From Euclid to Riemann and Beyond¹¹1Dedicated to the memory of Marcel Berger (14 April 1927–15 October 2016).

– How to describe the shape of the universe –

Since the inception of civilization that emerged in a number of far-flung places around the globe like ancient Egypt, Mesopotamia, ancient India, ancient China, and so on, mankind has struggled to understand the universe and especially how the world came to be as it is.²²2To know the birth and evolution of the universe is “the problem of problems” at the present day as seen in the Big Bang theory, the most prevailing hypothesis of the birth. Such attempts are seen in mythical tales about the birth of the world. “Chaos”, “water”, and the like, were thought to be the fundamental entities in its beginning that was to grow gradually into the present state. For example, the epic of Atrahasis written about 1800 BCE contains a creation myth about the Mesopotamian gods Enki (god of water), Anu (god of sky), and Enlil (god of wind). The Chinese myth in the Three Five Historic Records (the 3rd century CE) tells us that the universe in the beginning was like a big egg, inside of which was darkness chaos.³³3The term “chaos” (χάος) is rooted in the poem Theogony, a major source on Greek mythology, by Hesiod (ca. ​750 BCE–ca. ​650 BCE). The epic poet describes Chaos as the primeval emptiness of the universe.

The question of the origin is tied to the question of future. Hindu cosmology has a unique feature in this respect. In contrast to the didactics in monotheistic Christianity describing the end of the world as a single event (with the Last Judgement) in history, the Rig veda, one of the oldest extant texts in Indo-European language composed between 1500 BCE–1200 BCE, alleges that our cosmos experiences a creation-destruction cycle almost endlessly (the view celebrated much later by Nietzsche). Furthermore, some literature (e.g. the Bhagavata Purana composed between the 8th and the 10th century CE or as early as the 6th century CE) mentions the “multiverse” (infinitely many universes), which resuscitated as the modern astronomical theory of the parallel universes.

In the meantime, thinkers in colonial towns in Asia Minor, Magna Graecia, and mainland Greece, cultivated a love for systematizing phenomena on a rational basis, as opposed to supernatural explanations typified by mythology and folklore in which the cult of Olympian gods and goddesses are wrapped. Many of them were not only concerned with fundamental issues arising from everyday life, as represented by Socrates (ca. ​470 BCE–399 BCE), but also labored to mathematically understand multifarious phenomena and to construct an orderly system. They appreciated purity, universality, a certainty and an elegance of mathematics, the characteristics that all other forms of knowledge do not possess. Legend has it that Plato (ca. ​428 BCE–ca. ​348 BCE) engraved the phrase “Let no one ignorant of geometry enter here” at the entrance of the Academeia (’Αϰαδημία) he founded in ca. ​387 BCE in an outskirt of Athens. Whether or not this is historically real, Academeia indubitably put great emphasis on mathematics as a prerequisite of philosophy.⁴⁴4Among Plato’s thirty-five extant dialogues, there are quite a few in which the characters (Socrates in particular) discuss mathematical knowledge in one form or another; say, Hippias major, Meno, Parmenides, Theaetetus, Republic, Laws, Timaeus, Philebus. In the Republic, Book VII, Socrates says, “Geometry is fully intellectual and a study leading to the good because it deals with absolute, unchanging truths, so that it should be part of education.”

Eventually, Greek philosophers began to speculate about the structure of the universe by deploying geometric apparatus. Exemplary is the spherical model, with the earth at the center, proposed by Plato himself, and his two former students Eudoxus of Cnidus (ca. ​408 BCE–ca. ​355 BCE) and Aristotle from Stagira (384 BCE–322 BCE).⁵⁵5Pythagoras of Samos (ca. ​580 BCE–ca. ​500 BCE) was the first to declare that the earth is a sphere, and that the universe has a soul and intelligence. Plato was a devout Pythagorean, originally meaning the member of a mathematico-religious community created by Pythagoras (ca. ​530 BCE) in Croton, Southern Italy. In his dialogue Timaeus, Plato explored cosmogonical issues, and deliberated the nature of the physical world and human beings. He referred to the demiurge (δημιουργ´ος) as the creator of the world who chose a round sphere as the most appropriate shape that embraces within itself all the shapes there are. Meanwhile, Eudoxus proposed a sophisticated system of homocentric spheres rotating about different axes through the center of the Earth, as an answer to his mentor who set the question how to reduce the apparent motions of heavenly bodies to uniform circular motions. This was stated in the treatise On velocities—now lost, but Aristotle knew about it. Eudoxus may have regarded his system simply as an abstract geometrical model; Aristotle took it to be a description of the real world, and organized it into a kind of fixed hierarchy, conjoining with his metaphysical principle (Metaphysics, XII).

Aristotle’s spherical model (Fig. ​1) was refined later as the sophisticated geocentric theory by the Alexandrian astronomer Ptolemy (ca. ​100 CE–ca. ​170 CE). His vision of the universe was set forth in his mathematico-astronomical treatise Almagest, and had been accepted for more than twelve hundred years in Western Europe and the Islamic world until Copernicus’ heliocentric theory emerged (Sect. ​3) because his theory succeeded, up to a point, in describing the apparent motions of the sun, moon, and planets.⁶⁶6The title “Almagest” was derived from the Arabic name meaning “greatest.” The original Greek title is Mathematike Syntaxis (Μαϑηματιϰὴ Σ´υνταξις). It was rendered into Latin by Gerard of Cremona (ca. ​1114–1187) from the Arabic version found in Toledo (1175), and became the most widely known in Western Europe before the Renaissance. What is notable in his system is the use of numerous epicycles (ἐπίϰυϰλος), where an epicycle is a small circle along which a planet is assumed to move, while each epicycle in turn moves along a larger circle (deferent).⁷⁷7The reasonable accuracy of Ptolemy’s system results from that an “almost periodic” motion (in the sense of H. A. Bohr)—a presumable nature of planetary motions—can be represented to any desired degree of approximation by a superposition of circular motions. This schema—a coinage of the Greek doctrine—was first set out by Apollonius of Perga (ca. ​262 BCE–ca. ​190 BCE), and developed further by Hipparchus of Nicaea (ca. ​190 BCE–ca. ​120 BCE).

At all events, an intriguing (and arguable) feature of their model is the idea of unreachable “outermost sphere.” In Aristotle’s concept (Physics, VIII, 6), it is the domain of the Prime Mover (τ`ο πρὼτη ἀϰίνητον), a variant of Plato’s notion in his cosmological argument unfolded in the Timaeus, which caused the outermost sphere to rotate at a constant angular velocity.

We shall come back to the spherical model in Sect. ​3 after discussing a relevant issue, and recount at some length how this peculiar model had an effect on philosophical and religious aspects of cosmology.

Besides the kinematical nature of celestial bodies, what seems to lie in the background of Aristotle’s view about the universe is his philosophical thought about infinity. Actually the issue of infinity was a favorite subject for Greek philosophers, dating back to the pre-Socratic period.⁸⁸8Anaximander of Miletus (ca. ​610 BCE–ca. ​546 BCE) was the first who contemplated about infinity, and employed the word apeiron (ἄπειρον meaning “unlimited”) to explain all natural phenomena in the world. Anaxagoras of Clazomenae (ca. ​510 BCE–ca. ​428 BCE) wrote the book About Nature, in which he says, “All things were together, infinite in number.” Assuming the infinite divisibility of matter, he avers, “There is no smallest among the small and no largest among the large, but always something still smaller and something still larger.”

Deliberating over his predecessors’ ​vision, Aristotle distinguished between actual and potential infinity. Briefly speaking, actual infinity is “things” that are completed and definite, thereby being transcendental in nature, while potential infinity is “things” that continue without terminating; more and more elements can be always added, but with no recognizable ending point, thus being what can be somehow corroborated within the scope of the capacity of human deed or thought. For a variety of possible reasons, Aristotle rejects actual infinity, claiming that only potential infinity exists, and captures space as something infinitely divisible into parts that are again infinitely divisible, and so on (Physics, III). His doctrine, which had satisfied nearly all scholars for a long time,⁹⁹9G. W. F. Hegel, a pivotal figure of German idealism, defended Aristotle’s perspective on the infinite in his Wissenschaft der Logik (1812–1816), though he used the terms “true (absolute) infinity” and “spurious infinity (schlecht Unendlichkeit)” instead. was employed by himself to find a way out of Zeno’s paradoxes, especially the paradox (παράδοξος) of Achilles and the Tortoise.¹⁰¹⁰10Zeno of Elea (ca. ​490 BCE–ca. ​430 BCE) was a student of Parmenides (ca. ​515 BCE–ca. ​450 BCE) who contended that the true reality is absolutely unitary, unchanging, eternal, “the one.” To vindicate his teacher’s tenet, Zeno offered the four arguable paradoxes “The Dichotomy,” “Achilles and the Tortoise,” “The Arrow,” and “The Stadium.” During the Warring States period (476 BCE–221 BCE) in China, the School of Names cultivated a philosophy similar to Parmenides’. Hui Shi (ca. ​380 BCE–ca. ​305 BCE) belonging to this school says, “Ultimate greatness has no exterior, ultimate smallness has no interior.” This, in a rehashed form, says, “The fastest runner [Achilles] in a race can never overtake the slowest [tortoise], because the pursuer must first get to the point whence the pursued started, so that the slowest must always hold a lead” (Physics, VI; 9, 239b15). To quote Aristotle’s reaction in response to the quibble, Zeno’s argument exploits an ambiguity in the nature of ‘infinity’ because Zeno seems to insist that Achilles cannot complete an infinite number of his actions (getting the point where the tortoise was); that is, ‘complete’ is the word for actual infinity, but ‘infinite number of actions’ is the phrase for potential infinity.

Aristotle’s resolute rejection of actual infinity seems in part to come from the fact that mathematicians of those days had no adequate manner to treat continuous magnitude and could do, all in all, quite well without actual infinity. This is distinctively seen in the statement in Euclid’s Elements, Book IX, Prop. ​20, about the infinitude of prime numbers, which deftly asserts, “Prime numbers are more than any assigned multitude of prime numbers.”¹¹¹¹11Euclid does not seem to entirely refrain from the use of non-potential infinity. In the Elements, Book X, Def. ​3, he says, “there exist straight lines infinite in multitude which are commensurable with a given one” ([7], Book X, p.10).

On the other hand, it appears that Archimedes of Syracuse (ca. ​287 BCE–ca. ​212 BCE) had a prescient view of infinity, as adumbrated in the Archimedes Palimpsest,¹²¹²12In 1906, J. L. Heiberg, the leading authority on Archimedes, confirmed that the palimpsest, overwritten with a Christian religious text by 13th-century monks, included the Method of Mechanical Theorems, one of Archimedes’ lost works by that time. Our knowledge of Archimedes was greatly enriched by this fabulous discovery. a 10th-century Byzantine Greek copy housed at the Metochion of the Holy Sepulcher in Jerusalem. To our astonishment, in the 174-pages text (specifically in the Method, Prop. ​14), he duly compared two infinite collections of certain geometric objects by means of a one-to-one correspondence (henceforth OTOC); see [21]. This is surely related to the concept of actual infinity that has been revived in 19th century (Sect. ​18).

Archimedes was also a master of the method of exhaustion (or the method of double contradiction) which originated, however incomplete, with Antiphon the Sophist (ca. ​480 BCE–ca. ​411 BCE) and Bryson of Heraclea (born in ca. ​450 BCE), and exploited by Eudoxus to avoid flaws that may happen when we treat infinity in a naive way. Such a flaw is found in the claim by the two originators; they contended that it is possible to construct, with compass and straightedge, a square with the same area as a given circle $C$. Their argument (criticized roundly by Aristotle in the Posterior Analytics I, ​9, ​75b40) is as follows: From the correct fact that such a construction is possible for a given polygon (Euclid’s Elements, Book II, Prop. ​14), they elicited the incorrect consequence that the same is true for $C$ on the grounds that the regular polygon with $2^{n}$ edges inscribed in $C$ eventually coincides with $C$ as we let $n$ increase endlessly (needless to say, a passage to a limit does not necessarily preserve the given properties).¹³¹³13The approach taken by Antiphon and Bryson, however incorrect, was appropriately adopted by Archimedes, who obtained $223/71<\pi<22/7$ using two regular polygons of 96 sides inscribed and circumscribed to a circle (Measurement of the Circle).

Incidentally, the problem Antiphon and Bryson challenged is designated “squaring the circle”, one of the three big problems on constructions in Greek geometry; the other problems are “doubling the cube” and “trisecting the angle.” Anaxagoras is the first who worked on squaring the circle, while Hippocrates of Chios (ca. ​470 BCE–ca. ​410 BCE) squared a lune. This held out hope to square the circle for a time, but all attempts met with failure. It was in 1882 that this problem came to a conclusion; Ferdinand von Lindemann (1852–1939) proved the impossibility of such a construction by showing that $\pi$ $=$ $3.141592\cdots$ is a transcendental number; i.e., $\pi$ is not a solution of an algebraic equation with integral coefficients (1882). Doubling the cube and trisecting the angle are also impossible as P. L. Wantzel showed (1837).

