the acuteness which Simson brought to bear in his critical examination of the Elements, and of his great services to the study of Euclid, than is furnished by the admirable note on this proposition where he points out a serious flaw in the proof as given in the text.

For the first time Euclid is arguing about greater and less ratios, and it will be found by an examination of the steps of the proof that he assumes more with regard to the meaning of the terms than he is entitled to assume, having regard to the fact that the definition of greater ratio (Def. 7) is all that, as yet, he has to go upon. That we cannot argue, at present, about greater and less as applied to ratios in the same way as about the same terms in relation to magnitudes is indeed sufficiently indicated by the fact that Euclid does not assume for ratios what is in Book I. an axiom, viz. that things which are equal to the same thing are equal to one another; on the contrary he proves, in Prop. 11, that ratios which are the same with the same ratio are the same with one another.

Let us now examine the steps of the proof in the text. First we are told that

“A is greater than B.

For, if not, it is either equal to B or less than it.

Now A is not equal to B;

for in that case each of the two magnitudes A, B would have had the same ratio to C:

[v. 7]

but they have not:

therefore A is not equal to B.”

As Simson remarks, the force of this reasoning is as follows.

If A has to C the same ratio as B has to C,

then—supposing any equimultiples of A, B to be taken and any multiple of C—

by Def. 5, if the multiple of A be greater than the multiple of C, the multiple of B is also greater than that of C.

But it follows from the hypothesis (that A has a greater ratio to C than B has to C) that,

by Def. 7, there must be some equimultiples of A, B and some multiple of C such that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than the same multiple of C.

And this directly contradicts the preceding deduction from the supposition that A has to C the same ratio as B has to C;

therefore that supposition is impossible.

The proof now goes on thus:

Nor again is A less than B;

for, in that case, A would have had to C a less ratio than B has to C;

[v. 8]

but it has not;

therefore A is not less than B.”

It is here that the difficulty arises. As before, we must use Def. 7. “A would have had to C a less ratio than B has to C,” or the equivalent statement that B would have had to C a greater ratio than A has to C, means that there would have been some equimultiples of B, A and some multiple of C such that

(1) the multiple of B is greater than the multiple of C, but

(2) the multiple of A is not greater than the multiple of C,

and it ought to have been proved that this can never happen if the hypothesis of the proposition is true, viz. that A has to C a greater ratio than B has to C: that is, it should have been proved that, in the latter case, the multiple of A is always greater than the multiple of C whenever the multiple of B is greater than the multiple of C (for, when this is demonstrated, it will be evident that B cannot have a greater ratio to C than A has to C). But this is not proved (cf. the remark of De Morgan quoted in the note on v. Def. 7, p. 130), and hence it is not proved that the above inference from the supposition that A is less than B is inconsistent with the hypothesis in the enunciation. The proof therefore fails.

Simson suggests that the proof is not Euclid’s, but the work of some one who apparently “has been deceived in applying what is manifest, when understood of magnitudes, unto ratios, viz. that a magnitude cannot be both greater and less than another.”

The proof substituted by Simson is satisfactory and simple.

“Let A have to C a greater ratio than B has to C;

A is greater than B.

For, because A has a greater ratio to C than B has to C, there are some equimultiples of A, B and some multiple of C such that

the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it.

[v. Def. 7]

Let them be taken, and let D, E be equimultiples of A, B, and F a multiple of C, such that

Therefore D is greater than E.

And, because D and E are equimultiples of A and B, and D is greater than E,

therefore A is greater than B.

[Simson’s 4th Ax.]

Next, let C have a greater ratio to B than it has to A;

B is less than A.

For there is some multiple F of C and some equimultiples E and D of B and A such that

F is greater than E but not greater than D.

[v. Def. 7]

Therefore E is less than D;

and, because E and D are equimultiples of B and A,

therefore B is less than A.”

Proposition 11

Ratios which are the same with the same ratio are also the same with one another.

For, as A is to B, so let C be to D,

and, as C is to D, so let B be to F;

I say that, as A is to B, so is E to F.

For of A, C, E let equimultiples G, H, K be taken, and of B, D, F other, chance, equimultiples L, M, N.

Then since, as A is to B, so is C to D,

and of A, C equimultiples G, H have been taken,

and of B, D other, chance, equimultiples L, M,

therefore, if G is in excess of L, H is also in excess of M,

if equal, equal,

and if less, less.

Again, since, as C is to D, so is E to F,

and of C, E equimultiples H, K have been taken,

and of D, F other, chance, equimultiples M, N,

therefore, if H is in excess of M, K is also in excess of N,

if equal, equal,

and if less, less.

But we saw that, if H was in excess of M, G was also in excess of L; if equal, equal; and if less, less;

so that, in addition, if G is in excess of L, K is also in excess of N,

if equal, equal,

and if less, less.

And G, K are equimultiples of A, E,

while L, N are other, chance, equimultiples of B, F;

therefore, as A is to B, so is E to F.

Therefore etc.

Q.E.D.

The idiomatic use of the imperfect in quoting a result previously obtained is noteworthy. Instead of saying “But it was proved that, if H is in excess of M, G is also in excess of L”, the Greek text has “But if H was in excess of M, G was also in excess of L” .

This proposition is tacitly used in combination with v. 16 and v. 24 in the geometrical passage in Aristotle, Meteorologica iii. 5, 376 a 22–26.

Proposition 12

If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.

