From τὰ φυσικά (ta physika) to physics – XIX

Although Islamic scholars made substantial contributions to mechanics, astronomy, and especially optics along the road from the Greek ta physika to modern physics, it was in the realm of mathematics that they made what was probably their greatest contribution to the development of that discipline. 

Greek science was to a great extent dominated by geometry, first and foremost the work of Euclid but also that of Apollonius and Archimedes. This continued to be the case during the Middle Ages and Greek geometry also loomed large in Islamic scientific culture. However, one characteristic of the new science developed in the seventeenth century in Europe was the rejection of the synthetic mathematics of Euclidian geometry for the newly emerging analytical mathematics that would become known as calculus. The roots of this change are to be found in the new streams of mathematics inherited from Islamic sources.

Islamic mathematicians developed three new streams of mathematics, arithmetic, algebra, and trigonometry all three of which they had in turn acquired from their predecessors in India. Of course, all three streams existed in one form or another in Ancient Greece but what the Islamic scholars acquired from India was of an entirely different calibre to what had gone before in Ancient Greece.

The arithmetic that Islamic science acquired from India was, of course, the place value decimal number system of which the eighteenth-century French mathematician, physicist, astronomer Pierre-Simon Laplace once wrote:

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

The place value decimal number system evolved over a period of several centuries finally reaching a semi-complete form with the introduction of zero as a number in the Brāhmasphuṭasiddhānta of the mathematician and astronomer Brahmagupta (c. 598–c. 668 CE).

Brahmagupta?

In this work Brahmagupta presents the place value decimal number system including positive, negative numbers and zero, the rules of the four fundamental operations (addition, subtraction, multiplication, and division) in a form that would be at home in a modern elementary arithmetic textbook. The one exception being his attempt to define division by zero, which as we all know ids a no,no. 

Verse from chapter XVIII of the Brāhmasphuṭasiddhānta describing the rules for zero as a number

Brahmagupta’s texts were translated into Arabic in about 750 by Abū ʿAbdallāh Muḥammad bin Ibrāhīm bin Ḥabīb al-Fazārī (died early ninth century) together with Yaʿqūb ibn Ṭāriq  (died c. 796) as ‘Az-Zīj ‛alā Sinī al-‛Arab or the Sindhind. 

The earliest Arabic text on the Hindu numerals was written by Muḥammad ibn Musá al-Khwārizmī (c. 780–c. 850) The kitab al-jam’ wa’l-tafriq al-ḥisāb al-hindī (Addition and subtraction According to the Hindu Calculation) probably written about 800 CE. It didn’t survive in Arabic but there is a Latin translation made in the twelfth century.

First page of the Latin translation Source: Wikimedia Commons

 Abū Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (c. 801–873) wrote his kitāb fī isti’māl al-‘adād al-hindī (On the Use of the Hindu Numerals) around the same time, which also didn’t survive.

Al-Kindi on an Iraqi stamp from 1962 Source: Wilimedia Commons

The earliest surviving works are Kitāb al-uṣūl fī l-ḥisāb al-hindī (“Book of the Principles of Hindu reckoning”) by Abul-Hasan Kūshyār ibn Labbān (971–1029), and Kitāb al‑takmila fī l-ḥisāb (“The completion of arithmetic”) by Ibn Ṭāhir al-Baghdādī (d. 1037). The Kitāb al-fuṣūl fī l-ḥisāb al-hindī (“The book of chapters on Hindu arithmetic”) by Abū l‑Ḥasan al-Uqlīdisī (fl. c. 950) is the first text to describe decimal fractions, which the Indian mathematicians had not developed. 

However, Islamic scholars used a variety of number systems. They used the place value decimal number system written with number symbols but also written with Arabic letters as in an alpha-numerical number system. Beyond that they used a pure sexagesimal system, inherited from the Babylonians. They also followed Ptolemaeus with a so-called astronomical number system that used a decimal system for the whole numbers combined with sexagesimal fractions for the fraction part. One area in which the place value decimal number system was widely used was in what we would now term commercial arithmetic. Special applications that drifted towards algebra were the determination of profit or loss shares in trade deals and in the determination of  inheritance shares under the complex Islamic inheritance rules.

Algebra and arithmetic are closely linked and this was very much the case in the medieval Islamic adoption and development of algebra. In its origins algebra was restricted to what we would now term the theory of equations. We find aspects of this in virtually all pre-Islamic mathematical cultures, Egyptian, Babylonia, China, Indian and Greece. Whereas the first four all practiced a largely arithmetical approach to the solution of equations, the Greeks developed a geometrical algebra for such solutions. We still  retain elements of this when we talk about quadratic and cubic equations; for the Ancient Greek mathematicians such equations describing geometrical figures. 

The early Islamic mathematicians borrowed heavily from all of the earlier sources. Once again very influential was the Brāhmasphuṭasiddhānta of Brahmagupta. J. L. Berggren attributes the  creation of algebra to al-Khwārizmī:

Out of this dual heritage of solutions to problems asking for the discovery of numerical and geometrical unknowns Islamic civilisation created and named a science–algebra.^[1]

A Soviet postage stamp issued 6 September 1983, commemorating al-Khwārizmī’s (approximate) 1200th birthday Source: Wikimedia Commons

It is well-known that the term algebra is derived from the title of al- Khwārizmī’s book al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah (The Compendious Book on Calculation by Completion and Balancing), whereal-Jabr means “setting back in its place” or “restoration.” Al- Khwārizmī “uses the term to denote the operation of restoring a quantity subtracted from one side of the equation to the other side to make it positive.”^[2] The Latinised version of his name also provided us with the term algorithm. Although, Algorisme was originally the term for calculating with the Hindu-Arabic numer system. 

A page from al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah Source: Wikimedia Commons

Although al- Khwārizmī is the best known Islamic algebra author he is by no means the only one. The Mesopotamian polymath Thābit ibn Qurra (c. 830–901) gave a more general demonstration of the solution of quadratic equations than al- Khwārizmī. 

The prominent Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (c. 850–c. 930) was known as Al-ḥāsib al-miṣrī (The Egyptian Calculator). His most influential work was his Kitāb fī al-jabr wa al-muqābala (Book of Algebra), which superseded and expanded on al- Khwārizmī work. 

He wrote about al_Khwārizmī:

I have studied with great attention the writings of the mathematicians, examined their assertions, and scrutinized what they explain in their works; I thus observed that the book by Muḥammad ibn Mūsā al-Khwārizmī known as Algebra is superior in the accuracy of its principle and the exactness of its argumentation. It thus behooves us, the community of mathematicians, to recognize his priority and to admit his knowledge and his superiority, as in writing his book on algebra he was an initiator and the discoverer of its principles, …(Wikipedia)

Kitāb fī al-jabr wa al-muqābala 

The first chapter teaches algebra by solving problems of application to geometry, often involving an unknown variable and square roots. The second chapter deals with the six types of problems found in Al-Khwarizmi’s book, but some of which, especially those of x², were now worked out directly instead of first solving for x and accompanied with geometrical illustrations and proofs. The third chapter contains examples of quadratic irrationals as solutions and coefficients. The fourth chapter shows how these irrationalities are used to solve problems involving polygons. The rest of the book contains solutions for sets of indeterminate equations, problems of application in realistic situations, and problems involving unrealistic situations intended forrecreational mathematics. (Wikipedia)

Like that of al- Khwārizmī, Abū Kāmil’s work would filter through to Europe in the later Middle Ages, as did the work of the Persian mathematician and engineer Abū Bakr Muḥammad ibn al Ḥasan al-Karajī (c. 935–c. 1029). His three principal surviving works are mathematical: Al-Badi’ fi’l-hisab (Wonderful on calculation), Al-Fakhri fi’l-jabr wa’l-muqabala (Glorious on algebra), and Al-Kafi fi’l-hisab (Sufficient on calculation). Whereas the work of al- Khwārizmī and , Abū Kāmil were still anchored in the algebraic geometry of the Greeks, al-Karajī went as long way to making it a numerical discipline. 