In the background of the method of exhaustion is the premise that a given quantity $\alpha$ (not necessarily numerical) can be made smaller than another one given beforehand by successively halving $A$. This, in modern terms, says that for any quantity $\beta$, there exists a natural number $n$ with $\alpha/2^{n}<\beta$; thus the premise goes along well with Anaxagoras’ view of “limitless smallness”¹⁴¹⁴14The word “exhaustion” for this reasoning was first used by Grégoire de Saint-Vincent (1584–1667); Opus Geometricum Quadraturae Circuli et Sectionum Conti, 1647, p. ​739. and is, though restricted to very special situations, regarded as a harbinger of the predicate calculus in modern logic—the branch of logic that deliberately deals with quantified statements such as “there exists an $x$ such that $\cdots$” or “for any $x$, $\cdots$”— and particularly the $\epsilon$-$\delta$ argument invented for the rigorous treatment of limits in the 19th century which adequately avoids endless processes (Remark 19.1). A variant of this premise is​: “Given two quantities, one can find a multiple of either which will exceed the other.” This is what we call the Axiom of Archimedes since it was explicitly formulated by Archimedes in his work On the Sphere and Cylinder.¹⁵¹⁵15The Elements, Book V, Def. ​4 states, “Magnitudes (μέγεϑος) are said to have a ratio to one another which can, when multiplied, exceed one another.” A similar statement appears in Aristotle’s Physics VIII, 10, 266b 2. The name “Axiom of Archimedes” was given by O. Stolz in 1883, and was adopt by David Hilbert (1862–1943) in a modern treatment of Euclidean geometry (Remark 10.1 (2)). Eudoxus and Archimedes combined these premises with reductio ad absurdum (proof by contradiction) in a judicious manner,¹⁶¹⁶16Reductio ad absurdum is a reasoning in the Eleatics philosophy initiated by Parmenides. Incidentally, the Chinese word for “contradiction” is “máodùn,” literally “spear-shield,” stemming from an anecdote in the Han Feizi, an ancient Chinese text attributed to the political philosopher Han Fei (ca. ​280 BCE–ca. ​233 BCE). The story goes as follows. A dealer of spears and shields advertised that one of his spears could pierce any shield, and at the same time said that one of his shields could defend from all spear attacks. Then one customer queried the dealer what happens when the shield and spear he mentioned would be used in a fight. and established various results on area and volume by highly sophisticated arguments. Noteworthy is that, in his computations of ratios of areas or volumes of two figures, Archimedes made use of infinitesimals in a way similar to the one in integral calculus at the early stage (e.g. Quadrature of the Parabola).

Some 1500 years later, the issue of infinity was examined by two scholars. Thomas Bradwardine (ca. ​1290–ca. ​1349)—a key figure in the Oxford Calculators (a group of mathematics-oriented thinkers associated with Merton College)—employed the principle of OTOC in discussing the aspects of infinite. His observation is construed, in modern terms, as “an infinite subset could be equal to its proper subset” (​Tractatus de continuo, 1328–1335). This is a polemic work directed against atomistic thinkers in his time such as N. d’Autrécourt who, following the classical atomic concept, considered that matter and space were all made up of indivisible atoms, as opposed to Aristotle’s doctrine. After a while, Nicole Oresme (1320–1382), a significant scholar of the later Middle Ages, elaborated the thought that the collection of odd numbers is not smaller than that of natural numbers because it is possible to count the odd numbers by the natural numbers (​Physics Commentary, around 1345).¹⁷¹⁷17Oresme was in favor of the spherical model of the universe, but took a noncommittal attitude regarding the Aristotelian theory of the stationary Earth and a rotating sphere of the fixed stars (Livre du ciel et du monde and Questiones super De celo, Questiones de spera).

Meanwhile, the magic of infinity has bewildered Galileo Galilei (1564–1642). In his Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze (1638), written during house arrest as the result of the Inquisition, he says, in a similar vein to Oresme’s claim, “Even though the number of squares should be less than that of all natural numbers, there appears to exist as many squares as natural numbers because any natural number corresponds to its square, and any square corresponds to its square root” (Galileo’s paradox).

Let us turn to the issue of the universe. Were Aristotle’s spherical model correct, there would be its “outside.” So the question at once arises as to what the outside means after all. Is it something substantial or just speculative fabrication? Defending this ridiculous image of the universe, Aristotle explained away by saying, “there is neither place, nor void outside the heavens by reason that the heaven does not exist inside another thing” (On the Heavens, I, 9).

Aristotle’s model continued to be employed even in the time when the Christian tenet dominated the scholarly world in Europe. Especially, influenced by Thomas Aquinas (1225–1274), who incorporated extensive Aristotelian philosophy throughout his own theology, the scientific substratum in Christianity was synthesized with the Aristotelian physics.²⁰²⁰20In his masterwork Summa Theologiae, Aquinas asserts that God is perfect, complete, and embodies actual infinity. To reconcile his assertion with Aristotle’s doctrine, he says, “God is an infinite that has bounds.” After a while, the Florentine poet Dante Alighieri (1265–1321) reinforced Aristotle’s idea of “outside” in his unfinished work Convivio. He says, “in the supreme edifice of the universe, all the world is included, and beyond which is nothing; and it is not in space, but was formed solely in the Primal Mind.” He further offered an imaginal vision of an intriguing macrocosm in order to turn down the queer consequence of the Biblical concept of ascent to Heaven and descent to Hell which connotes that Hell is the center of the spherical universe (Sect. ​21).

An exception is Nicolaus Cusanus (Nicholas of Cusa, 1401–1464), a first-class scientist of his time. He alleges, refuting the prevalent outlook of the world, “The universe is not finite in the sense of physically unboundedness since there is no special center in the universe, and hence the outermost sphere cannot be a boundary.” At the same time, he argued finiteness of the universe, by which he meant to say “privatively infinite” because “the world cannot be conceived of as finite, albeit it is not infinite” (De Docta ignorantia; 1440). This rather contradictory dictum elaborated in the ground-breaking book is a consequence of his analysis of conceivability and a parallelism between the universe and God.

A hundred years later, a dramatic turnabout took place. The heliocentric theory proposed by Nicolaus Copernicus (1473–1543) in his De revolutionibus orbium coelestium brings out the apparent retrograde motion of planets better than Ptolemy’s theory. He already got his ideas—relying largely on Arabic astronomy typified by al-Batt$\overline{\rm a}$nī²¹²¹21His work Kit$\overline{\rm a}$b az-Zīj (Book of Astronomical Tables), translated into Latin as De Motu Stellarum by Plato Tiburtinus in 1116, was quoted by Tycho Brahe, Kepler and Galileo. (ca. ​858 CE–929 CE)—some time before 1541 (Commentariolus), but resisted, in spite of his friends’ persuasion, to make his theory public since he was afraid of being a target of contempt. It was in 1543, just before his death, that his work (with a preface which puts the accent on the hypothetical nature of the contents) was brought out.

Now, what did Copernicus think about the size of the universe? His model of the universe is spherical with the outermost consisting of motionless, fixed stars; thus being not much different from Aristotle’s model in this respect. Meanwhile, Giordano Bruno (1548–1600) argued against the outermost sphere (while accepting Copernicus’ theory), reasoning that the infinite power of God would not have produced a finite creation. He was quite explicit in his belief that there is no special place as the center of the universe (De l’infinito universo et mondi, 1584); thus indicating that the universe is endless, limitless, and homogeneous. His outspoken views, containing Hermetic elements, scandalized Catholics and Protestants alike.²²²²22Some modern scholars consider him as a magus, not a pioneer of science because Hermeticism is an ancient spiritual and magical tradition. Others, however, have argued that the magical world-view in Hermeticism was a necessary precursor to the Scientific Revolution. Consequently, he was condemned as an impenitent heretic and eventually burned alive at the stake after 8 years’ solitary confinement.

Bruno’s view was partly shared by his contemporary Thomas Digges (1546–1595). He was the first to expound the Copernican system in English. In an appendix to a new edition of his father’s book A Prognostication everlasting, Digges discarded Copernicus’ notion of a fixed shell of immovable stars, presuming infinitely many stars at varying distances (1576)—a more tenable reasoning than Bruno’s. What should be thought over here is that the religious atmosphere in England in his time was different from that in the Continent because of the English Reformation that started in the reign of Henry VIII..

Interrupting the chronological account, we shall go back to ancient Greece, where we find forerunners of Cusanus, Copernicus, and Bruno. Among them, Archytas of Tarentum (428 BCE–347 BCE) is a precursor of Bruno. He inferred that any place in our space looks the same, and hence no boundary can exist; otherwise we have a completely different sight at a boundary point. He thus concluded infiniteness of the universe.²³²³23Anaximander, belonging to the previous generation, conceived a mechanical model of the world based on a non-mythological explanatory hypothesis, and alleged that the Earth has a disk shape and is floating very still in the center of the infinite, not supported by anything, while Anaxagoras portrayed the sun as a mass of blazing metal, and postulated that Mind (νου̃ς, Nous) was the initiating and governing principle of the cosmos (ϰ´οςμος). The Pythagorean Philolaus (ca. ​470 BCE–ca. ​385 BCE) relinquished the geocentric model, saying that the Earth, Sun, and stars revolve around an unseen central fire. As alluded to in Archimedes’ work Sand Reckoner, Aristarchus of Samos (ca. ​310 BCE–ca. ​230 BCE)—apparently influenced by Philolaus—had speculated that the sun is at the center of the solar system for the reason that the geocentric theory is against his conclusion that the sun is much bigger than the moon. In his work On the Sizes and Distances, he figured out the distances and sizes of the sun and the moon, under the assumption that the moon receives light from the sun. In the Sand Reckoner, Archimedes himself harbored the ambition to estimate the size of the universe in the wake of Aristarchus’ heliocentric spherical model. To this end, he proposed a peculiar number system to remedy the inadequacies of the Greek one and expressed the number of sand grains filling a cosmological sphere. The presupposition he made is that the ratio of the diameter of the universe to the diameter of the orbit of the earth around the sun equals the ratio of the diameter of the orbit of the earth around the sun to the diameter of the earth.

Aristarchus’ heliocentric model was espoused by Seleucus of Seleucia, a Hellenistic astronomer (born ca. ​190 BCE), who developed a method to compute planetary positions. In the end, however, the Greek heliocentrism had been long forgotten. Even Hipparchus who undertook to find a more accurate distance between the sun and the moon took a step backwards.

Now in passing, we shall make special mention of Alexandria founded at the mouth of the Nile in 332 BCE by Alexander the Great, after his conquest of Egypt. It was Ptolemy ​I, the founder of the Ptolemaic dynasty, who raised the city to a center of Hellenistic culture, an offspring of Greek culture that flourished around the Mediterranean after the decline of Athens. He and his son Ptolemy ​II—both held academic activities in high esteem—established the Great Museum (Μουςει̃ον), where many thinkers and scientists from the Mediterranean world studied and collaborated with each other. Archimedes stayed there sometime in adolescence.

Prominent among them is Eratosthenes of Cyrene (ca. ​276 BCE–ca. ​195 BCE), the third chief librarian appointed by Ptolemy II and famous for an algorithm for finding all prime numbers.²⁴²⁴24Archimedes’ Method mentioned in fn. ​12 takes the form of a letter to Eratosthenes. Pushing further the view about the spherical earth, he adroitly calculated the meridian of the earth. The outcome was 46,620 km, about 16% greater than the actual value. His work On the measurement of the earth was lost, but the book On the circular motions of the celestial bodies by Cleomedes (died ca. ​489 CE) explains Eratosthenes’ deduction relying on a property of parallels and the observation that at noon on the summer solstice, the sun casts no shadow in Syene (now Aswan), while it casts a shadow on one fiftieth of a circle ($7.2$ degree) in Alexandria on the same degree of longitude as Syene (Fig. ​2). His resulting value is deduced from the distance between the two cities (5040 stades$=$925 km), which he inferred from the number of days that caravans require for travelling between them.

The Ptolemaic dynasty lasted until the Roman conquest of Egypt in 30 BCE, but even afterwards Alexandria maintained its position as the center of scientific activity (see Sect. ​11).

Finishing the excursion into ancient times, we shall turn back to the later stage of the Renaissance, the age that science was progressively separated from theology, and scientists came to direct their attention to scientific evidence rather than to theological accounts (thus science was becoming an occasional annoyance to the religious authorities). Representative in this period (religiously tumultuous times) are the Catholic Galileo and the Protestant Johannes Kepler (1571–1630); both gave a final death blow to Aristotelian/Ptolemaic theory.