Let any number of magnitudes A, B, C, D, E, F be proportional, so that, as A is to B, so is C to D and E to F;

I say that, as A is to B, so are A, C, E to B, D, F

For of A, C, E let equimultiples G, H, K be taken,

and of B, D, F other, chance, equimultiples L, M, N.

Then since, as A is to B, so is C to D, and E to F,

and of A, C, E equimultiples G, H, K have been taken,

and of B, D, F other, chance, equimultiples L, M, N,

therefore, if G is in excess of L, H is also in excess of M,

and K of N,

if equal, equal,

and if less, less;

so that, in addition,

if G is in excess of L, then G, H, K are in excess of L, M, N,

if equal, equal,

and if less, less.

Now G and G, H, K are equimultiples of A and A, C, E, since, if any number of magnitudes whatever are respectively equimultiples of any magnitudes equal in multitude, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.

[v. 1]

For the same reason

L and L, M, N are also equimultiples of B and B, D, F;

therefore, as A is to B, so are A, C, E to B, D, F.

[v. Def. 5]

Therefore etc.

Q.E.D.

Algebraically, if a:a′ = b:b′ = c:c′ etc., each ratio is equal to the ratio (a + b + c + . . .):(a′ + b′ + c′ + . . .).

This theorem is quoted by Aristotle, Eth. Nic. v. 7, 1131 b 14, in the shortened form “the whole is to the whole what each part is to each part (respectively).”

Proposition 13

If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.

For let a first magnitude A have to a second B the same ratio as a third C has to a fourth D,

and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth F; I say that the first A will also have to the second B a greater ratio than the fifth E to the sixth F.

For, since there are some equimultiples of C, E,

and of D, F other, chance, equimultiples, such that the multiple of C is in excess of the multiple of D,

while the multiple of E is not in excess of the multiple of F,

[v. Def. 7]

let them be taken,

and let G, H be equimultiples of C, E,

and K, L other, chance, equimultiples of D, F,

so that G is in excess of K, but H is not in excess of L;

and, whatever multiple G is of C, let M be also that multiple of A,

and, whatever multiple K is of D, let N be also that multiple of B.

Now, since, as A is to B, so is C to D,

and of A, C equimultiples M, G have been taken,

and of B, D other, chance, equimultiples N, K,

therefore, if M is in excess of N, G is also in excess of K,

if equal, equal,

and if less, less.

[v. Def. 5]

But G is in excess of K;

therefore M is also in excess of N.

But H is not in excess of L;

and M, H are equimultiples of A, E,

and N, L other, chance, equimultiples of B, F;

therefore A has to B a greater ratio than E has to F.

[v. Def. 7]

Therefore etc.

Q.E.D.

Alter the words “for, since” in the first line of the proof, Theon added “C has to D a greater ratio than E has to F,” so that “there are some equimultiples” began, with him, the principal sentence.

The Greek text has, after “of D, F other, chance, equimultiples,” “and the multiple of C is in excess of the multiple of D. . . .” The meaning being “such that,” I have substituted this for “and,” after Simson.

The following will show the method of Euclid’s proof.

there will he some equimultiples mc, me of c, e, and some equimultiples nd, nf of d, f, such that

Simson adds as a corollary the following:

“If the first have a greater ratio to the second than the third has to the fourth, but the third the same ratio to the fourth which the fifth has to the sixth, it may be demonstrated in like manner that the first has a greater ratio to the second than the fifth has to the sixth.”

This however scarcely seems to be worth separate statement, since it only amounts to changing the order of the two parts of the hypothesis.

Proposition 14

If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.

For let a first magnitude A have the same ratio to a second B as a third C has to a fourth D; and let A be greater than C;

I say that B is also greater than D.

For, since A is greater than C,

and B is another, chance, magnitude,

therefore A has to B a greater ratio than C has to B.

[v. 8]

But, as A is to B, so is C to D;

therefore C has also to D a greater ratio than C has to B.

[v. 13]

But that to which the same has a greater ratio is less;

[v. 10]

therefore D is less than B;

so that B is greater than D.

Similarly we can prove that, if A be equal to C, B will also be equal to D; and, if A be less than C, B will also be less than D.

Therefore etc.

Q.E.D.

Algebraically, if a : b = c : d,

then, according as a > = < c, b > = < d.

Simson adds the specific proof of the second and third parts of this proposition, which Euclid dismisses with “Similarly we can prove. . . .”

“Secondly if A be equal to C, B is equal to D; for A is to B as C, that is A, is to D;

therefore B is equal to D.

[v. 9]

Thirdly, if A be less than C, B shall be less than D.

For C is greater than A;

and, because C is to D as A is to B,

D is greater than B, by the first case.

Wherefore B is less than D.”

Aristotle, Meteorol. iii. 5, 376 a 11–14, quotes the equivalent proposition that, if a > b, c > d.

Proposition 15

Parts have the same ratio as the same multiples of them taken in corresponding order.

For let AB be the same multiple of C that DE is of F; I say that, as C is to F, so is AB to DE.

For, since AB is the same multiple of C that DE is of F, as many magnitudes as there are in AB equal to C, so many are there also in DE equal to F.

Let AB be divided into the magnitudes AG, GH, HB equal to C,

and DE into the magnitudes DK, KL, LE equal to F;

then the multitude of the magnitudes AG, GH, HB will be equal to the multitude of the magnitudes DK, KL, LE.

And, since AG, GH, HB are equal to one another,

and DK, KL, LE are also equal to one another,

therefore, as AG is to DK, so is GH to KL, and HB to LE.