Although Brahmagupta had dealt with negative numbers and the rules for calculating with negative quantities, they were largely ignored  by the early Islamic algebraists. Al-Samawʾal ibn Yaḥyā al-Maghribī (c. 1130–c. 1180), who was born in Baghdad into a Jewish family of North African origin he converted to Islam, introduced the rule of signs in his al-Bahir fi’l-jabr, (The brilliant in algebra), written when he was nineteen years old. He also dealt with the law of exponents and polynomial division. 

Binomial coefficients from Al-Samawal al-Maghribi al-Bahir fi’l-jabr,

Our final Islamic algebraist is Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (1048–1131) better known in English as Omar Khayyam (‘Umar al-Khayyāmī) as a poet but he was polymath, who did important work in mathematics and astronomy. His most important development in algebra was a geometrical general theory of cubic equations. 

‘Umar al-Khayyāmī) “Cubic equation and intersection of conic sections” the first page of a two-chaptered manuscript kept in Tehran University. Source: Wikimedua Commons

The third major innovative area of Islamic mathematics was trigonometry. Trigonometry had its origins in Greek astronomy, with Hipparchus (c. 190–c. 120 BCE) providing a table of chords of a circle to designate the size of angles.

Being astronomy, the application is, of course only to spherical triangles. His table did not survive but Ptolemaeus took it over in his Mathēmatikē Syntaxis know in Arabic as the Almagest. 

When the Indians took over many aspects of Ancient Greek astronomy they also acquired the cord measure of angles, which they halved to create the sine, a table of sines is presented in the Surya Siddhanta from the 4^thor fifth centuries.

English translation of the Surya Siddhanta by Rev. Ebenezer Burgess 1935 Source

This work also defines the cosine, versine and inverse sine. Early Islamic astronomers acquired their astronomy from both Ancient Greece and India but went on to use the Indian sine rather than the Greek cord measure for angles. 

The tangent function was known to various ancient cultures, outside of astronomy, as a means for determining the hight of structures. Because the shadow of a tall object creates a right angle triangle from which the tangent and cotangent can be used to determine the height of the object, the tangent became known as the shadow function. 

Once again the Persian mathematician al- Khwārizmī was a pioneer in this branch of mathematics producing sine, cosine, and tangent tables. Another Persian astronomer, mathematician and geographer, Habash al-Hasib al-Marwazi (766 – d. after 869) described and produced tables of the tangent and cotangent. 

The Syrian astronomer Abū ʿAbd Allāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī aṣ-Ṣābiʾ al-Battānī (before 858–929) defined and produced tables for the secant and cosecant. He was also the first to apply the trigonometric functions to plane triangles. In general, Islamic mathematicians introduced the use of trigonometrical functions into surveying and cartography. 

al-Battānī Source: Wikimedia Commons

Abū al-Wafāʾ Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī (940–998), born in Khorasan (in today’s Iran), was the first to present all six of the trigonometrical functions in his Kitab al‐Majisṭī . 

 Page of the manuscript of Kitab al-majisti by Abu al-Wafa. (Source)

Abū ʿAbd Allāh Muḥammad ibn Muʿādh al-Jayyānī (989–1079), an Arabic mathematician from al-Andalus produced his Kitab madschhulat qisiyy al-kura (The book of unknown arcs of the sphere) a treatise on spherical trigonometry. Al-Jayyānī’s work on spherical trigonometry contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.

A page from Al-Jayyānī’s work on spherical trigonometry

The Persian polymath, Nasīr al-Dīn al-Tusī (1201–1274), in his work asch-Schakl al-Qattāʿ (Treatise on the Quadrilateral) was the first to handle trigonometry as a mathematical disciple independent of astronomy. He dealt both with the cordal trigonometry of the Greeks as well as the six modern functions, introducing the law of tangents for spherical triangles and providing proofs of it and the law of sines. 

Throughout the medieval Islamic period from al- Khwārizmī in the eighth century to Ulugh Beg in the fifteenth, Islamic astronomers and mathematicians continually worked on developing new mathematical methods to calculate ever more accurate tables of trigonometrical functions. In general, they took the simple method developed by Hipparchus to determine the size of angles in astronomy and over many generations developed an entire branch of mathematics, which would continue to increase in importance after re-entering Europe.

It is difficult to overemphasise to contributions that Islamic mathematicians made in various areas of mathematics that they had inherited from their predecessors, developments that would play a significant role in the general development of science and physics in particular in later centuries.

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^[1] J. L. Berggren, Episodes in the Mathematics of Medieval Islam, Springer, New York, 2003, p. 102 

^[2] Berggren, p. 102

From τὰ φυσικά (ta physika) to physics – XIII

Just as the period of dominance of Aristotelian philosophy in antiquity was succeeded by the rise to dominance of Stoicism and Epicureanism, as I documented in the fifth episode of the series, so they too began to lose their hold on the world of thought in late antiquity. From the middle of the third till the middle of the seventh century CE thought in the ancient world was dominated by Neoplatonism. The term Neoplatonism is a neologism created to describe a renaissance of nominally Platonic thought that took place in this period. The term itself is to some extent misleading, whereas the terms Stoic, Peripatetic or Platonic signify a single school founder by a single philosopher with a set of doctrines developed by that founder, Neoplatonism doesn’t.  To quote the Stanford Encyclopedia of Philosophy:

Late antique philosophers now counted among “the Neoplatonists” did not think of themselves as engaged in some sort of effort specifically to revive the spirit and the letter of Plato’s dialogues. To be sure, they did call themselves “Platonists” and held Plato’s views, which they understood as a positive system of philosophical doctrine, in higher esteem than the tenets of the pre-Socratics, Aristotle, or any other subsequent thinker. However, and more importantly, their signature project is more accurately described as a grand synthesis of an intellectual heritage that was by then exceedingly rich and profound. In effect, they absorbed, appropriated, and creatively harmonized almost the entire Hellenic tradition of philosophy, religion, and even literature—with the exceptions of Epicureanism, which they roundly rejected, and the thoroughgoing corporealism of the Stoics. The result of this effort was a grandiose and powerfully persuasive system of thought that reflected upon a millennium of intellectual culture and brought the scientific and moral theories of Plato, Aristotle, and the ethics of the Stoics into fruitful dialogue with literature, myth, and religious practice. In virtue of their inherent respect for the writings of many of their predecessors, the Neoplatonists together offered a kind of meta-discourse and reflection on the sum-total of ideas produced over centuries of sustained inquiry into the human condition.

Plotinus (c. 204/5–270 CE) is regarded as the first of the Neoplatonists. Central to his philosophy and in fact to all of the Neoplatonists is monism expressed through the concepts of the One and Henosis.

Head in white marble. Ostia Antica, Museo, inv. 436. Neck broken through diagonally, head broken into two halves and reconstructed. Lower half of nose is missing. One of four replicas which were all discovered in Ostia. The identification as Plotinus is plausible but not proven. Source: Wikimedia Commons

Plotinus taught that there is a supreme, totally transcendent “One”, containing no division, multiplicity, or distinction; beyond all categories of being and non-being. His “One” “cannot be any existing thing”, nor is it merely the sum of all things (compare the Stoic doctrine of disbelief in non-material existence), but “is prior to all existents”. Plotinus identified his “One” with the concept of ‘Good’ and the principle of ‘Beauty’. (Wikipedia)

Henosis is the word for mystical “oneness”, “union”, or “unity” in classical Greek. In Platonism, and especially Neoplatonism, the goal of henosis is union with what is fundamental in reality: the One the Source, or Monad. (Wikipedia)

Plotinus was succeeded by his pupil Porphyry of Tyre (c. 234–c. 305 CE),

Porphire Sophiste, in a French 16th-century engraving Source: Wikimedia Commons

who was in turn succeeded by his pupil Iamblichus (c. 245–c. 325 CE).