Kepler discovered the three laws of planetary motion, based on the data of Mars’ motion recorded by Tycho Brahe (1546–1601).²⁵²⁵25Tycho Brahe, the last of the major naked-eye astronomers, rejected the heliocentricism. His model of the universe is a combination of the Ptolemaic and the Copernican systems, in which the planets revolved around the Sun, which in turn moved around the stationary Earth. His first and second laws were elaborated in the Astronomia nova (1609). The first law asserts that the orbit of a planet is elliptical in shape with the center of the sun being located at one focus. The second law says that a line segment joining the sun and a planet sweeps out equal areas in equal intervals of time. The third law stated in the Epitome Astronomiae Copernicanae (1617–1621) and the Harmonices Mundi (1619) maintains that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Before this epoch-making discovery, however, he attempted to explain the distances in the Solar system by means of “regular convex polyhedra” inscribing and circumscribed by spheres, where the six spheres separating those solids correspond to Saturn, Jupiter, Mars, Earth, Venus, and Mercury (Mysterium Cosmographicum, 1596; Fig. ​3). He also rejected the infiniteness of the universe, on the basis of an astronomical speculation on the one hand, and on the traditional scholastic doctrine on the other (for this reason, Kepler is portrayed as the last astronomer of the Renaissance, and not the first of the new age).

Galileo was not affected by Aristotelian prejudice that an experiment was an interference with the natural course of Nature. Without performing any experiment at all, medieval successors of Aristotle insisted, for instance, that a projectile is pushed along by the force they called “impetus,” and if the impetus is expended, the object should fall straight to the ground. At the apogee of his scientific career, Galileo had conducted experiments on projectile motion, and observed that their trajectories are always parabolic (1604–1608). He further investigated the relation between the distance $d$ an object (say, a ball) falls and the time $t$ that passes during the fall, and found the formula $d=\frac{1}{2}gt^{2}$ (in modern terms), where $g=9.8\leavevmode\nobreak\ \!{\rm m}/{\rm s}^{2}$, the gravitational acceleration (Discorsi). The crux of this formula is that the drop distance does not depend on the mass of a falling object, contrary to Aristotle’s prediction (Physics, IV, 8, 215a25).²⁶²⁶26The mean speed theorem due to the Oxford Calculators (nothing but the formula for the area of a trapezoid from today’s view) is a precursor of “the law of falling bodies.”

Galileo turned his eyes on the universe. With his handmade telescope, he made various astronomical observations and confirmed that four moons orbited Jupiter. This prompted him to defend the heliocentricism, for provided that Aristotle were right about all things orbiting earth, these moons could not exist (Sidereus Nuncius, 1610). As an inevitable consequence, his Copernican view led to the condemnation at the Inquisition (1616, 1633). After being forced to racant, he was thrown into a more thorny position than Kepler, and had to avoid dangerous issues that may provoke the Church. Regarding the size of the universe, he denied the existence of the celestial sphere as the limit, while he was reluctant to say definitely that the universe is infinite, because of censorship by the Church. In a letter to F. Liceti on February 10, 1640, Galileo says, “the question about the size of the universe is beyond human knowledge; it can be only answered by the Bible and a divine revelation.”

Next to Kepler and Galileo are Descartes and Pascal, the intellectual heroes of 17th-century France, who laid the starting point of the Enlightenment with their inquiries into truth and the limits of reason.

René Descartes (1596–1650), whose natural philosophy agrees in broad lines with Galileo’s one, defended the heliocentrism, by saying that it is much simpler and distinct, but hesitated to make his opinion public upon the news of the Inquisition’s conviction of Galileo (1633). As regards the size of the universe, he maintained at first that the universe is finite; but later became ambivalent. In a letter in 1649 to the rationalist theologian H. More of Cambridge, he conceded, after a long dispute, that the universe must have infinite expanse because one cannot think of the limit; for if the limit exists, one cannot help but think of the outside of space which must be the same as our space.

While, for Descartes, the self is prior and independent of any knowledge of the world, Blaise Pascal (1623–1662) pronounced that the order is reversed; that is, knowledge of the world is a prerequisite to knowledge of the self. In the Pensées (1670), in comparison with the disproportion between our justice and God’s, he writes, “Unity added to infinity does not increase it at all [….]: the finite is annihilated in the presence of the infinite and becomes pure nothingness” (fragment 418). Further, he says, “Nature is an infinite sphere of which the center is everywhere and the circumference nowhere” (fragment 199).²⁷²⁷27This sentence, reminiscent of Cusanus’ view, may be from the Liber XXIV philosophorum (Book of the 24 Philosophers), an influential philosophical and theological medieval text, usually attributed to Hermes Trismegistus, the purported author of the Corpus Hermeticum.

As described hitherto, the posture to pay attention to the universe is a steady tradition of European culture. In its background, philosophy was nurtured in the bosom of cosmogony, and could not be separated from religion because of the intimacy between them. Especially after the rise of Christianity, Western scholars had to be confronted, at the risk of their life in the worst case, with God as a creator. Even after the tension between science and religion was eased, scientists could not be entirely free from God, whatever His image is. Such milieu led quite naturally to probing questions about our universe in return.

In summary, “whether the universe is finite or infinite,” whatever it means, is an esoteric issue in religion, metaphysics, and astronomy in Europe that had intrigued and baffled mankind since dim antiquity. What matters most of all is whether the outside is necessary when we talk about finiteness of the universe.

Intuition for space surrounding us was the driving force for people to puff up their image of the universe. Needless to say, from the ancient times to the Middle Ages and even the early modern times, the mental image that nearly all people had is, if not being particularly conscious about it, the one described by Euclidean space, the model of our space named after the Alexandrian Euclid (Ε’υϰλείδης; ca. 300 BCE; see Sect. ​9 for the details). Even the advocates of the spherical model of the universe imagined Euclidean space as the entity embracing all. Synthetic geometry executed on this model is what we call Euclidean geometry. It started with a collection of geometrical results acquired in Egypt and Mesopotamia by empirical investigations or experience of land surveys and constructions of magnificent and imposing structures, and had been systemized, as the search of universals, through the efforts of Greek geometers.

Putting it briefly, Euclidean space is homogeneous in the sense that there is no special place, and it is isotropic; i.e., there is no special direction.²⁸²⁸28Euclidean space holds one more significant aspect expressed by a property of parallels, but its true meaning had not been comprehended for quite a while; see Sect. ​10. These features are not expressly indicated in Greek geometry, but are guaranteed by a property of congruence;²⁹²⁹29 Prop. ​4 in the Elements, Book I, is the first of the congruence propositions. namely any geometric figure can be rotated and moved to an arbitrary place while keeping its shape and size. What should be pinpointed here is that it was not until the 19th century that people explicitly conceptualized Euclidean space (or space as a mathematical object which fits in with our spatial intuition). Until then, geometry meant only Euclidean geometry, and the space where geometry is performed was considered as an a priori entity, or what amounts to the same thing, our space—a place of storage in which objects are recognized, and a place of manufacture in which objects are constructed—was not an entity for which we investigate whether our understanding of it is right or wrong, thereby all the propositions of geometry being considered “absolute truth.”

As stated above, Euclidean geometry had been a lofty edifice for nearly 2000 years that nobody could break down. Only the appearance changed when Descartes invented the so-called algebraic method. This epoch-making method is elucidated in his La Géométrie [5], one of the three essays attached to the philosophical and autobiographical treatise Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (1637).³⁰³⁰30La Géométrie consists of Book I (Des problèmes qu’on peut construire sans y employer que des cercles et des lignes droites), Book II (De la nature des lignes courbes), and Book III (De la construction des problèmes solides ou plus que solides). The other two essays are La Dioptrique and Les Météores.

Descartes’ prime concern was, though he was trained in religion with the still-authoritarian nature, to find principles that one can know as true without any scruples. In the self-imposed search for certainty, he linked philosophy with science, and had the confidence that certainty could be found in the mathematical proofs having the apodictic character. Upon his emphasis on lucid methodology and dissatisfaction with the ancient arcane method, as already glimpsed in his Regulae ad Directionem Ingenii (1628), he attempted to create “universal mathematics” by bridging the gap between arithmetic and geometry that used to be thought of as different terrains. Indeed, the Greeks definitely distinguished geometric quantities from numerical values, and even thought that length, area, and volume belong to different categories, thus lumping them together makes no sense. Under such shackles (and being devoid of symbolic algebra), the “equality,” “addition/subtraction,” and the “large/small relation” for two figures in the same category were defined by means of geometric operations.³¹³¹31The Greeks had difficulty to handle irrationals in their arithmetic and were forced to replace algebraic manipulations by geometric ones ([22]). Actually, geometry had been thought of as far more general than arithmetic as seen in Aristotle’s words, “We cannot prove geometric truths by arithmetic” (Posterior Analytics, I, 7; see also Plato’s Philebus, 56d). The word “algebra,” stemmed from the Arabic title al-Kit$\overline{a}b$ al-mukhtasar fi his$\overline{a}$b al-jabr walmuq$\overline{a}$bala (The Compendious Book on Calculation by Completion and Balancing) of the book written approximately 830 CE by al-Khw$\overline{\rm a}$rizmī (ca. ​780–ca. ​850) (Remark 2.1 (2)), was first imported into Europe in the early Middle Ages as a medical term, meaning “the joining together of what is broken.” In addition, algorithm, meaning a process to be followed in calculations, is a transliteration of his surname al-Khw$\overline{\rm a}$rizmī. Specifically, they considered that two polygons (resp. polyhedra) are “equal” if they are scissors-congruent; i.e., if the first can be cut into finitely many polygonal (resp. polyhedral) pieces that can be reassembled to yield the second.

In his essay La Géométrie, Descartes made public that all kinds of geometric magnitude can be unified by representing them as line segments, and introduced the fundamental rules of calculation in this framework.³²³²32A letter dated March 26, 1619 to I. Beeckman indicates that Descartes already held a rough scheme at the age of 22. Pierre de Fermat (1601–1665) argued a similar idea in his Ad locos plano et solidos isagoge (1636), though not published in his life time. His idea is to fix a unit length, with which he defines addition, subtraction, multiplication, division, and the extraction of roots for segments, appealing to the proportional relations of similar triangles and the Pythagorean Theorem, as indicated in Fig. ​4. Then he adopts a single axis to represent these operations; thereby replacing the Greek geometric algebra by “numerical” algebra. On this basic format, he handles various problems for a class of algebraic curves. Priding himself on his invention, he says, in a fond familiar letter dated November 1643 to Elisabeth (Princess Palatine of Bohemia), that extra effort to make up the constructions and proofs by means of Euclid’s theorems is no more than a poor excuse for self-congratulation of petty geometers because such effort does not require any scientific mind.

Afterwards, Franciscus van Schooten (1615–1660), who met Descartes in Leiden (1632) and read the still-unpublished La Géométrie, translated the French text into Latin, and endeavored to disseminate the algebraic method to the scientific community (1649). In the second edition (1659–61), he added annotations and transformed Descartes’ approach into a systematic theory, which made it far more accessible to a large number of audience.

Descartes’ method, along with algebraic notations dating back to François Viète (1540–1603) who made a momentous step towards modern algebra,³³³³33Viète, In artem analyticem isagoge (1591). was inherited as analytic geometry afterwards. With this progression, geometric figures were to be transplanted to algebraic objects in the coordinate plane $\mathbb{R}^{2}$ or the coordinate space $\mathbb{R}^{3}$.³⁴³⁴34It was Leibniz who first introduced the “coordinate system” in its present sense (De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorum usu, Acta Eruditorum, 11 (1692), 168–171). Incidentally, Oresme used constructions similar to rectangular coordinates in the Questiones super geometriam Euclidis and Tractatus de configurationibus qualitatum et motuum (ca. ​1370), but there is no evidence of linking algebra and geometry. What is more, this new discipline was integrally connected with the calculus initiated by Issac Newton (1642–1726) and Gottfried Wilhelm von Leibniz (1646–1716), independently and almost simultaneously, which is to be a vital necessity when we talk about the shape of our universe.

Newton learnt much from the Exercitationes mathematicae libri quinque (1657) by van Schooten and Clavis Mathematicae (1631) by William Oughtred (1574–1660) when he was a student of Trinity College, Cambridge. “Fluxion” is Newton’s underlying term in his differential calculus, meaning the instantaneous rate of change of a fluent, a varying (flowing) quantity. His idea, to which Newton was led by personal communication with Issac Barrow (1630–1677) and also by Wallis’ book Arithmetica infinitorum (1656),³⁵³⁵35John Wallis (1616–1703) propagated Descartes’ idea in Great Britain through his work. is stated in two manuscripts; one is of October 1666 written when he was evacuated in a neighborhood of his family home at Woolsthorpe during the Great Plague, and another is the Tractatus De Methodis Serierum et Fluxionum (1671). In his calculus, Newton made use of power series in a systematic way.