[v. 7]

Therefore, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents;

[v. 12]

therefore, as AG is to DK, so is AB to DE.

But AG is equal to C and DK to F;

therefore, as C is to F, so is AB to DE.

Therefore etc.

Q.E.D.

Proposition 16

If four magnitudes be proportional, they will also be proportional alternately.

Let A, B, C, D be four proportional magnitudes, [5]

so that, as A is to B, so is C to D;

I say that they will also be so alternately, that is, as A is to C, so is B to D.

For of A, B let equimultiples E, F be taken,

and of C, D other, chance, equimultiples G, H.

Then, since E is the same multiple of A that F is of B,

and parts have the same ratio as the same multiples of them,

[v. 15]

therefore, as A is to B, so is E to F.

But as A is to B, so is C to D;

therefore also, as C is to D, so is E to F.

[v. 11]

Again, since G, H are equimultiples of C, D,

therefore, as C is to D, so is G to H.

[v. 15]

But, as C is to D, so is E to F;

therefore also, as E is to F, so is G to H.

[v. 11]

But, if four magnitudes be proportional, and the first be greater than the third, the second will also be greater than the fourth;

if equal, equal;

and if less, less.

[v. 14]

Therefore, if E is in excess of G, F is also in excess of H,

if equal, equal,

and if less, less.

Now E, F are equimultiples of A, B,

and G, H other, chance, equimultiples of C, D;

therefore, as A is to C, so is B to D.

[v. Def. 5]

Therefore etc.

Q.E.D.

Taking equimultiples ma, mb of a, b, and equimultiples nc, nd of c, d, we have, by v. 15,

Aristotle tacitly uses the theorem in Meteorologica iii. 5, 376 a 22–24.

The four magnitudes in this proposition must all be of the same kind, and Simson inserts “of the same kind” in the enunciation.

This is the first of the propositions of Eucl. v. which Smith and Bryant (Euclid’s Elements of Geometry, 1901) prove by means of vi. 1 so far as the only geometrical magnitudes in question are straight lines or rectilineal areas; and certainly the proofs are more easy to follow than Euclid’s. The proof of this proposition is as follows.

To prove that, If four magnitudes of the same kind [straight lines or rectilineal areas] be proportionals, they will be proportionals when taken alternately.

Let P, Q, R, S be the four magnitudes of the same kind such that

P : Q = R : S;

then it is required to prove that

P : R = Q : S.

First, let all the magnitudes be areas.

Construct a rectangle abcd equal to the area P, and to bc apply the rectangle bcef equal to Q.

Also to ab, bf apply rectangles ag, bk equal to R, S respectively.

Then, since the rectangles ac, be have the same height, they are to one another as their bases.

[vi. 1]

Hence (by the converse of vi. 1 ) the rectangles ag, bk have the same height, so that k is on the line hg.

Hence the rectangles ac, ag have the same height, namely ab; also be, bk have the same height, namely bf.

Secondly, let the magnitudes be straight lines AB, BC, CD, DE.

Construct the rectangles Ab, Bc, Cd, De with the same height.

Proposition 17

If magnitudes be proportional componendo, they will also be proportional separando.

Let AB, BE, CD, DF be magnitudes proportional componendo, so that, as AB is to BE, so is CD to DF;

I say that they will also be proportional separando, that is, as AE is to EB, so is CF to DF.

For of AE, EB, CF, FD let equimultiples GH, HK, LM, MN be taken,

and of EB, FD other, chance, equimultiples, KO, NP.

Then, since GH is the same multiple of AE that HK is of EB,

therefore GH is the same multiple of AE that GK is of AB.

[v. 1]

But GH is the same multiple of AE that LM is of CF;

therefore GK is the same multiple of AB that LM is of CF.

Again, since LM is the same multiple of CF that MN is of FD,

therefore LM is the same multiple of CF that LN is of CD.

[v. 1]

But LM was the same multiple of CF that GK is of AB;

therefore GK is the same multiple of AB that LN is of CD.

Therefore GK, LN are equimultiples of AB, CD.

Again, since HK is the same multiple of EB that MN is of FD,

and KO is also the same multiple of EB that NP is of FD,

therefore the sum HO is also the same multiple of EB that MP is of FD.

[v. 2]

And, since, as AB is to BE, so is CD to DF,

and of AB, CD equimultiples GK, LN have been taken,

and of EB, FD equimultiples HO, MP,

therefore, if GK is in excess of HO, LN is also in excess of MP,

if equal, equal,

and if less, less.

Let GK be in excess of HO;

then, if HK be subtracted from each,

GH is also in excess of KO.

But we saw that, if GK was in excess of HO, LN was also in excess of MP;

therefore LN is also in excess of MP,

and, if MN be subtracted from each,

LM is also in excess of NP;

so that, if GH is in excess of KO, LM is also in excess of NP.

Similarly we can prove that,

if GH be equal to KO, LM will also be equal to NP,

and if less, less.

And GH, LM are equimultiples of AE, CF,

while KO, NP are other, chance, equimultiples of EB, FD;

therefore, as AE is to EB, so is CF to FD.

Therefore etc.

Q.E.D.