Source: Wikimedia Commons

Both Theon of Alexandria (c. 335–c. 405 CE) and his daughter Hypatia (c. 360–c. 415 CE) were Neoplatonists but their philosophy differed from that of the acolytes of Iamblichus, which dominated Neoplatonic thought in Alexandria during their time. 

The Neoplatonic philosopher-mathematicians produced commentaries on and annotated editions of the major Greek mathematical works. Theon was a textbook editor, who produced annotated edition of Euclid’s Elements, Euclid’s Data, his Optics and Ptolemaios’ Mathēmatikē Syntaxis. Theon’s edition of the Elements was, until the nineteenth century, the only surviving edition.

Theon of Alexandria is best known for having edited the existing text of Euclid’s Elements, shown here in a ninth-century manuscript Vatican Library via Wikimedia Commons

We have no surviving works by Hypatia but the Suda, a tenth-century Byzantine encyclopaedia of the ancient Mediterranean world lists three mathematical works for her, which it states have all been lost. The Suda credits her with commentaries on the Conic Sections of the third-century BCE Apollonius of Perga, the “Astronomical Table” and the Arithemica of the second- and third-century CE Diophantus of Alexandria. Alan Cameron, however, argues convincingly that she in fact edited the surviving text of Ptolemaeus’ Handy Tables, (the second item on the Suda list) normally attributed to her father Theon as well as a large part of the text of the Almagest her father used for his commentary.  Only six of the thirteen books of Apollonius’ Conic Sections exist in Greek; historians argue that the additional four books that exist in Arabic are from Hypatia, a plausible assumption. So once again, what we have is that Hypatia was like her father a textbook editor.

Proclus Lycius (412–185) wrote a commentary on Euclid’s Elements. According to Thomas Heath in volume one of his edition of The Thirteen Books of Euclid’s Elements:

It is well known that the commentary of Proclus on Eucl. Book I is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus.

First Latin edition of one of the major works by Proclus Lycaeus (412-485), founder and head of the neo-Platonic school of Athens: a commentary on the first book of Euclid’s “Elements of Geometry”, Source: Wikimedia Commons

Pappus of Alexandria (fl. 320) produced an encyclopaedic compendium of ancient Greek geometry, astronomy , and mechanics in eight books entitled, Synagoge (Συναγωγή) or Collection. This work, whilst highly important as a record of the history of Greek mathematics, remained virtually unknown until the sixteenth century when it was translated and published by Federico Commandino (1509–1575) in 1588. It became influential in the seventeeth century. The Suda credits him with a commentary on the first four books of Ptolemaios’ Mathēmatikē Syntaxis, now lost. He also wrote commentaries on Euclid’s Elements fragments of which are preserved in Proclus and on Ptolemaios’ Ἁρμονικά (Harmonika), now lost.

Title page of Pappus’s Mathematicae Collectiones, translated into Latin by Federico Commandino (1588). Source: Wikimedia Commons

Apart from small odds and ends, such as Pappus’ hexagon theorem in projective geometry, these Neoplatonic philosopher-mathematicians produced very little original work. However, their role in recording and conserving Greek mathematical works should not be underestimated.

The non-mathematical Neoplatonic philosophers also contributed almost nothing new to the roots of the discipline of physics that I have sketched in the previous episodes of this series but their obsessively inclusive, eclectic agglomeration of the works of earlier Greek philosophers, in particular Plato and Aristotle, meant that these works that had slid into the background during the dominance of Stoicism and Epicureanism was once again brought into the foreground and passed on down to future generations. 

All three of the monotheistic religions, Judaism, Christianity, and Islam took a strong interest in Neoplatonism because of its strongly monist core and often became first acquainted with the works of Plato, Aristotle, and other earlier Greek philosophers through Neoplatonic sources rather than through the originals. In the history of science transmission of sources often takes indirect roots.

Above I said that Neoplatonic philosophers also contributed almost nothing new to the roots of the discipline of physics, however, there is one very notable exception, the sixth century Christian, Neoplatonist John Philoponus (c. 490–c. 570) of Alexandria. Philoponus was a pupil and sometime amanuensis of the Neoplatonist philosopher Ammonius Hermiae (C. 440­–c. 520),who was also from Alexandria but had studied in Athens under Proclus before returning to Alexandria to teach. He lectured on Plato, Aristotle and Porphyry of Tyre, as well as on astronomy and geometry. As is often the case most of his supposed numerous writings have not survived. He is known to have lectured and written extensively over Aristotle as did Philoponus his pupil. However, whereas Ammonius seems to have been positive in his assessments of Aristotle, Philoponus was highly critical. 

Amongst his voluminous writings Philoponus wrote extensive critiques of almost all of Aristotle’s texts of which in our context a couple are of great importance. As a Christian Philoponus rejected Aristotle’s concept of an eternal cosmos, replacing it with a cosmos created by God in its entirety in one moment. Because his cosmos was a single unified whole he rejected Aristotle’s division of the cosmos into supralunar and sublunar regions. The cosmos was overall the same and subject to the same laws. In this he was following the Stoics, and his philosophy is heavily influenced by Stoic concepts. Philoponus also anticipates Descartes in stating that bodies have extension in space.

Most important in the history of physics Philoponus rejects both Aristotle’s concept of fall and his concept of projectile motion. It seems that, unlike Galileo, Philoponus really did drop objects of differing weight from a tower and concluded that they fall almost at the same speed:

“if one lets fall simultaneously from the same height two bodies differing greatly in weight, one will find that the ratio of their times of motion does not correspond does not correspond to the ration of their weights, but that the difference in time is a very small one” (In Physica, 683, 17).^[1]

He dismisses Aristotle’s theory of projectile motion and produces what would later become known as the theory of impetus an important precursor to the theory of inertia.

“some incorporeal kinetic power is imparted by the thrower to the object thrown “and that” if an arrow or a stone is projected by force in a void, the same things will happen much more easily, nothing being necessary except the thrower” (ibid, 641, 29).

Denying Aristotle’s distinction between sublunar and supralunar motion, Philoponus also applied his impetus concept to the motion of the planets.

Because of his deviant religious views on the nature of the Trinity, Philoponus was declared anathema at the Third Council of Constantinople, which limited the reception of his anti-Aristotelian dynamics in late antiquity, but his works were translated into Syriac and Arabic where they would have a significant influence as we shall see in future episodes.

Philoponus was the first philosopher to go beyond the dynamics of Aristotle and his concepts are the beginnings of the path that would eventually lead to the modern theories of that branch of physics.

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^[1] In Physica, H. Vitelli, ed. (Berlin, 1887)

Christmas Trilogy 2023 Part 1: Whose shoulders?

There are a series of legendary quotes that are associated with Isaac Newton and one of the most oft repeated is his, If I have seen further it is by standing on the shoulders of Giants. This leads people to inquire, who perchance those giants were and almost invariable with reference to his Philosophiæ Naturalis Principia Mathematica (1687) they state with authority, Galileo Galilei. Apart from the fact that it simply isn’t true there is another basic problem with this attribution, Newton’s statement has absolutely nothing to do with his Principia and was in fact written eight years before the Principia was published.

Source: Wikimedia Commons

When Newton published his first ever paper, A Serie’s of Quere’s Propounded by Mr. Isaac Newton, to be Determin’d by Experiments, Positively and Directly Concluding His New Theory of Light and Colours; and Here Recommended to the Industry of the Lovers of Experimental Philosophy, as they Were Generously Imparted to the Publisher in a Letter of the Said Mr. Newtons of July 8.1672 in the Philosophical Transactions of the Royal Society, he got attacked on all sides most virulently by Robert Hooke (1635–1703), then England’s leading authority on all things optical. Hooke’s criticism boiled down to his claim that the paper contained very little of worth and what it did contain he had already discovered himself. He also objected very strongly, as did also Christiaan Huygens (1629–1695), that Newton had used his experiments to promote a corpuscular theory of light, whereas his two critics both propagated a wave theory. There ensued various heated debates with Isaac giving as good as he got and the whole thing culminated in Newton threatening to withdraw from the Royal Society. 