Meanwhile, Leibniz almost completed calculus while staying in Paris (1673–1676) as a diplomat of the Electorate of Mainz in order to get Germany back on its feet from the exhaustion caused by the Thirty Years’ War. Although the aspired end of diplomacy was not attained, he had an opportunity to meet Christiaan Huygens (1629–1695), a leading scientist of his time, who happened to be invited by Louis XIV as a founding member of the Académie des Sciences. Very helpful for him was the suggestion by Huygens to read Pascal’s Lettre de Monsieur Dettonville$\cdots$ (1658). Further, his perusal of Descartes’ work was an assistance in strengthening the basis of his thought. Around the time when Leibniz returned to Germany, he improved the presentation of calculus, and brought it out as two papers; Nova methodus pro maximis et minimis, and De geometria recondita et analysi indivisibilium atque infinitorum.³⁶³⁶36His papers were published in Acta Eruditorum, 3 (1684), 467–473 and 5 (1686), 292–300. In the second paper, he states, “With this idea, geometry will make a far more greater strides than Viète’s and Descartes’.”

Their calculus, though still something of mystery (Remark 15.1 (2)), provided a powerful tool which enabled to handle more complicated geometric figures than the ones that Greek geometers treated in an ad hoc manner.³⁷³⁷37In the Elements, Book III, Def. ​2, Euclid says, “a straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.” Curiously, in the Metaphysics, III, 998a, Aristotle reports that Protagoras (ca. ​490 BCE–ca. ​420 BCE) argued against geometry that a straight line cannot be perceived to touch a circle at only one point. Indeed, differential calculus provides a unified recipe to find tangents of a general curve (say, transcendental curves Descartes did not deal with), and integral calculus (called the “inverse tangent problem” by Leibniz) allows more latitude in calculating the area of a general figure without any ingenious trick. Paramountly important is the discovery that tangent (a local concept) and area (a global concept) are linked through the fundamental theorem of calculus (FTC).³⁸³⁸38I. Barrow, who strongly opposed to Descartes’ method in contrast with Wallis, knew the FTC in a geometric form (Prop. ​11, Lecture 10 of his Lectiones Geometricae delivered in 1664–1666). A transparent proof was given by Leibniz in his De geometria, 1686.

There are other lines of evolution of Descartes’ method and its offspring. Analytic geometry paved the way for higher-dimensional geometry (Sect. ​14), which was to be incorporated into Riemann’s theory of curved spaces (Sect. ​15). Algebraic geometry is regarded as the ultimate incarnation of Cartesian geometry, which came of age from the late 19th century to the early 20th century. Descartes’ method also prompted to set up mathematical theories in a strictly logical manner, not relying on geometry, but relying on arithmetic wherein concepts involved are entirely framed in the language of real numbers or systems of such (Sect. ​16). This signifies in some sense that mathematicians eventually become aware of mathematical reality—a sort of Plato’s world of Forms (είδος or its cognates)—distinguished from physical reality or the empirical world.

Leaving Euclidean geometry aside for a moment, we shall touch on projective geometry, the embryonic subject that came up during the Renaissance period and had matured in the 19th century. This new approach is exclusively concerned with quantity-independent properties such as “three points are on a line” (collinearity) and “three lines intersect at a point” (concurrency) that are invariant under projective transformations (the left of Fig. ​5).⁴³⁴³43 Imitating this wording, one may say that Euclidean geometry is a geometry that treats invariant properties under congruence transformations. Such a perspective is ascribed to Christian Felix Klein (1849–1925), who promulgated his scheme in the booklet Vergleichende Betrachtungen über neuere geometrische Forschungen (“Erlangen Program” in short) in 1872.

Projective geometry has a historical link to (linear) perspective, an innovative skill in drawing contrived by Filippo Brunelleschi(1377–1446), which allowed Renaissance artists to portray a thing in space and landscapes as someone actually might see them.⁴⁴⁴⁴44One of the earliest who have used perspective is Masaccio. He limned the San Giovenale Triptych (1422) based on the principles he learned from Brunelleschi, which emblematizes the transition from medieval mysticism to the Renaissance spirit. The theoretical aspect of perspective was investigated by Leon Battista Alberti (1404–1472) and Piero della Francesca (1416–1492). In the preface of his book Della Pittura (1435) dedicated to Brunelleschi, Alberti emphatically writes, “I used to marvel and at the same time to grieve that so many excellent and superior arts and sciences from our most vigorous antique past could now seem lacking and almost wholly lost. $\cdots$ Since this work [due to Brunelleschi] seems impossible of execution in our time, if I judge rightly, it was probably unknown and unthought of among the Ancients” (translated by J. R. Spencer). Piero’s book De Prospectiva Pingendi (1475) exemplifies the effective symbiosis of geometry and art; he actually had profound knowledge of Greek geometry and made a transcription of a Latin translation of Archimedes’ work.

The “Renaissance Man” Leonardo da Vinci (1452–1519)—stimulated by Alberti’s book—fully deserves his reputation as a true master of perspective. His technique is particularly seen in his study for A Magis adoratur (ca. ​1481). He says, “Perspective is nothing else than seeing a place or objects behind a plane of glass, quite transparent, on the surface of which the objects behind the glass are to be drawn” ([30]). Albrecht Dürer (1471–1528), a weighty figure of the Northern Renaissance who shared da Vinci’s pursuit of art, discussed in his work Underweysung der Messung (1525) an assortment of mechanisms for drawing in perspective from models (the right of Fig. ​6). This work, in which he touched on “Doubling the cube,” is the first on advanced mathematics in German.

Subsequently, Federico Commandino (1506–1575) published the work entitled Commentarius in planisphaerium Ptolemaei (1558). This, a commentary on Ptolemy’s work Planisphaerium, includes an account of the stereographic projection of the celestial sphere, and is a work on perspective from a mathematical viewpoint (concurrently, he translated several work of ancient scholars).

The origin of projective geometry can be traced back to the work of Apollonius of Perga on conic sections and Pappus of Alexandria (ca. ​290 CE–ca. ​350 CE).⁴⁶⁴⁶46The first who studied conic sections is Menaechmus (ca.​380 BCE–ca.​320 BCE), who used them to solve “doubling the cube.” The names ellipse (ἔλλειψις), parabola (παραβολη) and hyperbola (´υπερβολη) for conic curves were introduced by Apollonius. In his Collection, Book VII, where Pappus enunciated his hexagon theorem, he made use of a concept equivalent to what we call now the cross-ratio, which was to enrich projective geometry from the quantitative side since it is invariant under projective transformations. Here the cross-ratio of collinear points $A$, $B$, $C$ and $D$ is defined as $[A,B,C,D]$ $=\displaystyle(AC\cdot BD)/(BC\cdot AD)$, where each of the distances is signed according to a consistent orientation of the line.⁴⁷⁴⁷47The cross-ratio of real numbers $x_{1},x_{2},x_{3},x_{4}$ is defined by $[x_{1},x_{2},x_{3},x_{4}]\!:=(x_{3}-x_{1})(x_{4}-x_{2})/(x_{3}-x_{2})(x_{4}-x_{1})$. The projective invariance of cross-ratios amounts to the identity $[T(x_{1}),T(x_{2}),T(x_{3}),T(x_{4})]=[x_{1},x_{2},x_{3},x_{4}]$, where $T(x)=(ax+b)/(cx+d)$ $(ad-bc\neq 0)$.

Projective geometry as a solid discipline was essentially initiated by Girard Desargues (1591–1661), a coeval of Descartes. In 1639, motivated by a practical purpose pertinent to perspective, he published the Brouillon project d’une atteinte aux événemens des rencontres du Cone avec un Plan.⁴⁸⁴⁸48The celebrated Desargues’ theorem showed up in an appendix of the book Exemple de l’une des manières universelles du S.G.D.L. touchant la pratique de la perspective published in 1648 by his friend A. Bosse. Pascal took a strong interest in Desargues’ work, and studied conic sections in depth at the age of 16. Philippe de La Hire (1640–1718) was affected by Desargues’ work, too. He is most noted for the work Sectiones Conicae in novem libros distributatae (1685). However, partly because Descartes’ method spread among the mathematical community in those days and because of his peculiar style of writing, his work remained unrecognized until his work was republished in 1864 by Noël-Germinal Poudra. Without being aware of Desargues’ work, Jean-Victor Poncelet (1788–1867) laid a firm foundation for projective geometry in his masterpiece Traité des propriétés projectives des figures (1822).

In any case, projective geometry falls within classical geometry at this moment; but it has an interesting hallmark in view of infiniteness. Guidobaldi del Monte (1505–1607) in the Perspectivae Libri VI (1600) and Kepler in his Astronomiae pars optica (1604) proposed the idea of points at infinity. Subsequently, having in mind the “horizon” in perspective drawing (Fig. ​5), Desargues appends points at infinity to the plane as idealized limiting points at the “end,” and argues that two parallel lines intersect at a point at infinity. Later, the “plane plus points at infinity” came to be termed the projective plane (Sect. ​16).

Not surprisingly, Euclidean geometry predominated in physics for a long time. Salient is Newton who adopted Euclidean geometry as the base of his grand work Philosophiae Naturalis Principia Mathematica (1687, 1713, 1726; briefly called the Principia), in which he formulated the law of inertia,⁴⁹⁴⁹49The law of inertia was essentially discovered by Galileo during the first decade of the 17th century though he did not understand the law in the general way (the term “inertia” was first introduced by Kepler in his Epitome Astronomiae Copernicanae). The general formulation of the law was devised by Galileo’s pupils and Descartes (the Principles of Philosophy, 1644). the law of motion, the law of action-reaction and the law of universal gravitation.

It was in 1666, exactly the same year when he worked out calculus, that Newton found a clue leading to the law of universal gravitation. Hagiography has it that he was inspired, in a stroke of genius, to formulate the law while watching the fall of an apple from a tree. Leaving aside whether this is a fact or not, all we can definitely say is, he concluded that the force acting on objects on the ground (say, apples) acts on the moon as well; he thus found the extraordinarily significant connection between the terrestrial and celestial which had been thought of as being independent of each other.

To go further with his subsequent observation, let $O$ be the center of the earth, and let $P$ be the position of the moon. We denote by $v$ and $a$ the speed and acceleration of $P$, respectively. The acceleration is directed towards $O$, and $a=v^{2}/R$, where $R$ is the distance between $O$ and $P$. If the orbital period of the moon is $T$, then $vT=2\pi R$, so that $a=4\pi^{2}R/T^{2}$. Next, applying Kepler’s third law to a circular motion, we have $R^{3}=cT^{2}$ $(c>0)$. Hence $a=4\pi^{2}cR^{-2}$ and $ma=4\pi^{2}mcR^{-2}$, which is the force acting on $P$ in view of the law of motion, and the law of universal gravitation follows. Conversely, based on his laws, Newton derived Kepler’s laws through far-reaching deductions.⁵⁰⁵⁰50Newton tried to reconfirm his theory of gravitation by explaining the motion of the moon observed by J. Flamsteed, but the 3-body problem for the Moon, Earth and Sun turned out be too much complicated to accomplish his goal (he planned to carry a desired result as a centerpiece in the new edition of the Principia).

The Newton’s cardinal importance in the history of cosmology lies in his understanding of physical space. According to him, Euclidean space is an “absolute space,” with which one can say that an object must be either in a state of absolute rest or moving at some absolute speed. To justify this, he assumed that the fixed stars can be a basis of “inertial frame.” His bucket experiment,⁵²⁵²52When a bucket of water hung by a long cord is twisted and released, the surface of water, initially being flat, is eventually distorted into a paraboloid-like shape by the effect of centrifugal force. This shape shows that the water is rotating as well, despite the fact that the water is at rest relative to the bucket. From this, Newton concluded that the force applied to water does not depend on the relative motion between the bucket and the water, but it results from the absolute rotation of water in the stationary absolute space (fn. ​187). elucidated in the Scholium to Book 1 of the Principia, is an attempt to support the existence of absolute motion.

Setting aside the theoretical matters, what is peculiar about the Principia is that Newton adhered to the Euclidean style, and used laboriously the traditional theory of proportions in Book V of the Elements and the results in Apollonius’ Conics even though he was well acquainted with Descartes’ work, which was, as a matter of course, more appropriate for the presentation. He testified, writing about himself in the third person, “By the help of the Analysis, Mr. ​Newton found out most of Propositions of his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of Heavens might be founded upon good Geometry. And this makes it now difficult for unskillful Men to see the Analysis by which those Propositions were found out” (Phil. ​Trans. ​R. ​Soc., 29 (1715), 173–224).