I have already noted the somewhat strange me of the participles of and to convey the sense of the technical and , or what we denote by componendo and separando. is, literally “if magnitudes compounded he proportional, they will also be proportional separated,” by which is meant “if one magnitude made up of two parts is to one of its parts as another magnitude made up of two parts is to one of its parts, the remainder of the first whole is to the part of it first taken as the remainder of the second whole is to the part of it first taken.” In the algebraical formula above a, c are the wholes and b, a − b and d, c − d are the parts and remainders respectively. The formula might also he stated thus:

in which case a + b, c + d are the wholes and b, a and d, c the parts and remainders respectively. Looking at the last formula, we observe that “separated,” is used with reference not to the magnitudes a, b, c, d but to the compounded magnitudes a + b, b, c + d, d.

As the proof is somewhat long, it will be useful to give a conspectus of it in the more symbolical form. To avoid minuses, we will take for the hypothesis the form

a + b is to b as c + d is to d.

Take any equimultiples of the four magnitudes a, b, c, d, viz.

ma, mb, mc, md,

and any other equimultiples of the consequents, viz.

nb and nd.

Then, by v. 1, m(a + b), m(c + d) are equimultiples of a + b, c + d,

and, by v. 2, (m + n)b, (m + n)d are equimultiples of b, d.

Therefore, by Def. 5, since a + b is to b as c + d is to d,

according as m (a + b) > = < (m + n) b, m (c + d) > = < (m + n) d.

Subtract from m (a + b), (m + n) b the common part mb, and from m (c + d), (m + n) d the common part md; and we have,

according as ma > = < nb, mc > = < nd.

But ma, mc are any equimultiples of a, c, and nb, nd any equimultiples of b, d,

therefore, by v. Def 5,

a is to b as c is to d.

Smith and Bryant’s proof follows, mutatis mutandis, their alternative proof of the next proposition.

Proposition 18

If magnitudes be proportional separando, they will also be proportional componendo.

Let AE, EB, CF, FD be magnitudes proportional separando, so that, as AE is to EB, so is CF to FD;

I say that they will also be proportional componendo, that is, as AB is to BE, so is CD to FD.

For, if CD be not to DF as AB to BE, then, as AB is to BE, so will CD be either to some magnitude less than DF or to a greater.

First, let it be in that ratio to a less magnitude DG.

Then, since, as AB is to BE, so is CD to DG,

they are magnitudes proportional componendo;

so that they will also be proportional separando.

[v. 17]

Therefore, as AE is to EB, so is CG to GD.

But also, by hypothesis,

as AE is to EB, so is CF to FD.

Therefore also, as CG is to GD, so is CF to FD.

[v. 11]

But the first CG is greater than the third CF;

therefore the second GD is also greater than the fourth FD.

[v. 14]

But it is also less: which is impossible.

Therefore, as AB is to BE, so is not CD to a less magnitude than FD.

Similarly we can prove that neither is it in that ratio to a greater;

it is therefore in that ratio to FD itself.

Therefore etc.

Q.E.D.

In the enunciation of this proposition there is the same special use of and as there was of and in the last enunciation. Practically, as the algebraical form shows, might have been lefi out.

The following is the method of proof employed by Euclid.

which relations respectively contradict v. 14.

Simson pointed out (as Saccheri before him saw) that Euclid’s demonstration is not legitimate, because it assumes without proof that to any three magnitudes, two of which, at least, are of the same kind, there exists a fourth proportional. Clavius and, according to him, other editors made this an axiom. But it is far from axiomatic; it is not till vi. 12 that Euclid shows, by construction, that it is true even in the particular case where the three given magnitudes are all straight lines.

In order to remove the defect it is necessary either (1) to prove beforehand the proposition thus assumed by Euclid or (2) to prove v. 18 independently of it.

Saccheri ingeniously proposed that the assumed proposition should be proved, for areas and straight lines, by means of Euclid vi. 1, 2 and 12. As he says, there was nothing to prevent Euclid from interposing these propositions immediately after v. 17 and then proving v. 18 by means of them. vi. 12 enables us to construct the fourth proportional when the three given magnitudes are straight lines; and vi. 12 depends only on vi. 1 and 2. “Now,” says Saccheri, “when we have once found the means of constructing a straight line which is a fourth proportional to three given straight lines, we obviously have the solution of the general problem ‘To construct a straight line which shall have to a given straight line the same ratio which two polygons have (to one another).’” For it is sufficient to transform the polygons into two triangles of equal height and then to construct a straight line which shall be a fourth proportional to the bases of the triangles and the given straight line.

The method of Saccheri is, as will be seen, similar to that adopted by Smith and Bryant in proving the theorems of Euclid v. 16, 17, 18, 22, so far as straight lines and rectilineal areas are concerned, by means of vi. 1.

De Morgan gives a sketch of a general proof of the assumed proposition that, B being any magnitude, and P and Q two magnitudes of the same kind, there does exist a magnitude A which is to B in the same ratio as P to Q.

“The right to reason upon any aliquot part of any magnitude is assumed; though, in truth, aliquot parts obtained by continual bisection would suffice: and it is taken as previously proved that the tests of greater and of less ratio are never both presented in any one scale of relation as compared with another.”

“(1) If M be to B in a greater ratio than P to Q, so is every magnitude greater than M, and so are some less magnitudes; and if M be to B in a less ratio than P to Q, so is every magnitude less than M, and so are some greater magnitudes. Part of this is in every system: the rest is proved thus. If M be to B in a greater ratio than P to Q, say, for instance, we find that 15M lies between 22B and 23B, while 15P lies before 22Q. Let 15M exceed 22B by Z; then, if N be less than M by anything less than the 15th part of Z, 15N is between 22B and 23B: or N, less than M, is in a greater ratio to B than P to Q. And similarly for the other case.