A couple of years later tempers had cooled down somewhat with Newton and Hooke once again on talking terms or better said on corresponding terms between Cambridge and London and in an attempt to ameliorate, Newton wrote the following in a letter to Hooke dated 5 February 1675:

What Des-Cartes [sic] did was a good step. You have added much several ways, & especially in taking the colours of thin plates into philosophical consideration. If I have seen further it is by standing on the shoulders of Giants.

I’m not going to go into a lot of detail here but the expression standing on the shoulders of Giants was not original to Newton but in various forms has existed since at least the twelfth century, attributed by John of Salisbury (c. 1110–1180) in his Metalogicon in 1159 to Bernard of Chartres (died after 1124): 

 “Bernard of Chartres used to compare us to dwarfs perched on the shoulders of giants. He pointed out that we see more and farther than our predecessors, not because we have keener vision or greater height, but because we are lifted up and borne aloft on their gigantic stature.”

There are other claims as to the origins of the concept. There is a wonderfully entertaining book on the saying by the sociologist Robert K Merton (1910–2003), On the Shoulders of Giants: The Post Italianate Edition(1965), which I heartily recommend for long train journeys or flights.

But let us return to the good Isaac and the question, if the phrase standing on the shoulders of Giants were to be applied to his Principia, if not Galileo, who would those giants be?

The first rather paradoxical answer is René Descartes (1596–1650).

Source: Wikimedia Commons

Why paradoxical? It is paradoxical because the Principia is fundamentally an anti-Cartesian text starting with the title. Descartes book on the principles of nature is his Principia Philosophiae published in 1644, from which Newton borrowed the principle of inertia, his first law of motion. Newton deliberately borrowed the title but added Mathematica to show that his work was based on mathematics and not philosophical waffle like Descartes’ volume. 

Source: Wikimedia Commons

As a young man Newton had been a follower of Descartes, who was considered modern in opposition to the Aristotelian philosophy still taught at Cambridge. Descartes was one of the primary sources from which he learnt the new mathematics, although using the vastly expanded Latin edition of La Géométrie, produced by Frans van Schooten the Younger (1615–1660) rather than the French original. As I have pointed out on numerous occasions, it was van Schooten who first introduced Cartesian coordinates and not Descartes. But I digress.

As he matured, Newton distanced himself from Descartes vortex physics and the second of the three volumes of his Principia is devoted to a systematic deconstruction of Descartes theories. So, Descartes is very much a father of the Principia but in a negative not a positive sense. Just as Descartes was trying to refute Aristotle in his Principia, so Newton refuted Descartes in his.

In a positive sense the shoulders on which the Principia is held aloft more than any others are those of Johannes Kepler (1571–1630), whose three laws of planetary motion build the backbone of Newton’s astronomy.

Source: Wikimedia Commons

Interestingly, Newton claims the first two laws for himself, stating that Kepler had not proved them but he, Newton, had. What he is saying in that Kepler merely derived the laws empirically from Tycho’s data, whereas he derived them mathematically from his three laws of motion and centripetal force, to which I will return shortly. Newton then rather contradicts himself by crediting the third law to Kepler. Why contradicts? The third law is also an empirical law derived from observational data that had be shown to be valid for the then known planets in the cosmos, the moons of Jupiter and the moons of Saturn. This was important for Newton because using his laws of motion he proved that the third law and the inverse square law of gravity were mathematically equivalent. It followed from this proof that because the third law was empirically true the law of gravity must also be true. 

Returning to the concept of centripetal force, Newton uses this throughout Book I of Principia and not the force of gravity is his mathematical derivation of the behaviour of masses in motion under the influence of forces. The concept of centripetal force or rather its inverse, centrifugal force was derived by Christiaan Huygens (1629–1695), our next set of shoulders, and published in his Horologium Oscillatorium Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (The Pendulum Clock: or Geometrical demonstrations concerning the motion of pendula as applied to clocks) (1673) from whence Newton borrowed it.

Christiaan Huygens

Source: Wikimedia Commons

From the same work Newton also took his second law of motion and Huygens’ correct determination of the acceleration due to the force of gravity, which Galileo had got wrong. It is fairly certain that others derived the inverse square law of gravity in the 1660s by combining Huygens work with Kepler’s third law. However, Newton claimed to have done the necessary derivation independently without borrowing from Huygens.

Huygens was also the first to astronomically survey Saturn with a telescope, discovering the largest of Saturn’s moons, Titan, in 1655. Giovanni Domenico Cassini (1625–1712) followed close on his heels discovering four more moons–Tethys, Dione, Rhea, Iapetus–between 1673 and 1686. Newton was eager to know whether the moons of Saturn provided another confirmation of Kepler’s third law.

Giovanni Domenico Cassini Source: Wikimedia Commons

The man who provided that information was Edmond Halley (1656–1742).

Portrait of Halley (c. 1690) Source: Wikimedia Commons

More than anybody else Halley carried Principia on his shoulders. It was Halley, who with his visit to Cambridge in August 1684 to ask Newton what the shape of planetary orbits would be if there were an inverse square law of gravity, provoked Newton to return to a problem he had abandoned almost twenty years earlier. It was Halley who urged Newton to write down his ideas for publication. It was Halley who persuaded The Royal Society to publish those ideas. It was Halley, who calmed the waves when Hooke provoked Newton by claiming the he and he alone should be credited with the discovery of the law of gravity, leading to Newton threatening to abandon Book III of Principia, The System of the World, the most important book. In the end, it was also Halley, who shouldered the cost of publishing the Principia, when the Royal Society ran out of money. 

However, Halley also made scientific contribution to the book. As already noted above he supplied the important information that the moons of Saturn also conformed to Kepler’s third law. When Newton assumed that comets also conformed to the inverse squared law of gravity and Kepler’s laws, it was Halley who did the necessary research to confirm the assumption. The flight paths of comets played a major role in Book III. 

Another set of shoulders on which the Principia rests are those of John Flamsteed (1646–1719), the Astronomer Royal. Without Flamsteed’s observational data accumulated since his appointment in 1675, Newton could not have written Book III. Flamsteed was also the first note that two comets were actually one and the same comet that had orbited the sun and returned. This was the point when the orbits of comets became important for a universal law of gravity. Although, it took Newton some time to accept that Flamsteed was right.

Portrait by Thomas Gibson, 1712 Source: Wikimedia Commons

There are two giants, who never get mentioned in Principia, but who provide the scaffolding that permeates the entire book. There is a widespread myth that Newton used calculus to write Principia, he didn’t. Although he played a leading role in the development of analysis, Newton lost faith in its ability to provide solid proof, so all of the mathematics of Principia is classical Greek synthetic geometry, albeit somewhat updated by Newton. This means the geometry of Euclid (fl. 300 BCE) and because of the constant use of circles, ellipses, and parabolas the Conics of Apollonius of Perga (c. 240–c. 190 BCE). As to why Newton never mentions them, in his brief discussion of Dana Densmore’s Newton’s “Principia” : The Central Argument–Translations, Notes, and Expanded Proofs (Green Lion Press, 1995) I. B.  Cohen writes:

Many readers will be grateful to Densmore for providing the missing steps and unstated assumptions in Newton’s constructions and often laconic proofs. They‚ will also be greatly aided by the geometric rules, methods, and procedures from Euclid and Apollonius that Densmore supplies, which Newton supposes readers will know.^[1]

Returning to my opening paragraph, Newton, of course give Galileo credit every time that he uses either the laws of fall or the parabola law of projectile motion but in comparison to some of those I have sketched above his overall contribution to the Principia is relatively small and doesn’t really qualify him as a shouldering giant. 