Apart from the style of the Principia, this epoch-making work (and its offspring) was “the theory of everything” in the centuries to follow as far as the classical description of the world is concerned. Indeed, Newton’s laws seemed to tap all the secrets of nature, and hence to be the last word in physics. It was in the 19th century that physical phenomena inexplicable by the Newtonian mechanics were discovered one after another. A notable example is electromagnetic phenomena which are in discord with his mechanics in the fundamental level; see Remark 17.2 ​(4). Moreover various physicochemical phenomena called for entirely new explanations as well, and eventually led to quantum physics.

Newton was a pious Unitarian. His theological thought had him say, “Space is God’s boundless uniform sensorium.”⁵³⁵³53The medieval tradition had so much effect on Newton that he was deeply involved in alchemy and regarded the universe as a cryptogram set by God. As J. M. Keynes says, he was not the first of the age of reason, but the last of the magicians (Newton the Man, 1947). This gratuitous comment to his own comprehensive theory provoked a criticism from Leibniz, who said that God does not need a sense organ to perceive objects. Moreover, on the basis of “the principle of sufficient reason” and “the principle of the identity of indiscernibles,” Leibniz claimed that space is merely relations between objects, thereby no absolute location in space, and that time is order of succession. His thought was unfolded in a series of long letters between 1715 and 1716 to a friend of Newton, S. Clarke.⁵⁴⁵⁴54Clarke (1717), A Collection of Papers, which passed between the late Learned Mr. Leibniz, and Dr. Clarke, In the Years 1715 and 1716. Having said that, however, the existence of God is an issue which Leibniz could not sidestep. To be specific, his principle of sufficient reason made him assert that nothing happens without a reason, and that all reasons are ex hypothesi God’s reasons. One may ask, for instance, “Why would God have created the universe here, rather than somewhere else?” That is, when God created the universe, He had an infinite number of choices. According to Leibniz, He would choose the best one among different possible worlds. As will be explained later (Sect. ​12), his insistence, though having a strong theological inclination, is relevant to a fundamental physical principle.

At all events, the period that begins with Newton and Leibniz corresponds to the commencement of the close relationship between mathematics and physics. From then on, both disciplines have securely influenced each other.

Immanuel Kant (1724–1804), who stimulated the birth of German idealism, was influenced by the rationalist philosophy represented by Descartes and Leibniz on the one hand, and troubled by Hume’s thoroughgoing skepticism on the other.⁵⁵⁵⁵55The empiricist David Hume is a successor of F. Bacon, T. Hobbes, J. Locke, and G. Berkeley. He says that an orderly universe does not necessarily prove the existence of God (An Enquiry Concerning Human Understanding, 1748). Hobbes, an Aristotle’s critic, described the world as mere “matter in motion,” maybe the most colorless depiction of the universe since the ancient atomists, but he did not abandon God in his cosmology (Leviathan, 1651). He felt the need to rebuild metaphysics to argue against Hume’s view. Being also dissatisfied with the state of affairs surrounding metaphysics, in contrast to the scientific model cultivated in his days, Kant strove to lay his philosophical foundation of reason and judgement on secure grounds.

He was an enthusiast of Newtonian physics framed on Euclidean geometry, and asked himself, “Are space and time real existences? Are they only determinations or relations of things as Leibniz insists?” His inquiry brought him to the conclusion that Euclidean geometry is the inevitable necessity of thought and inherent in nature because our space (as an “absolute” entity) is a “sacrosanct” framework for all and any experience. He also held that space (as a “relative” entity) is the “subjective constitution of the mind” (see Sect. ​22), He stressed further that mathematical propositions are formal descriptions of the a priori structures of space and time.⁵⁶⁵⁶56Affected by the science in his time, Kant developed further the nebular hypothesis proposed by E. Swedenborg in 1734 (this hypothesis that the solar system condensed from a cloud of rotating gas was discussed later by Laplace in 1796).

In the work Allgemeine Naturgeschichte und Theorie des Himmels (1755), Kant discussed infiniteness of the universe. He believed that the universe is infinite, because God is the “infinite being,” and creates the universe in proportion to his power (recall Bruno’s view). In turn, he argues that it is not philosophically possible to decide whether the universe is infinite or finite; that is, we are incapable of perceiving so large distance, because the mind is finite.⁵⁷⁵⁷57Hegel dismissed the claim that the universe extends infinitely.

As Kant emphasized in the Kritik der reinen Vernunft (1781), abstract speculation must be pursued without losing touch with reality grasped by intuition; otherwise the outcome could be empty. This is not least the case for scientific knowledge. Yet intuition about our space turned out to be a clinging constraint on us; it was not easy to free ourselves from it.

In Christianity, God is portrayed to be omnipresent, yet His whereabouts is unknown. Nonetheless, the laity usually personify God (if not a person literally), and believe that He “dwells” outside the universe (or empyrean). Imagining such a deity is not altogether extravagant from the view that we will take up in investigating a model of the universe though we do not posit the existence of God. What is available in place of God is, of course, mathematics.⁵⁸⁵⁸58Robert Grosseteste (ca. ​1175–1253) in medieval Oxford says, “mathematics is the most supreme of all sciences since every natural science ultimately depends on it” (De Luce, 1225). He maintained, by the way, that stars and planets in a set of nested spheres around the earth were formed by crystallization of matter after the birth of the universe in an explosion. Thus, we pose the question, “Is it mathematically possible to tell how the universe is curved without mentioning any outside of the universe?”

The answer is “Yes.” To explain why, let us take a look at smooth surfaces as 2D toy models of the universe.⁵⁹⁵⁹59Throughout, “smooth” means “infinite differentiable.” That is, we human beings are supposed to be confined in a surface. Here space in which the surface is located is thought of as the “outside,” and is called the ambient space. A person in the outside plays the role of God incarnate. In our story, we allow him to exist for the time being; that is, we investigate surfaces in an extrinsic manner. At the final stage, we are disposed to remove him (and even the outside) from the scene, and to let the inhabitants in the surface find a way to understand his universe. In other words, what we shall do is, starting from an extrinsic study of surfaces, to look for some geometric concepts with which one can talk about how surfaces are curved without the assistance of the outside. Such concepts will be called intrinsic, one of the most vital key terms in this essay.

A few words about intrinsicness. Imagine that an inhabitant confined in a surface wants to examine the structure of his universe. The only method he holds is, like us, measuring the distance between two points and the angle between two directions. In our universe, we use the light ray to this end, which in a uniform medium propagates along shortest curves (straight lines) in space; a very special case of Fermat’s principle of least time saying that the path taken between two points by a light ray is the path which can be traversed in the least time.⁶⁰⁶⁰60This principle was stated in a letter (January 1, 1662) to M. C. de la Chambre. Employing it, he deduced the law of refraction. The first discoverer of the law is Harriot (July 1601), but he died before publishing. W. Snellius rediscovered the law (1621). Descartes derived the law in La Dioptrique under a wrong setup; actually he believed that light’s speed was instantaneous. The finite propagation of light was demonstrated by O. Rømer in 1676. Hence shortest curves on a surface are regarded as something similar to paths traced by light rays (see Sect. ​12), and it is reasonable to say that a quantity (or a concept) attached to a surface is “intrinsic” if it is derived from distance and angle measured by using shortest curves.

The progenitor of the intrinsic theory of surfaces is Johann Carl Friedrich Gauss (1777–1855). In 1828, he brought out the memoir Disquisitiones generales circa superficies curvas, where he formulated bona fide intrinsic curvature, which came to be called Gaussian curvature later.

Just for the comparison purpose, let us primarily look at the curvature of plane curves. Subsequent to the pioneering studies by Kepler (Opera, vol. ​2, p. ​175) and Huygens (1653-4), Newton used his calculus to compute curvature.⁶¹⁶¹61Tractus de methodis serierum et fluxionum (1670-1671). His approach is to compare a curve with uniformly curved figures; i.e., circles or straight lines. To be exact, given a point $p$ on a smooth curve $C$, he singles out the circle (or the straight line) $C_{0}$ that has the highest possible order of contact with $C$ at $p$, and then defines the curvature of $C$ at $p$ to be the reciprocal of the radius of $C_{0}$ (when $C_{0}$ is a straight line, the curvature is defined to be $0$); thus the smaller the circle $C_{0}$ is, the more curved $C$ is. To simplify his computation, take a coordinate system $(x,y)$ such that the $x$-axis is the tangent line of $C$ at $p$, and express $C$ around $p$ as the graph associated with a smooth function $y\!=\!f(x)$ with $f(0)\!=\!f^{\prime}(0)\!=\!0$ and $f^{\prime\prime}(0)\!\geq\!0$. Compare $f(x)$ $=$ $\frac{1}{2}f^{\prime\prime}(0)x^{2}+\frac{1}{3!}f^{\prime\prime\prime}(\theta x)x^{3}$ $\leavevmode\nobreak\ (0\!<\!\theta\!<\!1)$ (Taylor’s theorem) with the function $f_{0}(x)$ $=\!R-\sqrt{R^{2}-x^{2}}\!=\!\frac{1}{2}R^{-1}x^{2}+\cdots$ corresponding to the circle $C_{0}$ of radius $R$, tangent to the $x$-axis at $x=0$. Thus $C_{0}$ has the second order of contact with $C$ at $(0,0)$ if and only if $R=1/f^{\prime\prime}(0)$. Therefore the curvature is equal to $f^{\prime\prime}(0)$.

In the 18th century, Continental mathematicians polished calculus as an effectual instrument to handle diverse problems in sciences. Leonhard Euler (1707–1783) was among those who helped to brings calculus to completion and employed it as a tool for the extrinsic geometry of surfaces. His particular interest was the curvature of the curve obtained by cutting a surface with the plane spanned by a tangent line and the normal line. This study brought him to the notion of principal curvature.⁶²⁶²62Recherches sur la courbure des surfaces, Mém. ​Acad. ​R. ​Sci. ​Berlin, 16 (1767), 119-143. Then in 1795, Gaspard Monge (1746–1818) published the book Application de l’analyse à la géométrie, an early influential work on differential geometry containing a refinement of Euler’s work. However neither Euler nor Monge arrived at intrinsic curvature. Here came Gauss, who found how to formulate it with absolute confidence in its significance.

What comes to mind when defining the curvature of a surface $S$ at $p$ is the use of a coordinate system $(x,y,z)$ with origin $p$ such that the $xy$-plane is tangent to $S$ at $p$,⁶³⁶³63Gauss’s original definition uses the map (Gauss map) from $S$ to the unit sphere which associates to each point on $S$ its oriented unit normal vector (Disquisitiones generales, Art. ​6). with which $S$ is locally expressed as a graph of a smooth function $f(x,y)$ defined around $(x,y)\!=\!(0,0)$ with $f(0,0)\!=\!f_{x}(0,0)\!=\!f_{y}(0,0)\!=\!0$, where $\displaystyle f_{x}\!=\!\partial f/\partial x$ and $\displaystyle f_{y}\!=$ $\partial f/\partial y$. Then we obtain $f(x,y)\!=\!\frac{1}{2}(ax^{2}+2bxy+cy^{2})$ $+r(x,y)$, where $a\!=\!f_{xx}(0,0)$, $b\!=\!f_{xy}(0,0)$, $c\!=\!f_{yy}(0,0)$, and $r(x,y)$ is of higher degree than the second. The shape of the graph (and hence the shape of $S$ around $p$) is roughly determined by the quantity $ac-b^{2}$; it is paraboloid-like (resp. hyperboloid-like) if $ac-b^{2}\!>\!0$ (resp. $ac-b^{2}\!<\!0$) (Fig. ​7). Given all this, we define the curvature $K_{S}(p)$ to be $ac-b^{2}$, which is independent of the choice of a coordinate system. For example, $K_{S}\!\equiv\!R^{-2}$ for the sphere of radius $R$.

To account the exact meaning of intrinsicness of the Gaussian curvature, we consider another surface $S^{\prime}$, and let $\varPhi$ be a one-to-one correspondence from $S$ to $S^{\prime}$ preserving the distance of two points and the angle between two directions (such $\varPhi$ is called an isometry in the present-day terms; see Sect. ​20). If we would adopt the approach mentioned above on intrinsicness, Gauss’s outcome could be rephrased as “$K_{S^{\prime}}(\varPhi(p))=K_{S}(p)$.” In particular, $K_{S}\equiv 0$ for a cylindrical surface $S$, since $S$, which is ostensibly curved, is deformed without distortion to a strip in the plane. This example articulately signifies that the Gaussian curvature could be possibly defined without the aid of the outside.