(2) M can certainly be taken so small as to be in a less ratio to B than P to Q, and so large as to be in a greater; and since we can never pass from the greater ratio back again to the smaller by increasing M, it follows that, while we pass from the first designated value to the second, we come upon an intermediate magnitude A such that every smaller is in a less ratio to B than P to Q, and every greater in a greater ratio. Now A cannot he in a less ratio to B than P to Q, for then some greater magnitudes would also be in a less ratio; nor in a greater ratio, for then some less magnitudes would be in a greater ratio; therefore A is in the same ratio to B as P to Q. The previously proved proposition above mentioned shows the three alternatives to be the only ones.”

Alternative proofs of V 18.

Simson bases his alternative on v. 5, 6. As the 18th proposition is the converse of the 17th, and the latter is proved by means of v. 1 and 2, of which v. 5 and 6 are converses, the proof of v. 18 by v. 5 and 6 would be natural; and Simson holds that Euclid must have proved v. 18 in this way because “the 5th and 6th do not enter into the demonstration of any proposition in this book as we have it, nor can they he of any use in any proposition of the Elements,” and “the 5th and 6th have undoubtedly been put into the 5th book for the sake of some propositions in it, as all the other propositions about equimultiples have been.”

Simson’s proof is however, as it seems to me, intolerably long and difficult to follow unless it be put in the symbolical form as follows.

Suppose that a is to b as c is to d;

it is required to prove that a + b is to b as c + d is to d.

Take any equimultiples of the last four magnitudes, say

m (a + b), mb, m (c + d), md,

and any equimultiples of b, d, as

if equal, equal; and if less, less.

I. Suppose nb not greater than mb, so that nd is also not greater than md.

II. Suppose nb greater than mb.

Since m(a + b), mb, m(c + d), md are equimultiples of (a + b), b, (c + d), d,

so that ma, mc are equimultiples of a, c.

[v. 5]

Again nb, nd are equimultiples of b, d,

and so are mb, md;

therefore (n − m) b, (n − m) d are equimultiples of b, d and, whether n − m is equal to unity or to any other integer [v. 6], it follows, by Def. 5, that since a, b, c, d are proportionals,

if equal, equal; and if less, less.

(1) If now m (a + b) is greater than nb, subtracting mb from each, we have

(2) Similarly it may be proved that,

But (under 1. above) it was proved that, in the case where nb is not greater than mb,

Hence, whatever be the values of m and n, m (c + d) is always greater than, equal to, or less than nd according as m (a + b) is greater than, equal to, or less than nb.

Therefore, by Def. 5,

a + b is to b as c + d is to d.

Todhunter gives the following short demonstration from Austin (Examination of the first six books of Euclid’s Elements).

“Let AE be to EB as CF is to FD:

AB shall be to BE as CD is to DF.

For, because AE is to EB as CF is to FD,

therefore, alternately,

AE is to CF as EB is to FD.

[v. 16]

And, as one of the antecedents is to its consequent, so is the sum of the antecedents to the sum of the consequents:

[v. 12]

therefore, as EB is to FD, so are AE, EB together to CF, FD together;

Therefore, alternately,

AB is to BE as CD is to FD.”

The objection to this proof is that it is only valid in the case where the proposition v. 16 used in it is valid, i.e. where all four magnitudes are of the same kind.

Smith and Bryant’s proof avails where all four magnitudes are straight lines, where all four magnitudes are rectilineal areas, or where one antecedent and its consequent are straight lines and the other antecedent and its consequent rectilineal areas.

First, let all the magnitudes be areas.

Construct a rectangle abcd equal to A, and to be apply the rectangle bcef equal to B.

Also to ab, bf apply the rectangles ag, bk equal to C, D respectively.

Then, since the rectangles ac, be have equal heights bc, they are to one another as their bases.

[vi. 1]

Therefore [vi. 1, converse] the rectangles ag, bk have the same height, so that k is on the straight line hg.

Secondly, let the magnitudes A, B be straight lines and the magnitudes C, D areas.

Let ab, bf be equal to the straight lines A, B, and to ab, bf apply the rectangles ag, bk equal to C, D respectively.

Then, as before, the rectangles ag, bk have the same height.

Thirdly, let all the magnitudes be straight lines.

Apply to the straight lines C, D rectangles P, Q having the same height.

Proposition 19

If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.

For, as the whole AB is to the whole CD, so let the part AE subtracted be to the part CF subtracted;

I say that the remainder EB will also be to the remainder FD as the whole AB to the whole CD.

For since, as AB is to CD, so is AE to CF,

alternately also, as BA is to AE, so is DC to CF.

[v. 16]

And, since the magnitudes are proportional componendo, they will also be proportional separando,

[v. 17]

that is, as BE is to EA, so is DF to CF,

and, alternately,

as BE is to DF, so is EA to FC.

[v. 16]

But, as AE is to CF, so by hypothesis is the whole AB to the whole CD.

Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD.

[v. 11]

Therefore etc.

[PORISM. From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.]

Q.E.D.

Algebraically, if a : b = c : d (where c < a and d < b), then

(a − c):(b − d) = a : b.