Galileo Galilei, portrait by Francesco Porcia Source: Wikimedia Commons

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^[1] Isaac Newton The Principia Mathematical Principles of Natural Philosophy: A New Translation by I. Bernard Cohen and Anne Whitman assisted by Julia Budenz. Preceded by A Guide to Newton’s Principia by I. Bernard Cohen p. 296

Magnetic Variations – III Robert Norman

Robert Norman’s The Newe Attractive (1581) was the most scientific study of magnetism and the magnetic compass between Petrus Peregrinus’ Epistola de magnete from 1269 and William Gilbert’s De Magnete from 1600 and like the former featured strongly in the latter.

The Newe Atractive 1592 edition Source

As is all too often the case with comparatively minor Renaissance figures we know next to nothing about Robert Norman. His dates of birth and death are unknown and all that is known about his origins is that they were humble. According to his own account he spent eighteen or twenty years at sea before he settled down at Ratliffe (Ratcliff) part of the Manor and Ancient Parish of Stepney on the north bank of the Thames between Limehouse (to the east) and Shadwell (to the west), as an instrument maker and self-styled ‘hydrographer’. 

The Hamlet (administrative sub-division) of Ratcliff in Joel Gascoyne’s 1703 map of the Parish of Stepney Source: Wikimedia Commons

Ratcliffe in earlier times was also known as “sailor town”. It was originally known for shipbuilding but from the fourteenth century more for fitting and provisioning ships. In the sixteenth century various voyages of discovery were supplied and departed from Ratcliffe, including those of Willoughby and Frobisher.

Wikipedia

Norman’s principal claim to fame is as the discoverer of the second deviation of the magnetic compass needle, after variation or declination, magnetic dip or inclination. This, as observed by Norman, was the fact that the compass needles that he made did not sit horizontally on the middle point but the north end dip down at the north end, as he described it in chapter three of his The Newe Attractive:

“…rising alwaies to finish and end the, before I touched the needle I found continually that after I touched the Irons … the North point … would bende under the Horizon…”

The modern definition:

Magnetic dip, dip angle, or magnetic inclination is the angle made with the horizontal by the Earth’s magnetic field lines. This angle varies at different points on the Earth’s surface. Positive values of inclination indicate that the magnetic field of the Earth is pointing downward, into the Earth, at the point of measurement, and negative values indicate that it is pointing upward. The dip angle is in principle the angle made by the needle of a vertically held compass. (Wikipedia) 

Strictly speaking Norman was not the first to discover magnetic dip, that honour goes to the Franconian astronomer, mathematician and instrument maker, Georg Hartmann (1489–1564), who discovered it in 1544 and described it, with a lot of other information on magnetism and compasses, in a letter he wrote to Duke Albrecht of Prussia (1490–1568). However, he never published his discovery, and the letter to Albrecht only became known in the nineteenth century, so the laurels for the discovery are usually awarded to Norman. On a side note, Hartmann measured the magnetic variation of Rome in 1510 finding it to be 6°. 

Georg Hartmann Source: Astronomie in Nürnberg

Norman first noticed the dip on a six-inch compass needle that he had manufactured and initially thought that it had been somehow spoilt during the making process. He devised a series of experiments to try and find the cause and discovered that the needle was OK, and the cause was some attractive power of the Earth. Having discovered that dip was a natural phenomenon and constructed a dip-circle and measured the angle of dip for London that he measured accurately as 71° 51’. 

Figure of a dip circle, illustrating magnetic dip Robert Norman – Page 17 of The Newe Attractive via Wikimedia Commons

The discovery of magnetic dip and Norman’s invention of the dip-circle to measure it led to speculation that dip could be used to determine latitude by overcast skies in the same way that it had been hoped to determine longitude by magnetic variation. Although, the dip-circle became a standard piece of the navigator’s equipment throughout the seventeenth century its use to determine latitude never came about. 

Having dealt with the phenomenon of magnetic dip in a scientific manner, Robert Norman also turned his attention to magnetic variation. He dismissed the widespread idea that variation was by proportion around the globe and could thus be used to determine longitude citing the observed vagaries of variation. His comments were based on twenty years of experience at sea and the fact that the only people who gave him reliable figures for variation were those engaged in the Muscovy trade, and these did not in any way support the thesis. His book appears to have been the first publication to have an illustration of a compass card with a true north south meridian and a true east west line and then a compass north south line and a false east west line explain and indicating variation. 

Source

One important aspect of Norman’s studies of the magnetic compass is that he changed the perception of what actually took place when a compass needle stopped swinging. In the first post in this series, we briefly touched upon the supposed places to which the needle was drawn or attracted, the North Pole, the Pole star, a magnetic mountain or island etc. Norman saw it differently, to quote William Gilbert in his De Magnete:

Robert Norman, an Englishman, posits a point and place to which magnet looks (but whereto it is) not drawn : toward which magnetised iron, according to him is collimated but does not attract it. 

Source

Norman also instructed mariners to ensure that their compasses and marine charts had been made by the same people in the same locations. This was to ensure that they were based on the same value for magnetic variation. A compass combined with a marine chart from two different locations based on different variation values could and did lead to serious navigation problems on the open sea. He included a table of five different sorts of sailing compasses with their corresponding marine charts. 

David Waters, The Art of Navigation (Henry C. Taylor, 1958) p. 155

The Newe Attractive contained other material useful to navigators. The 1585 second edition contained a Regiment of the Seas “exactlie calculated unto the minute” valid for thirty years and presented in the same form as Medina and William Bourne, which contained a wealth of useful information. 

Robert Norman worked closely with William Borough (bap. 1536–1598), who I dealt with in the last episode, and who was comptroller of the queen’s ships, supplying him with instruments and knowledge. The Newe Attractive was dedicated to William Borough

Source

and as I wrote in the episode on Borough the book contained Borough’s A Discourse on the Variation, which was specifically written to be included as an appendix.

Source

Source

This treated the problem of variation “both Practically and Mathematically,” for the enlightenment of the simple and also the learned sort of mariner. Borough’s text contains a lot of polemic on the necessity of learning mathematics for navigation and also urging mariners to determine and record compass variation on their voyages. For this purpose, Robert Norman designed and constructed a new, improved variation compass to make the task of determining variation easier. Borough also strongly supported Norman’s rejection of the idea that variation was by proportion around the globe.

Source

Source

The combined Norman/Borough book went through new expanded and improved editions in 1585, 1592, 1611, and 1614.

In 1584, Norman published a second book, The Safegard of Sailers, or, Great Rutter, a manual of coastal sailing mostly translated from Dutch sources but with additional content of his own.

Title page of the 1671 edition Source

Source

This book was dedicated to Charles Howard, Earl of Nottingham and Lord High Admiral of England. 

Charles Howard (1536-1624), 1st Earl of Nottingham *oil on canvas *208.5 x 139.5 cm *ca. 1620 *inscribed b.l.: Carolus Baro. Howard de Effingham, Comes Nottingham, summus Angliae Admirallus – Ductor Classium 1588 -. Obijt anno 1624. Aetat. 88 Source: Wikimedia Commons

The main thing that distinguishes Robert Norman from other English writers on navigation, magnetism, and the compass in the sixteenth century is the systematic series of experiment that he designed and carried out first, to determine if magnetic dip was a real natural phenomenon and secondly to conceive and construct the dip circle to measure dip. In his ODNB article on Robert Norman, Jim Bennett^[1] wrote: 

Norman has attracted considerable interest on account of his self-conscious adoption of an experimental approach and his unusual application of instruments. He was deploying his dip circle at a time when instruments were associated not with natural philosophy but with applications of mathematics to practical arts. He was sensitive that, as an ‘unlearned mechanician’, he would scarcely have been expected to concern himself with an area of practical mathematics relevant to natural philosophy, but he vigorously asserted the worth of investigations by practical men, who had the relevant art ‘at their finger ends’, while their more learned critics were ‘in their studies amongest their bookes’. Norman saw himself and his fellow mechanics as heirs to the vernacular tradition of mathematical publication, exemplified by the works of Robert Recorde and Billingsley’s English translation of Euclid. 