Executing lengthy computations, Gauss confirmed the intrinsicness of $K_{S}$ (see Sect. ​17).⁶⁴⁶⁴64Crucial in his argument is the commutativity of partial differentiation. Euler and A. C. Clairaut knew it, but the first correct proof was given by K. H. A. Schwarz in 1873. Since the goal is more than a pleasing astonishment, Gauss named his outcome “Theorema Egregium (remarkable theorem)” (Disquisitiones generales, Art. ​12). Actually, it is no exaggeration to say that this theorem is a breakthrough not only in geometry, but also in cosmology. The following is his own words (1825) unfolded in a letter to the P. A. Hansen:

This research is deeply entwined with much else, I would like to say, with metaphysics of space and I find it difficult to shake off the consequences of this, such as for instance the true metaphysics of negative or imaginary quantities (Werke, XII, p. ​8. See [3]).⁶⁵⁶⁵65Even in the 18th century, there were a few who did not accept negative number. For example, F. Maseres said, “[negative numbers] darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple” (1758). Although limited to the practical use, negative numbers already appeared in the Jiuzhang Suanshu (The Nine Chapters on the Mathematical Art) written in the period of the Han Dynasty (202 BCE–8 CE). The reason why the Chinese accepted negative numbers is, in all likelihood, that they had the concept of yin (negative or dark) and yang (positive or bright).

Although the Theorema Egregium has a deep metaphysical nature as Gauss put it, his incipient motivation came from the practical activity pertaining to the geodetic survey of Hanover to link up with the existing Danish grid that started in 1818 and continued for 8 years. He traveled across the state in order to cover the whole country. Fueled by this arduous activity, he brought out many papers on the geometry of curves and surfaces from about 1820 onwards. The Theorema Egregium is a culmination of his intensive study in this field.

It is off the subject, but while looking at the background behind Gauss’s opinion about complex numbers quoted above, we shall prepare some material related to what we shall describe later (Sect. ​21).

Gerolamo Cardano (1501–1576), who stood out above the rest at that time as an algebraicist, acknowledged the existence of complex numbers in his Ars Magna (1545) that contains the first printed solutions to cubic and quartic equations. He observed that his formula may possibly involve complex numbers even when applied to a cubic equation possessing only real solutions.⁶⁶⁶⁶66For example, his formula applied to the cubic equation $x^{3}-15x-4\leavevmode\nobreak\ \!(=(x-4)(x+2+\sqrt{3})(x+2-\sqrt{3}))=0$ yields a solution $x$ $=$ $\sqrt[3]{2+\sqrt{-124}}+\sqrt[3]{2-\sqrt{-124}}$. Yet, he did not understand the nature of complex numbers and even thought that they are useless. Subsequently, Rafael Bombelli (1526–1572) took down the rules for multiplication of complex numbers (binomio) in his Book I of L’Algebra, while Descartes used the term “imaginary number” (nombre imaginaire) with the meaning of contempt. Wallis insisted that imaginary numbers are not unuseful and absurd when properly understood by using a geometric model just like negative numbers (Algebra, Vol. II, Chap. LXVI, 1673). Meanwhile, Euler persuaded himself that there is an advantage for the use of imaginary numbers (1751), and introduced the symbol $i$ for the imaginary unit (1755). It was around 1740 that he found the earth-shaking formula $e^{\sqrt{-1}\theta}=\cos\theta+\sqrt{-1}\sin\theta$.

In 1799, Gauss gained his doctorate in absentia from the University of Helmstedt. The subject of his thesis is the fundamental theorem of algebra,⁶⁷⁶⁷67Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. However, his proof had a gap. He gave three other proofs (1816, 1849) (see Sect. ​21 for a “topological” proof). which, with a long history of inspiring many mathematicians, says that every polynomial equation with complex coefficients has at least one complex root.⁶⁸⁶⁸68Albert Girard (1595–1632) was the first to suggest the fundamental theorem of algebra (FTA) (Invention Nouvelle en l’Algèbre, 1629). Meanwhile, Leibniz claimed that $x^{4}+1=0$ affords a counterexample for the FTA (1702). He assumed that the square root of $i$ should be a more complicated “imaginary” entity, not able to be expressed as $a+bi$; but the truth is that $x=\pm\frac{1}{\sqrt{2}}(1\pm i)$ (any double sign). Euler (1742) and D’Alembert (1746) counted on the FTA when they dealt with indefinite integrals of rational functions. As the title of his thesis indicates, he shied away from the use of imaginary numbers, notwithstanding that Gauss already had a tangible image. In a letter to F. W. Bessel, dated December 18, 1811, he communicated, “first I would like to ask anyone who wishes to introduce a new function into analysis to explain whether he wishes it to be applied merely to real quantities, and regard imaginary values of the argument only as an appendage, or whether he agrees with my thesis that in the realm of quantities the imaginaries $a+b\sqrt{-1}$ have to be accorded equal rights with the reals. Here it is not a question of practical value; analysis is for me an independent science, which would suffer serious loss of beauty and completeness, and would have constantly to impose very tiresome restrictions on truths which would hold generally otherwise, if these imaginary quantities were to be neglected…” (Werke, X, p. ​366–367; see [15]).

In 1833, William Rowan Hamilton (1805-1865) conceived the idea to express a complex number $a+b\sqrt{-1}$ by the point $(a,b)$ in the coordinate plane $\mathbb{R}^{2}$,⁶⁹⁶⁹69Gauss had been in possession of the geometric representation of complex numbers since 1796. In 1797, C. Wessel presented a memoir to the Copenhagen Academy of Sciences in which he announced the same idea, but it did not attract attention. In 1806, J. R. Argand went public with the same formulation. which motivated him to construct 3D “numbers” with arithmetical operations similar to complex numbers, but the quest was a dead end. Alternatively, he discovered quaternions, the 4D numbers at the expense of commutativity of multiplication (1843). Specifically, the multiplication is defined by

It may be said, incidentally, that Gauss had already discovered quaternions in 1818 (Mutationen des Raumes, Werke, VIII, 357–361). What he observed is, though his original expression is slightly different, that the rotation group ${\rm SO}(3)$ is parameterized by $(a,b,c,d)$ with $a^{2}\!+\!b^{2}\!+\!c^{2}\!+\!d^{2}\!=\!1$ as

and $A(a_{1},b_{1},c_{1},d_{1})A(a_{2},b_{2},c_{2},d_{2})\!=\!A\big{(}(a_{1},b_{1},c_{1},d_{1})\cdot(a_{2},b_{2},c_{2},$ $d_{2})\big{)}$. Note here that quaternions $\alpha=a+bi+cj+dk$ with $\|\alpha\|=1$ form a group, which is identified with the spin group ${\rm Spin}(3)$, pertinent to the quantum version of angular momentum that describes internal degrees of freedom of electrons. Additionally, the kernel of the homomorphism $\alpha\mapsto A(\alpha)$ is a homomorphism of ${\rm Spin}(3)$ onto ${\rm SO}(3)$ whose kernel is $\{\pm 1\}$ (Remark 21.1).

Gauss is one of the discoverers of non-Euclidean geometry (a geometry consistently built under the assumption that there are more than one line through a point which do not meet a given line), but he did not publish his work and only disclosed it in letters to his friends.⁷²⁷²72For instance, his discovery was mentioned in a letter to F. A. Taurinus dated November 8, 1824 (Werke, VIII, p. ​186). His hesitation in part came from the atmosphere in those days; that is, his new geometry was entirely against the predominant Kantianism. He wanted to avoid controversial issues,⁷³⁷³73In a letter to Bessel dated January 27, 1829, Gauss writes, “I fear the cry of the Boetians [known as vulgarians] if I were to voice my views” (Werke, VIII, p. ​200). and requested his friends not to make it public.

The other discoverers of the new geometry are Nikolai Ivanovich Lobachevsky (1792–1856) and János Bolyai (1802–1860).⁷⁴⁷⁴74Strictly speaking, both Lobachevsky and Bolyai discovered the 3D geometry, while Gauss dealt only with the 2D case (Sect. ​11). In 1826, Lobachevsky stated publicly that the new geometry exists.⁷⁵⁷⁵75Lobachevsky, Exposition succincte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles;O nachalakh geometrii, Kazanskii Vestnik, (1829–1830). Fourteen years later, as this research did not draw attention, he issued a small book in German containing a summary of his work to appeal to the mathematical community,⁷⁶⁷⁶76Geometrischen Untersuchungen zur Theorie der Parallellinien (1840). and then put out his final work Pangéométrie ou précis de géométrie fondée sur une théorie générale et rigoureuse des parallèles, a year before his death (1855); see [17]. Meanwhile, in a break with tradition, Bolyai independently began his head-on approach to the non-Euclidean properties as early as 1823. In a letter to his father Farkas, on November 3, 1823, János says with confidence, “I discovered a whole new world out of nothing,” and made public his discovery in a 26-pages appendix Scientiam spatii absolute veram exhibens to his father’s book.⁷⁷⁷⁷77 Tentamen juventutem studiosam in elementa Matheseos Purae (1832).

Gauss was a perfectionist by all accounts. He did not publish many outcomes, fearing that they were never perfect enough; his motto was “pauca sed matura—few, but ripe.” After his death it was discovered that many results credited to others had been already worked out by him (remember Cauchy’s integral theorem; Remark 8.1). The extraordinariness of Gauss may be narrated by the joke: Suppose you discovered something new. If Gauss would be still alive and would browse your result, then he would say, “Ah, that’s in my paper,” and surely would take an unpublished article out of a drawer in his desk. This was no joke for János Bolyai. Gauss received a copy of his father’s book in 1832, and endorsed the discovery in the response. To János’ disappointment, his cool reply reads, “The entire contents of your son’s work coincides almost exactly with my own meditation, which has occupied my mind for from thirty to thirty-five years” (Werke, VIII, p. ​220). So much disappointed, János wholly withdrew from scientific activity, though Gauss wrote to Gerling (February 14, 1832), “I consider the young geometer Bolyai a genius of the first rank” (see [4]). As regards Lobachevsky, it was not until 1841 that his work was recognized by Gauss who looked over the Geometrischen Untersuchungen by chance. Gauss was very impressed with it, and praised Lobachevsky as a clever mathematician in a letter to his friend J. F. Encke (February 1841; Werke, VIII, p. ​232).⁷⁸⁷⁸78F. K. Schweikart, a professor of law and an uncle of Taurinus, had a germinal idea of non-Euclidean geometry (“astral geometry” in his term), and asked for Gauss, through Gerling, to comment on the idea in 1818. In response to the request (March 16, 1819), Gauss communicated, with compliments, that he concurred in Schweikart’s observation (Werke, VIII, p. ​181). However Schweikart neither elaborated his idea nor published any result.

Now, what is known about Euclid and his Elements? It is popularly thought that Euclid was attached to Plato’s Academeia or at least was affiliated with it, and that he was temporarily a member of the Great Museum during the reign of Ptolemy I. Truth to tell, very little is known about his life. Archimedes, whose life, too, overlapped the reign of Ptolemy I, mentioned briefly Euclid in his On the Sphere and the Cylinder I.2, though being a fiction made by the posterity. In the Collection VII, Pappus records that Apollonius spent a long time with the disciples of Euclid in Alexandria. Apollonius himself referred to Euclid’s achievement with the statement that his method surpasses Euclid’s Conics. Proclus Diadochus (412 CE–485 CE) and Joannes Stobabeing (the 5th century CE) recorded the oft-told (but not trustworthy) anecdotes about Euclid. The sure thing is that he is the author of the definitive mathematical treatise Elements (Στοιχει̃α),⁷⁹⁷⁹79Besides the Elements, at least five works of Euclid have survived to today; Data, On Divisions of Figures, Catoptrics, Phaenomena, Optics. There are a few other works that are attributed to Euclid but have been lost: Conics, Porisms, Pseudaria, Surface Loci, Mechanics. which is a crystallization of the plane and space geometry known in his day (ca. ​300 BCE) and was so predominant that all earlier texts were driven out. It is solely to the Elements, one of the most influential books in long history, which Euclid owes his abiding fame in spite of his shadowy profile.

The Elements, consisting of thirteen books, starts with the construction of equilateral triangles and ends up with the classification of the regular solids; tetrahedron, cube, octahedron, dodecahedron, and icosahedron (Book XIII, Prop. ​465). This suggests the influence of Pythagorean mysticism that put great emphasis on aesthetic issues, which, nowadays, are described in terms of symmetry (ςυμμετρία).⁸⁰⁸⁰80“Symmetry” is explicated by group actions, with which both spatial homogeneity and isotropy are described. Klein’s Erlangen Program is also explained in this framework. Aetius (1st- or 2nd-century CE) says that Pythagoras discovered the five regular solids (De Placitis), while Proclus argues that it was the historical Theaetetus (ca. ​417 BCE–ca. ​369 BCE) who theoretically constructed the five solids. Plato’s Timaeus (53b5–6) provides the earliest known description of these solids as a group. Nowadays, regular polyhedra are discussed often in relation to finite subgroups (polyhedral groups) of ${\rm SO}(3)$ or ${\rm O}(3)$. For example, the icosahedral group is the rotational symmetry group of the icosahedron (see Remark 21.1). Moreover, it has an orderly organization consisting of 23 fundamental definitions, 5 postulates, 5 axioms, and 465 propositions. The “reductio ad absurdum” is made full use of as a powerful gambit (Prop. ​6 in Book I is the first one proved by reductio ad absurdum).