The “Porism” at the end of this proposition is led up to by a few lines which Heiberg brackets because it is not Euclid’s habit to explain a Porism, and indeed a Porism, from its very nature, should not need any explanation, being a sort of by-product appearing without effort or trouble, (Proclus). But Heiberg thinks that Simson does wrong in finding fault with the argument leading to the “Porism,” and that it does contain the true demonstration of conversion of a ratio. In this it appears to me that Heiberg is clearly mistaken, the supposed proof on the basis of Prop. 19 being no more correct than the similar attempt to prove the inversion of a ratio from Prop. 4. The words are: “And since it was proved that, as AB is to CD, so is EB to FD,

alternately also, as AB is to BE, so is CD to FD:
therefore magnitudes when compounded are proportional.

But it was proved that, as BA is to AE, so is DC to CF, and this is convertendo.”

It will be seen that this amounts to proving from the hypothesis a : b = c : d that the following transformations are simultaneously true, viz.:

The former is not proved from the latter as it ought to be if it were intended to prove conversion.

The inevitable conclusion is that both the “Porism” and the argument leading up to it are interpolations, though no doubt made, as Heiberg says, before Theon’s time.

The conversion of ratios does not depend upon v. 19 at all but, as Simson shows in his Proposition E (containing a proof already given by Clavius), on Props. 17 and 18. Prop. E is as follows.

If four magnitudes be proportionals, they are also proportionals by conversion, that is, the first is to its excess above the second as the third is to its excess above the fourth.

Let AB be to BE as CD to DF:

then BA is to AE as DC to CF.

Because AB is to BE as CD to DF,

by division [separando],

AE is to EB as CF to FD,

[v. 17]

and, by inversion,

BE is to EA as DF to FC.

[Simson’s Prop. B directly obtained from v. Def. 5]

Wherefore, by composition [componendo],

BA is to AE as DC to CF.

[v. 18]

Proposition 20

If there be three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first be greater than the third, the fourth will also be greater than the sixth; if equal, equal; and, if less, less.

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, so that,

and let A be greater than C ex aequali;

I say that D will also be greater than F; if A is equal to C, equal; and, if less, less.

For, since A is greater than C,

and B is some other magnitude,

and the greater has to the same a greater ratio than the less has,

[v. 8]

therefore A has to B a greater ratio than C has to B.

But, as A is to B, so is D to E,

and, as C is to B, inversely, so is F to E;

therefore D has also to E a greater ratio than F has to E.

[v. 13]

But, of magnitudes which have a ratio to the same, that which has a greater ratio is greater;

[v. 10]

therefore D is greater than F.

Similarly we can prove that, if A be equal to C, D will also be equal to F; and if less, less.

Therefore etc.

Q.E.D.

Though, as already remarked, Euclid has not yet given us any definition of compounded ratios, Props. 20–23 contain an important part of the theory of such ratios. The term “compounded ratio” is not used, but the propositions connect themselves with the definitions of ex aequali in its two forms, the ordinary form defined in Def. 17 and that called perturbed proportion in Def. 18. The compounded ratios dealt with in these propositions are those compounded of successive ratios in which the consequent of one is the antecedent of the next, or the antecedent of one is the consequent of the next.

Prop. 22 states the fundamental proposition about the ratio ex aequali in its ordinary form, to the effect that,

with the extension to any number of such ratios; Prop. 23 gives the corresponding theorem for the case of perturbed proportion, namely that,

Each depends on a preliminary proposition, Prop. 22 on Prop. 20 and Prop. 23 on Prop. 21. The course of the proof will be made most clear by using the algebraic notation.

The preliminary Prop. 20 asserts that,

For, according as a is greater than, equal to, or less than c,

the ratio a : b is greater than, equal to, or less than the ratio c : b,

[v. 8 or v. 7]

the ratio d : e is greater than, equal to, or less than the ratio, f : e,

[by aid of v. 13 and v. 11]

and therefore d is greater than, equal to, or less than f.

[v. 10 or v. 9]

It is next proved in Prop. 22 that, by v. 4, the given proportions can be transformed into

whence, by v. 20,

so that, by Def. 5,

a : c = d : f.

Prop. 23 depends on Prop. 21 in the same way as Prop. 22 on Prop. 20, but the transformation of the ratios in Prop. 23 is to the following:

and Prop. 21 is then used.

Simson makes the proof of Prop. 20 slightly more explicit, but the main difference from the text is in the addition of the two other cases which Euclid dismisses with “Similarly we can prove.” These cases are:

“Secondly, let A be equal to C; then shall D be equal to F.

Because A and C are equal to one another,

Next, let A be less than C; then shall D be less than F.

For C is greater than A,

and, as was shown in the first case,

C is to B as F to E,

and, in like manner,

B is to A as E to D;

therefore F is greater than D, by the first case; and therefore D is less than F.”

Proposition 21

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth will also be greater than the sixth; if equal, equal; and if less, less.

Let there be three magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two are in the same ratio, and let the proportion of them be perturbed, so that,

and let A be greater than C ex aequali;

I say that D will also be greater than F; if A is equal to C, equal; and if less, less.

For, since A is greater than C,

and B is some other magnitude,

therefore A has to B a greater ratio than C has to B.

[v. 8]

But, as A is to B, so is E to F,

and, as C is to B, inversely, so is E to D.

Therefore also E has to F a greater ratio than E has to D.

[v. 13]

But that to which the same has a greater ratio is less;

[v. 10]

therefore F is less than D;

therefore D is greater than F.

Similarly we can prove that,

if A be equal to C, D will also be equal to F;

and if less, less.

Therefore etc.

Q.E.D.

Simson’s alterations correspond to those which he makes in Prop. 20. After the first case he proceeds thus.

“Secondly, let A be equal to C; then shall D be equal to F.