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^[1] Jim Bennett was a truly great historian of scientific instruments and history of science museum curator, first in Cambridge at the Whipple and then in Oxford at the History of Science Museum. Sadly he died last Saturday, 28 October 2023, whilst I was using his article to write this blog post.

From τὰ φυσικά (ta physika) to physics – VIII

Euclid, Archimedes of Syracuse, and Apollonius of Perga are the three most significant mathematicians of Ancient Greece. The names of Euclid and Archimedes are very well known outside of the world of mathematics. However, although as a mathematician Apollonius can be put into the same class as Archimedes his name is probably not even known to many mathematicians let alone non-mathematicians. He didn’t write a legendary textbook like Euclid and there are no war machines or bathtub scenes like Archimedes. However, the one major work of his that has survived, Conics, played a very significant role in the history of mathematics, particularly in the Early Modern Period, and he also made a significant contribution to Greek astronomy; a contribution acknowledge by Ptolemaios in his Mathēmatikē Syntaxis, aka the Almagest.

As is unfortunately the case with many Ancient Greek mathematicians, we know almost nothing biographical about Apollonius. Eutocius of Ascolon (c. 480s – c. 520s CE) was a mathematician who wrote commentaries on several of Archimedes’ works and on Conics. In his commentary on Conics, which is dedicated to Anthemius of Tralles (2^nd half of the 5^th century), a mathematician and one of the architects of the Hagia Sophia in Constantinople, Eutocius wrote:

Apollonius, the geometrician, … came from Perga in Pamphylia in the times of Ptolemy III Euergetes, so records Herakleios the biographer of Archimedes ….

The dates usually given for Apollonius (c. 240–c. 190 BCE) are guestimates based on this brief passage and a few other hints. 

Perga was a Greek city in Pamphylia, it was the capital city of the Roman Province of Pamphylia Secunda, now Anatolia. Its ruins lie about 15 kilometres east of Antalya.

Perga: Roman rule of Perga began in 188 BC, and most of the surviving ruins today date from this period. After the collapse of the Roman Empire, Perge remained inhabited until Seljuk times, before being gradually abandoned. Source: Wikimedia Commons

Although probably born in Perga, the available sources indicate that Apollonius probably lived, studied and wrote in Alexandria. Apollonius was, however, a popular name in Ancient Greece and the “of Perga” tag helps to differentiate the mathematician from the other more or less well known Apolloniuses–Wikipedia lists forty-six writers, artists, theologians, orators, philosophers, etc., as well as an additional twenty physicians. 

Unfortunately, as with many other figures in Ancient Greece, the vast majority of Apollonius’ work has been lost and is only known to us by references in the work of other scholars. Pappus of Alexandria (c. 290 – c. 350 BCE) the indefatigable promoter of Greek mathematics lists and attempts to reconstruct six treatises, in Book VII of his Synagoge (Συναγωγή) or Collection (c. 340).

• Λόγου ἀποτομή, De Rationis Sectione (“Cutting of a Ratio”)
• Χωρίου ἀποτομή, De Spatii Sectione (“Cutting of an Area”)
• Διωρισμένη τομή, De Sectione Determinata (“Determinate Section”)
• Ἐπαφαί, De Tactionibus (“Tangencies”)
• Νεύσεις, De Inclinationibus (“Inclinations”)
• Τόποι ἐπίπεδοι, De Locis Planis (“Plane Loci”). Wikipedia

Another seven treatises are referred to by other writers.

The six treatises listed by Pappus are all in two books and are all geometrical and are extensions and developments of the existing propositions from Euclid and other sources and all belong to the higher geometry.  Just to give one example:

Tangencies (ʾєπαøαí), in two books, deals with the general problem characterized by Pappus as follows: “Given three elements, either points, lines or circles (or a mixture), to draw a circle tangent to each of the three elements (or through them if they are points).” There are ten possible different combinations of elements, and Apollonius dealt with all eight that had not already been treated by Euclid.^[1]

The other no longer extant works mentioned by other sources cover a wider range of topics:

• Περὶ τοῦ πυρίου, On the Burning-Glass, a treatise probably exploring the focal properties of the parabola.
• Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus)
• A comparison of the dodecahedron and the icosahedron inscribed in the same sphere.
• Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius’s criticisms and suggestions for the improvement of Euclid’s Elements. 
• Ὠκυτόκιον (“Quick Bringing-to-birth”), in which, according to Eutocius, Apollonius demonstrated how to find closer limits for the value of π than those of Archimedes, who calculated 3+¹⁄₇ as the upper limit and 3+¹⁰⁄₇₁ as the lower limit.
• an arithmetical work on a system both for expressing large numbers in language more everyday than that of Archimedes’ The Sand Reckoner and for multiplying these large numbers.
• a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus’ comm. on Eucl. x., preserved in Arabic and published by Woepke, 1856). Wikipedia

These various works, both the geometrical and the non-geometrical, demonstrate that Apollonius can easily be considered on a level with Archimedes. It is interesting to note that Apollonius, like Menaerchmus (380­320 BCE) and Archimedes before him, used an implied window as a common grid, in his geometric figures, similar to a Cartesian coordinate system. It, however, only consisted of what we now refer to as the first quadrant because Greek mathematician did no use negative numbers. Apollonius used his coordinate system in his work on the conic sections, the only work of Apollonius that has largely survived to which we will now turn.

Conics is a work in eight books of which only Books I to IV survived in the original Greek. Books V to VII survived in Arabic translations and Book VIII has been lost. Conics is, however, a very important work in the history of mathematics and was recognised as such in antiquity, in the Islamic Middle Ages and very much so in Europe in the Early Modern Period. 

Apollonius was not the first Greek mathematician to address the conic sections. He mentions that Conon of Samos (C. 280–c. 220 BCE) had worked on them and that his work was the basis for his own Book IV. We also saw, in the previous post in this series, that Archimedes did a lot of work on parabolas. What Apollonius did, in his Conics, was to change the basic definition of the conic sections to the one we still use today and to provide a systematic, comprehensive study of them.

The three curves now known as parabola, hyperbola, and ellipse were obtained by cutting a right circular cone by a plane at right angles to a generator of the cone. According to whether the cone has a right angle, an obtuse angle, or an acute angle at its vertex, the resultant section is respectively a parabola, a hyperbola, or an ellipse. These sections were therefore named by the earlier Greek investigators “section of a right-angled cone,” “section of an obtuse-angled cone,” and “section of an acute-angled cone,” respectively; those appellations are still given to them by Archimedes (although we know that he was well aware that they can be generated by methods other than the above). With the above method of generation, it is possible to characterize each of the curves by what is known in Greek as a σύμπτωμα, i.e., a constant relationship between certain magnitudes which vary according to the position of an arbitrary point taken on the curve (this corresponds to the equation of the curve in modern terms).