As often said, only a few theorems are thought of having been discovered by Euclid himself. The theorem of angle-sum (Book I, Prop. ​32), the Pythagorean Theorem (Book I, Prop. ​47), and perhaps the triangle inequality (Book I, Prop. ​20) are attributed to the Pythagoreans. The theory of proportions (Book V) and the method of exhaustion (Book X, Prop. ​1, Book XII) are, as we mentioned before, greatly indebted to Eudoxus.

We shall talk about a checkered history of the Elements, originally written on fragile papyrus scrolls.⁸¹⁸¹81In the tiny fragment “Papyrus Oxyrhynchus 29” housed in the library of the University of Pennsylvania, one can see a diagram related to Prop. ​5 of Book II, which can be construed in modern terms as $ab+\big{(}\frac{a+b}{2}-b\big{)}^{2}=\big{(}\frac{a+b}{2}\big{)}^{2}$. The book survived in Alexandria and Byzantium under the control of the Roman Empire. In contrast, Roman people—the potentates of the world during this period—by and large expressed little interest in pure sciences as testified by Marcus Tullius Cicero (BCE 106–BCE 43), who confesses, “Among the Greeks, nothing was more glorious than mathematics. We, however, have limited the usability of this art to measuring and calculating” (Tusculanae Disputationes, ca. ​45 BCE). Luckily, Alexandria is far away from the central government of the Empire, disintegrating in scandal and corruption. Situated as it was and thanks to the Hellenistic tradition, pure science flourished there. The people involved were Heron (ca. ​10 CE–ca. ​70 CE), Menelaus (ca. ​70 CE–ca. ​140 CE), Ptolemy, Diophantus (ca. ​215 CE–ca. ​285 CE), Pappus, Theon Alexandricus (ca. ​335 CE–ca. ​405 CE), and Hypatia (ca. ​350 CE–415 CE).⁸²⁸²82Diophantus stands out as unique because he applied himself to some algebraic problems in the time when geometry was still in the saddle. Theon is known for editing the Elements with commentaries. His version was the only Greek text known, until a hand-written copy of the Elements in the 10th century was discovered in the Vatican library (1808). Hypatia, a daughter of Theon, is the earliest female mathematician in history, and is far better known for her awful death; she was dragged to her death by fanatical Christian mobs.

But even in the glorious city, scientific attitude declined with the lapse of time, as the academic priority had been given, if anything, to annotations on predecessor’s work, and eventually fell into a state of decadence. This was paralleled by the steady decay of the city itself that all too often suffered overwhelming natural and manmade disasters. After a long tumultuous times, the great city was finally turned over to Muslim hands (641 CE).

In the Dark Ages, Europeans could no longer understand the Elements. In this circumstance, Anicius Manlius Boethius (ca. ​480 CE–524 CE) and Flavius Magnus Aurelius Cassiodorus Senator (ca. ​485 CE–ca. ​585 CE) are very rare typical scholars in this period who were largely influential during the Middle Ages. Meanwhile, the Elements was brought from Byzantium to the Islamic world in ca. ​760 CE, and was taught in Bagdad, Cordova, and Toledo where there were large-scale libraries comparable with the Great Museum in Alexandria.⁸³⁸³83Worthy to mention is the House of Wisdom (Bayt al-Hikma) in Bagdad. It was founded by Caliph Harun al-Rashid (ca. ​763 CE–809 CE) as an academic institute devoted to translations, research, and education, and was culminated under his son Caliph al-Mamoon. Al-Khw$\overline{\rm a}$rizmī was a scholar at this institute.

After approximately 400 years of blank period in Europe, Adelard of Bath (Remark 2.1 ​(2)), who undertook a journey to Cordova around 1120 to delve into Arabic texts of Greek classics, obtained a copy of the book and translated it into Latin. Later on, Gerard (fn. ​6) translated it into Latin from another Arabic version procured in Toledo. His translation is considered being close to the Greek original text. Then, in the mid-renaissance, only 27 years later from the type printing of the Bible by Johannes Gutenberg in Mainz, the first printed edition—based on the Latin version from the Arabic by Campanus of Novara who probably had access to Adelard’s translation—was published in Venice. Thereafter, versions of the Elements in various languages had been printed. The first English-language edition was printed in 1570 by H. Billingsley. Then in 1607, Matteo Ricci and Xu Guangqi translated into Chinese the first six volumes of the Latin version published in 1574 by Christopher Clavius.

One of the first multicolored books The First Six Books of the Elements of Euclid by O. Byrne was printed in 1847 in London. A. De Morgan was very critical of its non-traditional style embellished by pictorial proofs, and judged this inventive attempt to be nonsense (A Budget of Paradoxes, 1872). At that time, there was considerable debate about how to teach geometry; say, the pros and cons of whether to adopt a new approach to Euclidean geometry in teaching. C. Dodgson in Oxford, known as Lewis Carroll, assumed a critical attitude to his colleague Playfair who tried to simplify the Euclid’s proofs by introducing algebraic notations. In the little book Euclid and his Modern Rivals printed in 1879, he argues, “no sufficient reasons have yet been shown for abandoning [Euclid’s Elements] in favour of any one of the modern Manuals which have been offered as substitutes.” G. B. Halsted says, in the translator’s introduction to J. Bolyai’s Scientiam spatii, “Even today (1895), in the vast system of examinations carried out by the British Government, by Oxford, and by Cambridge, no proof of a theorem in geometry will be accepted which infringes Euclid’s sequence of propositions.” As a matter of course, it is now seldom to make direct use of Euclid’s propositions for university examinations, but the book went through more than 2000 editions to date, and has been (and still is) the encouragement and guide of scientific thought. Furthermore, modern mathematics inherits much of its style from the Elements; in particular, many mathematical theories, even if not all of them, begin with axiomatic systems (Sect. ​18).

What about the original Greek text? It was Heiberg (fn. ​12) who tracked down all extant manuscripts—the aforementioned Greek Vatican manuscript is one of them—to produce a definitive Greek text together with its Latin translation and prolegomena (Euclidis Opera Omnia, 8 vols, 1883–1916). This is a trailblazing achievement that has become the basis of later researches on Euclid.

Euclid’s greatest contribution is his daring selection of a few postulates as major premises (if not completely adequate from today’s view), of which Greek geometers thought as self-evident truth requiring no proof. He built up a deductive system in which every theorem is derived from these postulates. Among his five postulates, the last one, called the Fifth Postulate (FP henceforward), constitutes the core of our story on non-Euclidean geometry. It says:

If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles ([7], Book I).

The theorem of angle-sum and the Pythagorean Theorem rely on this postulate. For example, what is required for the former is the fact that “if a straight line fall upon two parallel straight lines it will make the alternate angles equal to one another” (Prop. ​29 in Book I), which is a consequence of the FP.⁸⁴⁸⁴84In contrast to these two theorems, the triangle inequality is proved without invoking the FP (Book I, Prop. ​20).

The first four postulates have fairly simple forms; say, the first one declares, in today’s terms, that given two points, there exists a straight line through these points, whereas the necessity of the last postulate is by no means overt, owing not only to the intricacy of its formulation, but also to the fact that the converse of Prop. ​29 is proved without an appeal to the last postulate (Prop. ​27). Actually the last one had been believed to be redundant and hence to be a “proposition” all through modern times until the second decade of the 19th century ([2]).

An elaborate and stalwart attempt to vindicate Euclidean geometry by “proving” the FP was done by Girolamo Saccheri (1667-1733), an Italian Jesuit priest, who explored the consequences under the negation of the postulate, hoping to reach a contradiction. For this sake, he made use of quadrilaterals of which two opposite sides are equal to each other and perpendicular to the base, and set up the following three hypotheses (Fig. ​8);⁸⁵⁸⁵85Omar Khayyam (1048–1131) dealt with the FP in his Explanation of the Difficulties in the Postulates of Euclid. He is considered a predecessor of Saccheri.

(i) the hypothesis of the right angle: $\angle C=\angle D=\angle R$,

(ii) the hypothesis of the obtuse angle: $\angle C=\angle D>\angle R$, and

(iii) the hypothesis of the acute angle: $\angle C=\angle D<\angle R$,

Johann Heinrich Lambert (1728–1777)—conceivably familiar with Saccheri’s work because, in his Theorie der Parallellinien (1766), he quoted the work of G. S. Klügel who listed nearly 30 attempts to prove the FP including Saccheri’s work⁸⁷⁸⁷87Conatuum praecipuorum theoriam parallelarum demonstrandi recensio (1763).—investigated the FP in alignment with an idea resembling that of Saccheri ([2], p. ​44).⁸⁸⁸⁸88Lambert proved that the transfer time from $\boldsymbol{x}_{1}$ to $\boldsymbol{x}_{2}$ along a Keplerian orbit is independent of the orbit’s eccentricity and depends only on $\|\boldsymbol{x}_{1}\|+\|\boldsymbol{x}_{2}\|$ and $\|\boldsymbol{x}_{1}-\boldsymbol{x}_{2}\|$ (Über die Eigenschaften der Kometenbewegung, 1761). He was the first to express Newton’s second law of motion in the notation of differential calculus (Vis Viva, 1783). His work, published after the author’s death, bristles with far-sighted observations and highly speculative remarks (see [26]). One of his revelations is that, under the negation of the FP, the angular defect $\pi-(\angle A+\angle B+\angle C)$ for a triangle $\triangle ABC$ is proportional to the area of the triangle. This is inferred from the fact that if we define $m(P)$ for a polygon $P$ with $n$ sides by setting $m(P)=(n-2)\pi-(\text{the sum of inner angles of}\leavevmode\nobreak\ P)$, then $m(\cdot)$ and the area functional ${\rm Area}(\cdot)$ share the “additive property”; i.e., $m(P)=m(P_{1})+m(P_{2})$ for a polygon $P$ composed of two polygons $P_{1}$, $P_{2}$. With this observation in hand, he ratiocinated that non-Euclidean plane-geometry (if it exists) has a close resemblance to spherical geometry.

Let us briefly touch on spherical geometry, which is nearly as old as Euclidean geometry, and was developed in connection with geography and astronomy. In his Sphaerica, extant only in an Arabic translation, Menelaus introduced the notion of spherical triangle, a figure formed on a sphere by three great circular arcs intersecting pairwise in three vertices. He observed that the angle-sum of a spherical triangle is greater than $\pi$ ([28]). Greek spherical geometry gave birth to spherical trigonometry that deals with the relations between trigonometric functions of the sides and angles of spherical triangles. At a later time, it was treated in Islamic mathematics and then by Viète, Napier, and others. A culmination is the following fundamental formulas given by Euler.⁸⁹⁸⁹89Trigonometria sphaerica universa, ex primis principiis breviter et dilucide derivata, Acta Academiae Scientarum Imperialis Petropolitinae, 3 (1782), 72–86 (see [25]). The spherical cosine law (10.2) is seen in al-Batt$\overline{\rm a}$nī’s work Kit$\overline{\rm a}$b az-Zīj (fn. ​21).

where $\triangle ABC$ is a spherical triangle on the sphere of radius $R$, and $a,b,c$ are the arc-length of the edges corresponding to $A,B,C$, respectively.

Now, what Lambert paid attention to is the formula $(\angle A+\angle B+\angle C)-\pi=R^{-2}{\rm Area}(\triangle ABC)$, a refinement of Menelaus’ observation on spherical triangles discovered by Harriot (1603) and Girard (Invention nouvelle en algebra, 1629), and probably known by Regiomontanus.⁹⁰⁹⁰90The Latin name of Johannes Müller von Königsberg (1436–1476), the most important astronomer of the 15th century who may have arrived at the heliocentricism. Replacing the radius $R$ by $\sqrt{-1}R$ in this formula, we obtain $\pi-(\angle A+\angle B+\angle C)=R^{-2}{\rm Area}(\triangle ABC)$, which fits in with his outcome on the angular defect, and brought him to the speculation that an “imaginary sphere” (under a suitable justification) provides a model of “non-Euclidean plane,” but he did not pursue this idea any further.