Because A and C are equal,

Next, let A be less than C; then shall D be less than F.

For C is greater than A,

and, as was shown,

C is to B as E, to D,

and, in like manner,

B is to A as F to E;

therefore F is greater than D, by the first case,

and therefore D is less than F.”

The proof may be shown thus.

According as a > = < c, a : b > = < c : b.

But a : b = e : f, and, by inversion, c : b = e : d.

Therefore, according as a > = < c, e : f > = < e : d,

and therefore d > = < f.

Proposition 22

If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.

Let there be any number of magnitudes A, B, C, and others D, E, F equal to them in multitude, which taken two and two together are in the same ratio, so that,

I say that they will also be in the same ratio ex aequali,

< that is, as A is to C, so is D to F >.

For of A, D let equimultiples G, H be taken,

and of B, E other, chance, equimultiples K, L;

and, further, of C, F other, chance, equimultiples M, N.

Then, since, as A is to B, so is D to E,

and of A, D equimultiples G, H have been taken,

and of B, E other, chance, equimultiples K, L,

therefore, as G is to K, so is H to L.

[v. 4]

For the same reason also,

as K is to M, so is L to N.

Since, then, there are three magnitudes G, K, M, and others H, L, N equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if G is in excess of M, H is also in excess of N;

if equal, equal; and if less, less.

[v. 20]

And G, H are equimultiples of A, D,

and M, N other, chance, equimultiples of C, F.

Therefore, as A is to C, so is D to F.

[v. Def. 5]

Therefore etc.

Q.E.D.

Euclid enunciates this proposition as true of any number of magnitudes whatever forming two sets connected in the manner described, but his proof is confined to the case where each set consists of three magnitudes only. The extension to any number of magnitudes is, however, easy, as shown by Simson.

“Next let there be four magnitudes A, B, C, D, and other four E, F, G, H, which two and two have the same ratio, viz.:

Because A, B, C are three magnitudes, and E, F, G other three, which taken two and two have the same ratio,

by the foregoing case,

A is to C as E to G.

But C is to D as G is to H;

wherefore again, by the first case,

A is to D as E to H.

And so on, whatever be the number of magnitudes.”

Proposition 23

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they will also be in the same ratio ex aequali.

Let there be three magnitudes A, B, C, and others equal to them in multitude, which, taken two and two together, are in the same proportion, namely D, E, F; and let the proportion of them be perturbed, so that,

I say that, as A is to C, so is D to F.

Of A, B, D let equimultiples G, H, K be taken,

and of C, E, F other, chance, equimultiples L, M, N.

Then, since G, H are equimultiples of A, B,

and parts have the same ratio as the same multiples of them,

[v. 15]

therefore, as A is to B, so is G to H.

For the same reason also,

as E is to F, so is M to N.

And, as A is to B, so is E to F;

therefore also, as G is to H, so is M to N.

[v. 11]

Next, since, as B is to C, so is D to E,

alternately, also, as B is to D, so is C to E.

[v. 16]

And, since H, K are equimultiples of B, D,

and parts have the same ratio as their equimultiples,

therefore, as B is to D, so is H to K.

[v. 15]

But, as B is to D, so is C to E;

therefore also, as H is to K, so is C to E.

[v. 11]

Again, since L, M are equimultiples of C, E,

therefore, as C is to E, so is L to M.

[v. 15]

But, as C is to E, so is H to K;

therefore also, as H is to K, so is L to M,

[v. 11]

and, alternately, as H is to L, so is K to M.

[v. 16]

But it was also proved that,

as G is to H, so is M to N.

Since, then, there are three magnitudes G, H, L, and others equal to them in multitude K, M, N, which taken two and two together are in the same ratio, and the proportion of them is perturbed,

therefore, ex aequali, if G is in excess of L, K is also in excess of N;

if equal, equal; and if less, less.

[v. 21]

And G, K are equimultiples of A, D,

and L, N of C, F.

Therefore, as A is to C, so is D to F.

Therefore etc.

Q.E.D.

There is an important difference between the version given by Simson of one part of the proof of this proposition and that found in the Greek text of Heiberg. Peyrard’s MS. has the version given by Heiberg, but Simson’s version has the authority of other MSS. The Basel editio princeps gives both versions (Simson’s being the first). After it has been proved by means of v. 15 and v. 11 that,

as G is to H, so is M to N,

or, with the notation used in the note on Prop. 20,

ma : mb = ne : nf,

it has to be proved further that,

and it is clear that the latter result may be directly inferred from v. 4. The reading translated by Simson makes this inference:

“And because, as B is to C, so is D to E,

and H, K are equimultiples of B, D,

and L, M of C, E,

therefore, as H is to L, so is K to M.”

[v. 4]

The version in Heiberg’s text is not only much longer (it adopts the roundabout method of using each of three Propositions v. 11, 15, 16 twice over), but it is open to the objection that it uses v. 16 which is only applicable if the four magnitudes are of the same kind; whereas v. 23, the proposition now in question, is not subject to this restriction.

Simson rightly observes that in the last step of the proof it should be stated that “G, K are any equimultiples whatever of A, D, and L, N any whatever of C, F.”

He also gives the extension of the proposition to any number of magnitudes, enunciating it thus:

“If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last”; and adding to the proof as follows:

“Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which, taken two and two in a cross order, have the same ratio, viz.:

then A is to D as E to H.

Because A, B, C are three magnitudes, and F, G, H other three which, taken two and two in a cross order, have the same ratio,

wherefore again, by the first case,

A is to D as E to H.