[…]

Apollonius’ approach is radically different. He generates all three curves from the double oblique circular cone.^[2]

Source

In his preface to Book 1, Apollonius gives a brief general description of the contents of all eight books:

The first four books constitute an elementary introduction. The first contains the methods of generating the three sections and their basic properties developed more fully and more generally than in the writings of others; the second contains the properties of the diameter and axes of the sections, the asymptotes and other things…; the third contains many surprising theorems useful for the syntheses of solid loci and for determinations of the possibilities of solutions; of the latter the greater part and the most beautiful are new. It was the discovery of these that made me aware that Euclid has not worked out the whole of the locus for three and four lines,¹⁴ but only a fortuitous part of it, and that not very happily; for it was not possible to complete the synthesis without my additional discoveries. The fourth book deals with how many ways the conic sections can meet one another and the circumference of the circle, and other additional matters, neither of which has been treated by my predecessors, namely in how many points a conic section or circumference of a circle can meet another. The remaining books are particular extensions; one of them [V] deals somewhat fully with minima and maxima, another [VI] with equal and similar conic sections, another [VII] with theorems concerning determinations, another [VIII] with determinate conic problems.^[3]

That Conics was regarded as an important advanced textbook is fairly obvious from its history, alone the fact that it survived whereas the rest of Apollonius’ output didn’t, speaks volumes. In late antiquity, Hypatia of Alexandria (c. 350–370–415 CE) is said to have edited it and added commentaries. It is thought that the surviving Arabic version, a translation by Thābit ibn Qurra (c. 830–910 CE) commissioned by the Banū Mūsā, Abū Jaʿfar, (Muḥammad ibn Mūsā ibn Shākir (before 803–873); Abū al‐Qāsim, Aḥmad ibn Mūsā ibn Shākir (d. 9th century) and Al-Ḥasan ibn Mūsā ibn Shākir (d. 9th century)) is based on Hypatia’s version.

Pages from the 9th century Arabic translation of the Conics Source: Wikimedia Commons

In his Treatise on Algebra, the Persian mathematician Omar Khayyam (1048–1143), used the conic sections for one section of his geometrical solutions for the positive roots of all cubic equations, specifically utilising Apollonius Book II, proposition 12. Khayyam’s work is seen as an anticipation of Descartes’ analytical geometry. 

Omar Khayyam “Cubic equation and intersection of conic sections” the first page of a two-chaptered manuscript kept in Tehran University Source: Wikimedia Commons

The Conics received a new lease of life during the Early Modern Period in Europe. The Books I–IV of Conics were first published in Latin in a translation by Giovanni Battista Memo, professor for mathematics at Venice, by Bernadino Bindoni in Venice in 1537.

Memo Woodcut titlpeage of 1537 Apollonius of Perga (Sp Coll Hunterian R.3.2) Source

Opening from the 1537 Apollonius covered with marginalia (Sp Coll Hunterian R.3.2) Source

Federico Commandino (1509–1575) included a Latin translation of Books I–IV of the Conics, in his long list of translations of Greek mathematics, published in 1566.

Apollonius of Perga, Conicorum libri quattuor vnà cum Pappi Alexandrini lemmatibus, et commentariis Eutocii Ascolonitae. Sereni Antinsensis philosophi libri duo nunc primum in lucem editi. Quæ omnia nuper Federicus Commandinus (Bologna, 1566), p. 5. Source

The first Arabic manuscript of Books V-VII, in Europe, was acquired by the Medici in the early seventeenth century. This was translated into Latin with the help of Maronite scholar Abraham Ecchellensis (Ibrahim ibn Daud al-Haqili) (1605–1664) by Giovanni Alfonso Borelli (1608–1679) in 1658 and published in Florence by Giuseppe Cocchini in 1661.

Elementa conica Apollonii Paergei et Archimedis opera nova & breviori methodo demonstrata by Giovanni Alfonso BORELL

Isaac Barrow (1630–1677), the first Lucasian Professor of Mathematics at Cambridge University, published a Latin translation of Books I–IV in London in 1675. Edmond Halley (1656–1742) completed and published the translation of all seven books in 1706, begun by Edward Bernard (1638–1697), Savilian Professor of Astronomy at Oxford University, who had died before he could finish it. Halley learnt Arabic in order to do so. He later produced a new complete translation published in 1710. For his translation from the Arabic, Halley used a newly discovered ninth-century manuscript held by the Bodleian Library.

Title page Halley’s 1710 translation Source

Frontispiece Halley’s 1710 Translation Source

There is no known medieval Latin translation of the Conics but Witelo (c. 1230–late thirteenth, early fourteenth, century) included references to it in his Perspectiva. Witelo couldn’t read Greek so he must have had access to a Latin translation. The most likely translator would have been his friend William of Moerbeke (c. 1220–c. 1286), to whom he dedicated the Perspectiva.

The conic sections would continue to play an important role in geometrical optics and Johannes Kepler (1571–1630) included a section on them (Chapter IV, section 4) in his Astronomiae pars optica published in Prague in 1604. He went much further than Apollonius and Witelo and described for the first time a non-cone-based system of conics in which all possible types of conic sections appear. 

He argued that if a focus of a conic section were allowed to move along the line joining the foci, the geometric form would morph or degenerate, one into another. In this way, an ellipse becomes a parabola when a focus moves toward infinity, and when two foci of an ellipse merge into one another, a circle is formed. As the foci of a hyperbola merge into one another, the hyperbola becomes a pair of straight lines. (Wikipedia)

Another optician, Francisco Maurolico (1494–1575), produced an edited and corrected edition of Books I–IV of the Conics, which was published posthumously in 1654. 

Outside of optics the hyperbola is a feature of many sundials and books on sundials were one of the two most frequently published genre of mathematical books in the Early Modern Period:

Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the declination line). The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun’s rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a pelekinon by the Greeks, since it resembles a double-bladed axe. (Wikipedia)

More significantly for the development of modern physics in the Early Modern Period, Kepler of course, showed that the planets orbit the Sun on elliptical orbits. Later it was shown that under the effect of the force of gravity the trajectory of celestial objects, comets for example, are conic sections, either ellipses, parabolas, or hyperbolas. In terrestrial physics both Thomas Harriot (c. 1560–1621) and Galileo (1564–1642) showed that the path of a projectile is a parabola; the solution to a problem that had plagued scholars since antiquity.

A page from Galileo’s notebooks, showing an experiment to determine that the path of a projectile is parabolic Source

Apollonius also made a small but significant contribution to the evolution of the geometrical model of the cosmos. As we have seen in earlier episodes, the Aristotelian model of the celestial spheres was based on the homocentric spheres first posited by Eudoxus of Cnidus (c. 390–c. 340 BCE) and further developed by Callippus (c. 370–c. 300 BCE), and by Aristotle himself. However, there were other geometrical models, such as the eccentric system, in which the centre of the orbit is offset from the Earth, and the deferent/epicycle system in which the planet is carried on a small circle which in turn is carried on a large circle that orbits the Earth. The dominant system that came down to the Early Modern Period was the deferent/epicycle system propagated by Ptolemaios in his Mathēmatikē Syntaxis. There were, however, repeated attempts over the centuries to revitalise the homocentric spheres of Aristotle over the centuries. 

Basic deferent/epicycle model used by Ptolemaios Source: Wikimedia Commons

It is possible to demonstrate that the eccentric system and the deferent/epicycle system are mathematically equivalent. Ptolemaios does this in Book XII of the Mathēmatikē Syntaxis and write there:

In the definition of this kind of problem, there is a preliminary lemma demonstrated (for a single anomaly, that related to the sun) by a number of mathematicians, notably Apollonius of Perga, to the following effect. [There follows, pages of geometrical calculation in the middle of which Ptolemaios introduces Apollonius’ lemma]^[4]

Some historians have hypothesised from this reference that it was Apollonius, who first created the deferent/epicycle system, others think that it probably predates him. We have no evidence either way. 

Alone with his Conics, Apollonius made a major contribution to the mathematics required to launch modern physics in the Early Modern Period. 

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^[1] G. J. Toomer, Apollonius article in Dictionary of Scientific Biography.