We now go back to Gauss’s discovery of non-Euclidean geometry. In his letter to Taurinus in 1824, Gauss mentions, “The assumption that angle-sum [of triangle] is less than $180^{\circ}$ leads to a curious geometry, quite different from ours [the euclidean] but thoroughly consistent, which I have developed to my entire satisfaction. The theorems of this geometry appear to be paradoxical, and, to the uninitiated, absurd, but calm, steady reflection reveals that they contain nothing at all impossible” (see [1]).

Now, when did Gauss become aware of the new geometry? A few tantalizing pieces of evidence are in Gauss’s Mathematisches Tagebuch, a record of his discoveries from 1796 to 1814 (Werke, X).⁹²⁹²92Gauss’s diary—which contains 146 entries written down from time to time, and most of which consist of brief and somewhat cryptical statements—was kept by his bereaved until 1899. The 80th entry (October, 1797) announces the discovery of the proof of the FTA. The 72nd item written on July 28, 1797 says, “Plani possibilitatem demostravi” [“I have demonstrated the possibility of a plane”]. This seems to indicate Gauss’s interest in the foundation of Euclidean geometry, of which he never lost sight from that time onward. Following that, in the 99th entry dated September 1799, he wrote, “In principiis Geometriae egregios progressus fecimus” [“On the foundation of geometry, I could make a remarkable progress”]. It is difficult to imagine what this oblique statement means, but it could have something to do with non-Euclidean geometry.⁹³⁹³93In a letter to F. Bolyai, March 6, 1832 (Werke VIII, p. ​224), Gauss alluded that Kant was wrong in asserting that space is only the form of our intuition. In the same year, Gauss confided to F. Bolyai to the effect that he had doubt about the truth of geometry.⁹⁴⁹⁴94In the letter, he also said, “If one could prove the existence of a triangle of arbitrarily great area, all Euclidean geometry would be validated” (see [4]). However, his letter to F. Bolyai in 1804 witnessed that he was still suspicious of the new geometry (Werke, VIII, p. ​160). All of the available witnesses point to the possibility that it is in 1817 at the latest that Gauss disengaged himself from his preconceptions, and began to deal seriously with the new geometry. In a letter to his friend H. Olbers (April 28, 1817), he elusively says, “I am becoming more and more convinced that the necessity of our geometry cannot be proved” (Werke, VIII, p. ​177).

In all likelihood, Gauss had a project to bring out an exposition of the “Nicht-Euklidische Geometrie”. His unpublished notes about the new theory of parallels might be part of this exposition (Werke, VIII, p. ​202–209). A letter addressed to Schumacher on May 17, 1831 says, “In the last few weeks, I have begun to write up a few of my own meditations, which in part are already about 40 years old.”⁹⁵⁹⁵95In a letter to Schumacher of July 12, 1831, Gauss said that the circumference of a non-Euclidean circle of radius $r$ is $\pi k(e^{r/k}-e^{-r/k})$ where $k$ is a positive constant (Werke, VIII, p. ​215), which turns out to coincide with the radius $R$ of the imaginary sphere. His project was, however, suspended upon the receipt of a copy of J. Bolyai’s work. Accordingly, the detailed study of the new geometry was put in the hands of Bolyai and Lobachevsky, who established, under the negation of the FP, an analogue of trigonometry, with which they could be confident of the consistency of the new geometry ([17] and [24]).

The fundamental formulas in non-Euclidean trigonometry are given by

These are obtained from the trigonometric formulas established by Bolyai and Lobachevsky.⁹⁶⁹⁶96The non-Euclidean formulas were found by Taurinus (Geometriae Prima Elementa, 1826), but the outcome did not wholly convince him of the possibility of a new geometry. Notice here that, thanks to Euler’s formula, we get (11.1) and (11.2) by transforming $R$ into $\sqrt{-1}R$ in (10.1) and (10.2), thereby justifying Lambert’s anticipation of a new geometry on the imaginary sphere.

We should underline here that non-Euclidean geometry involves a positive parameter $R$ just like spherical geometry involving the radius of a sphere. As Lambert took notice of it, this implies that, once $R$ is selected, the absolute unit of length is determined.⁹⁷⁹⁷97In the ordinary geometry, only a relative meaning is attached to the choice of a particular unit in the measurement. The concept of “similarity” arises exactly from this fact. Actually, the non-existence of an absolute unit of length is equivalent to the FP. Failing to notice this fact, Legendre believed that he succeeded in proving the FP.⁹⁸⁹⁸98Léflexions sur différentes manières de démontrer la théorie des paralléles ou le théorème sur la somme des trois angles du triangle, 1833. In his futile attempt to prove the FP, Legendre rediscovered Saccheri’s result. The error in his “proof” was pointed out by Gauss in a letter to Gerling (1816; Werke, VIII, p.168, 175). (By 1808, Gauss realized that the new geometry must have an absolute unit of length; Werke VIII, p.165.)

Thus the issue of the FP, which had perplexed many able scholars for more than 2000 years, was almost settled down (the reason why we say “almost” will become clear later). More than that, mathematicians grew to realize that postulates in general are not statements which are regarded as being established, accepted, or self-evidently true, but a sort of predefined rules.

On reflection, Euclid’s choice of the FP shows his keen insight and deserves great veneration. Thomas Little Heath (1861–1940), a leading authority of Greek mathematics, remarked that “when we consider the countless successive attempts made through more than twenty centuries to prove the Postulate, many of them by geometers of ability, we cannot but admire the genius of the man who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable” ([7], p. ​202).

In Sect. ​7, we stressed the role of shortest curves in the intrinsic measurement in a surface. In order to outline another work of Gauss related to non-Euclidean geometry, we shall take a look at such curves from a general point of view.

The length of a curve $c(t)=(x(t),y(t),z(t))$, $a\leq t\leq b$, on a surface $S$ is given by $\ell(c)=\int_{a}^{b}\sqrt{\dot{x}(t)^{2}+\dot{y}(t)^{2}+\dot{z}(t)^{2}}\leavevmode\nobreak\ \!dt$. Thus our issue boils down to finding a curve $c$ that minimizes $\ell(c)$ among all curves on $S$ satisfying the boundary condition $c(a)=p$, $c(b)=q$. Johann Bernoulli raised the problem of the shortest distance between two points on a convex surface (1697). Following his senescent teacher Johann, Euler obtained a differential equation for shortest curves on a surface given by an equation of the form $F(x,y,z)=0$ (1732).

Finding shortest curves is a quintessential problem in the calculus of variations, a central plank of mathematical analysis originated by Euler, which deals with maximizing or minimizing a general functional (i.e., a function whose “variable” is functions or maps). A bud of this newly-born field is glimpsed in Fermat’s principle of least time, but the derivation of Snell’s law is a straightforward application of the usual extreme value problem. The starting point in effect was the brachistochrone curve problem raised by Johann Bernoulli (1696);⁹⁹⁹⁹99“Brachistochrone” is from the Greek words βράχιςτος (shortest) and χρ´ονος (time). The first to consider the brachistochrone problem is Galileo (Two New Sciences). the problem of finding the shape of the curve down which a sliding particle, starting at rest from $P_{1}=(x_{1},y_{1})$ and accelerated by gravity, will slip (without friction) to a given point $P_{2}=(x_{2},y_{2})$ in the least time. To be exact, the problem is to find a function $y=y(x)$ which minimizes the integral $\int_{x_{1}}^{x_{2}}\sqrt{\frac{1+y^{\prime}{}^{2}}{2gy}}\leavevmode\nobreak\ \!dx$.

In the paper Elementa Calculi Variationum read to the Berlin Academy on September 16, 1756, Euler unified various variational problems. At about the same time, Joseph-Louis Lagrange (1736–1813) simplified Euler’s earlier analysis and derived what we now call the Euler-Lagrange equation (E-L equation). Especially, his letter to Euler dated August 12, 1755 (when he was only 19 years old!) contains the technique used today. Here in general (however restricted to the case of one-variable), the E-L equation associated with the functional of the form

is given by

What Lagrange proved is that $\boldsymbol{q}(t)$ minimizing $\mathcal{F}$ is a solution of (12.2).

Applying the E-L equation to length-minimizing curves on a surface, we obtain an ordinary differential equation for $x(t),y(t),z(t)$. What should be aware of is that a solution of this equation is only a locally length-minimizing curve; not necessarily shortest. Such a curve is called a geodesic, the term (coming from geodesy) coined by Joseph Liouville (1809–1882).¹⁰⁰¹⁰⁰100De la ligne géodésique sur un ellipsoide quelconque, J. ​Math. ​Pures Appl., 9 (1844), 401–408. Strictly speaking, as a parameter of a geodesic, we adopt the one proportional to arc-length. We should note that $\ell(c)$ is left invariant under a parameter change.

A geodesic may be defined as a trajectory of force-free motion. To explain this, we regard a curve $c(t)$ in a surface $S$ as the trajectory of a particle in motion. The velocity vector $\dot{c}(t)$ is a vector tangent to $S$ at $c(t)$, but the acceleration vector $\ddot{c}(t)$ is not necessarily tangent to $S$. If the constraint force confining the particle to $S$ is the only force, then $\ddot{c}(t)$ must be perpendicular to the tangent plane of $S$ at $c(t)$ because the constraint force always acts vertically on $S$. Such a curve $c$ turns out to be a geodesic. Hence, if we denote by $\frac{D}{dt}\frac{dc}{dt}$ the tangential part of $\ddot{c}$, then $c$ is a geodesic if and only if $\frac{D}{dt}\frac{dc}{dt}=0$. Since this is a second-order ordinary differential equation (see (17.5) and Sect. ​20), we have a unique geodesic $c$ defined on an interval containing $0$ such that $c(0)=p$, $\dot{c}(0)=\boldsymbol{\xi}$ for given a point $p$ and a vector $\boldsymbol{\xi}$ tangent to $S$ at $p$.

There is an alternative derivation of the equation $\frac{D}{dt}\frac{dc}{dt}=0$, based on the principle of least action that was enunciated by Pierre Louis Moreau de Maupertuis (1698–1759) and occupies its position in the physicist’s pantheon,¹⁰¹¹⁰¹101Accord de différentes loix de la nature qui avoient jusqu’ici paru incompatibles, read to the French Academy on April 15, 1744. His paper Loi du repos des corps, Histoire Acad. ​R. ​Sci. ​Paris (1740), 170–176, is its forerunner. according to which a particle travels by the action-minimizing path between two points. The action he defined is the “sum of $mvs$” along the entire trajectory of a particle with mass $m$, where $s$ is the length of a tiny segment on which the velocity of the particle is $v$.¹⁰²¹⁰²102On reflection, the principle of least action seems to tell us that present events are dependent on later events in a certain manner; so, though a bit exaggerated, a natural occurrence seems to be founded on an intention directed to a certain end. For such a teleological and purposive flavor (dating back to Aristotle), Maupertuis thought that his cardinal principle proves a rational order in nature, and buttresses the existence of a rational God.

The obscurity of Maupertuis’ formulation was removed by Euler (1744)and Lagrange (1760). Specifically the action integral $\mathcal{S}$ applied to a particle in motion under a given potential energy $V$ (the concept conceived by Lagrange in 1773) is​: $\mathcal{S}\!=\!\int_{a}^{b}\Big{(}\frac{1}{2}m\|\dot{c}(t)\|^{2}-u(c(t))\Big{)}dt$. The E-L equation for $\mathcal{S}$ reduces to the Newtonian equation​: $\ddot{c}\!=\!-{\rm grad}\leavevmode\nobreak\ \!u$. (For example, the gravitational potential energy generated by the mass distribution $\mu$ is $u(\boldsymbol{x})\!=\!-G\int\|\boldsymbol{x}-\boldsymbol{y}\|^{-1}$ $d\mu(\boldsymbol{y})$.) This suggests us to define the action integral for a curve $c$ in $S$ by $E(c)\!=\!\int_{a}^{b}\|\dot{c}(t)\|^{2}\leavevmode\nobreak\ \!dt$. Indeed, the E-L equation for this functional coincide with $\frac{D}{dt}\frac{dc}{dt}=0$.

The minimum (maximum) problems come up in various scenes in geometry. For instance, the (classical) isoperimetric theorem whose origin goes back to ancient Greece states, “Among all planar regions with a given perimeter, the circle encloses the greatest area.” Needless to say, minimal surfaces are solutions of a minimum problem. In particular, finding a surface of the smallest area with a fixed bounday curve is known as Plateau’s problem, the problem raised by Lagrange in 1760 and investigated in depth in the 20th century onward.¹⁰³¹⁰³103 Plateau conducted experiments with soap films to study their configurations (1849). Minimum/maximum principles hold emphatic importance even in different fields; say, in potential theory (Remark 15.1 (1)), in statistical mechanics, in information science, in the design of crystal structures ([38]), and much else besides. Even more surprisingly, so too does in the characterization of the “shape” of the universe as will be explained later.