And so on, whatever be the number of magnitudes.”

Proposition 24

If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.

Let a first magnitude AB have to a second C the same ratio as a third DE has to a fourth F;

and let also a fifth BG have to the second C the same ratio as a sixth EH has to the fourth F;

I say that the first and fifth added together, AG, will have to the second C the same ratio as the third and sixth, DH, has to the fourth F.

For since, as BG is to C, so is EH to F,

inversely, as C is to BG, so is F to EH.

Since, then, as AB is to C, so is DE to F,

and, as C is to BG, so is F to EH,

therefore, ex aequali, as AB is to BG, so is DE to EH.

[v. 22]

And, since the magnitudes are proportional separando, they will also be proportional componendo;

[v. 18]

therefore, as AG is to GB, so is DH to HE.

But also, as BG is to C, so is EH to F;

therefore, ex aequali, as AG is to C, so is DH to F.

[v. 22]

Therefore etc.

Q.E.D.

This proposition is of the same character as those which precede the propositions relating to compounded ratios; but it could not be placed earlier than it is because v. 22 is used in the proof of it.

Inverting the second proportion to

and from this and the second of the two given proportions we obtain, by a fresh application of v. 22,

(a + b) : c = (d + e) : f.

The first use of v. 22 is important as showing that the opposite process to compounding ratios, or what we should now call division of one ratio by another, does not require any new and separate propositions.

Aristotle tacitly uses v. 24 in combination with v. 11 and v. 16, Meteorologica iii. 5, 376 a 22–26.

Simson adds two corollaries, one of which (Cor. 2) notes the extension to any number of magnitudes.

“The proposition holds true of two ranks of magnitudes whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest.”

Simson’s Cor. 1 states the corresponding proposition to the above with separando taking the place of componendo, viz., that corresponding to the algebraical form

(a − b) : c = (d − e) : f.

“COR. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition if division be used instead of composition.” That is, we use v. 17 instead of v. 18, and conclude that

(a − b) : b = (d − e) : e.

Proposition 25

If four magnitudes be proportional, the greatest and the least are greater than the remaining two.

Let the four magnitudes AB, CD, E, F be proportional so that, as AB is to CD, so is E to F, and let AB be the greatest of them and F the least;

I say that AB, F are greater than CD, E.

For let AG be made equal to E, and CH equal to F.

Since, as AB is to CD, so is E to F,

and E is equal to AG, and F to CH,

therefore, as AB is to CD, so is AG to CH.

And since, as the whole AB is to the whole CD, so is the part AG subtracted to the part CH subtracted,

the remainder GB will also be to the remainder HD as the whole AB is to the whole CD.

[v. 19]

But AB is greater than CD;

therefore GB is also greater than HD.

And, since AG is equal to E, and CH to F, therefore AG, F are equal to CH, E.

And if, GB, HD being unequal, and GB greater, AG, F be added to GB and CH, E be added to HD,

it follows that AB, F are greater than CD, E.

Therefore etc.

Q.E.D.

and a is the greatest of the four magnitudes and d the least,

a + d > b + c.

Simson is right in inserting a word in the setting-out, “let AB be the greatest of them and <consequently> F the least.” This follows from the particular case, really included in Def. 5, which Simson makes the subject of his proposition A, the case namely where the equimultiples taken are once the several magnitudes.

The proof is as follows.

There is an important particular case of this proposition, which is, however, not mentioned here, viz. the case where b = c. The result shows, in this case, that the arithmetic mean between two magnitudes is greater than their geometric mean. The truth of this is proved for straight lines in vi. 27 by “geometrical algebra,” and the theorem forms the for equations of the second degree.

Simson adds at the end of Book V. four propositions, F, G, H, K, which, however, do not seem to be of sufficient practical use to justify their inclusion here. But he adds at the end of his notes to the Book the following paragraph which deserves quotation word for word.

“The 5th book being thus corrected, I most readily agree to what the learned Dr. Barrow says, ‘that there is nothing in the whole body of the elements of a more subtile invention, nothing more solidly established, and more accurately handled than the doctrine of proportionals.’ And there is some ground to hope that geometers will think that this could not have been said with as good reason, since Theon’s time till the present.”

Simson’s claim herein will readily be admitted by all readers who are competent to form a judgment upon his criticisms and elucidations of Book V.

4. let EB be made that multiple of CG, . From this way of stating the construction one might suppose that CG was given and EB had to be found equal to a certain multiple of it. But in fact EB is what is given and CG has to be found, i.e. CG has to be constructed as a certain submultiple of EB.

5. “Let A, B, C, D be four proportional magnitudes, so that, as A is to B, so is C to D.” In a number of expressions like this it is absolutely necessary, when translating into English, to interpolate words which are not in the Greek. Thus the Greek here is: “, literally “Let A, B, C, D be four proportional magnitudes, as A to B, so C to D.” The same remark applies to the corresponding expressions in the next propositions, v. 17, 18, and to other forms of expression in v. 20–23 and later propositions: e.g. in v. 20 we have a phrase meaning literally “Let there be magnitudes. . .which taken two and two are in the same ratio, as A to B, so D to E,” etc.: in v. 21 “(magnitudes). . .which taken two and two are in the same ratio, and let the proportion of them be perturbed, as A to B, so E to F,” etc. In all such cases (where the Greek is so terse as to be almost-ungrammatical) I shall insert the words necessary in English, without further remark.