^[2] G. J. Toomer, Apollonius article in Dictionary of Scientific Biography

^[3] G. J. Toomer, Apollonius article in Dictionary of Scientific Biography

^[4] Ptolemy’s Almagest, Translated and Annotated by G. J. Toomer, Princeton University Press, Princeton 1998 p. 555

From τὰ φυσικά (ta physika) to physics – VII

No figure looms larger in the history of ancient mathematics than Archimedes, a name surrounded by a dense cloud of stories, myths, and legends. However, one should never lose sight of the fact that behind all the marvellous narratives about setting fire to enemy ships with burning mirrors and overturning others with giant levers, the real Archimedes was one of the greatest mathematicians who ever lived, and his work had a major impact on the newly emerging sciences in the sixteenth and seventeenth centuries. 

Bronze statue of Archimedes in Syracuse Source: Wikimedia Commons

Literally millions of words have been written over the centuries about Archimedes but if you ignore the posthumous hagiography, nearly all of which should be taken with a very large pinch of salt, then in realty we know almost nothing about the man. His supposed life dates c. 287–c. 212 BCE are based on something written by the Byzantine Greek historian, John Tzetzes (c.1110–1180 CE), so approximately fourteen hundred years after he lived. We do know that he was born in Syracuse on the island of Sicily, at the time a self-governing Greek city, and that he tells us in the Sand-Reckoner that his father was Phidias, an astronomer. Here, the reliable or solid facts end, the rest is just speculation. However, unlike Euclid nobody has ever doubted his existence.

Based on his surviving writings, Archimedes is credited with being a mathematician, engineer, astronomer, and inventor. He is usually referred to as a physicist, and although anachronistic, as I pointed out in the first episode of this series, the term in its modern usage was a nineteenth century minting by William Whewell, it is justified as several of his texts definitely fall withing the scope of modern physics, which played a significant role in his influence during the sixteenth and seventeenth centuries. 

I shall ignore all of the war machines that Archimedes, the engineer, created for the defence of Syracuse, as described in the posthumous hagiographies, and concentrate instead on his know contributions to mathematics and science. Of course, the most popular presentation of Archimedes is the story of the crown, the bathtub, and running down the street shouting Eureka (Ancient Greek: εὕρηκα, Romanised: héurēka).

There is nothing remotely like this story in Archimedes own writings. It is also thought not to be real because the water displacement method of determining density that it supposedly led to, would be extremely difficult to realise due to the problems of accurately measuring the volume of water displaced. We will come to what he probably did later but first, feeling frivolous, I can’t resist repeating a terrible joke I first heard in my dim and distant youth:

Archimedes running down the street:  Eureka! Eureka! …

Man, he passes: You dona smella too gooda youselfa!

The so-called Archimedean Screw, which as the name suggests was named after him, a widespread method of raising water deserves at least a mention although there is no direct proof that he actually invented it.

As with other figures from antiquity there are works referenced by other writers that are no longer extant, in Archimedes’ case seven such works are recorded. The works that still exist, in the currently assumed order in which they were written, are Measurement of a Circle, The Sand Reckoner, On the Equilibrium of Planes, Quadrature of the Parabola, On the Sphere and Cylinder, On Spirals, On Conoids and Spheroids, On Floating Bodies, Ostomachion, The Cattle Problem, and The Method of Mechanical Theorems.

The Ostomachion is a geometrical puzzle similar to Tangram and The Cattle Problem is algebraic puzzle that requires the solution of a number of simultaneous Diophantine equations. The Cattle Problem was addressed to Eratosthenes and other Alexandrian mathematicians. The puzzles need not detain us here, but it is interesting to note that Archimedes had a playful side.

Measurement of a Circle is only a short fragment containing three propositions: 

1) The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. 

2) The area of a circle is to the square on its diameter as 11 to 14.

3) The ratio of the circumference of any circle to its diameter is greater than

 but less than .

.

Proposition three is his approximation of π and proposition two is actually derived from it, so the order can not be original. Both propositions one and three are examples of Archimedes using the method of exhaustion.

Source: Wikimedia Commons

On the Equilibrium of Planes is in two books. Book one proves the law of the lever and contains propositions on the centre of gravity of the triangle and the trapezium. Book two has ten propositions on the centre of gravity of parabolic segments. This text would prove very important in the development of statics in the Early Modern Period.  

Quadrature of the Parabola has twenty-four propositions and culminates in the poof that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height. He provides two proofs. The first proof is achieved by an application of the method of exhaustion The second s the dissection of the parabolic segment into infinitely many triangles, as shown in the figure below and then summing the infinite series. A very clear demonstration of Archimedes’ mathematical genius

Archimedes’ second proof dissects the area using an arbitrary number of triangles. Source: Wikimedia Commons

On the Sphere and Cylinder contains Archimedes’ famous proof that a sphere has 2/3 the volume and surface area of its circumscribing cylinder. It contains the earliest know explanation of how to find the volumes and surface areas of the two solids. Once again, he uses the method of exhaustion to achieve his results.

A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases. Source: Wikimedia Commons

On Spirals defines what is now known as the Archimedean spiral, although it had been previously discussed by the astronomer and mathematician, Conon of Samos (c. 280–c. 220 BCE), as Archimedes acknowledges. He defines it thus:

If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.

The Archimedean spiral with three 360° turnings on one arm Source: Wikimedia Commons

He uses the spiral to trisect and angle and also to square a circle. The Archimedean spiral is often confused in popular writing with the Golden spiral and the Fibonacci spiral.

In thirty-two propositions in On Conoids and Spheroids, Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids. Once more making extensive use of the method of exhaustion. 

On Floating Bodies could be said to be the most important of Archimedes’ surviving text, or at least the one with the most impact. It is the earliest known text on hydrostatics. Once again in two books, it investigates the position that various solids will assume went floating in a fluid, according to their form and the variations in their specific gravities. The second book is devoted to the floating properties of various paraboloids. The book contains the first statement of Archimedes’ Principle:

Any object, totally or partially immersed in a fluid or liquid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

One page of Archimedes Palimpsest, On Floating Bodies The only know Greek manuscript of the text Source: Wikimedia Commons

It is probably through the application of the principle utilising a hydrostatic balance, that is a balance with the pans resting in containers of fluid, that Archimedes actually solved the problem of whether the crown was pure gold or an alloy. This was the suggestion put followed by a young Galileo in his first ever original work, sent to Guidobaldo del Monte in 1586 in order to win his patronage. 

As we will see later On Floating Bodies would go on to have a massive influence in the Early Modern period, when there was a major renaissance in interest in Archimedes’ work.

The brief sketches above demonstrate that Archimedes was a brilliant mathematician in what we would now term pure and applied mathematics or mathematical physics. There are two works that I haven’t dealt with yet, each of which is of a different nature to the works described above.

In The Sand Reckoner Archimedes develops a system for naming large numbers in order to determine the number of grains of sand needed to fill the cosmos. The normal Greek alpha-numerical number system being totally inadequate for the task. He proceeded by producing powers of the myriad (μυριάς — 10,000), nominally the largest number in the Greek counting system, so a myriad myriads (10⁸) became the next step in his system followed by myriad-myriad myriad-myriads (10¹⁶). He then proceeded to potentiate ( and so on… Having calculated a figure for the then accepted geocentric system, he went on to calculate a new lager figure for the heliocentric system of Aristarchus of Samos (c. 310–c. 230 BCE). This is one of the few references to Aristarchus’ system. Archimedes writes:

You are now aware that the “universe” is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account (τὰ γραφόμενα) as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the “universe” just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun on the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.

The other text The Method of Mechanical Theorems, which was first discovered in 1906 on a palimpsest, a parchment scraped off and reused. The Method is a letter that he wrote to Eratosthenes in Alexandria. In this work Archimedes explains how to use indivisibles (geometrical equivalents of infinitesimals) to calculate areas and volumes. Archimedes did not consider that this method produced rigorous proofs, so having determined the solution using it he then redetermined using the method of exhaustion. 

… certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge.

Although Archimedes was already a legend in antiquity because of his inventions his mathematical works received comparatively little attention. Isidore of Miletus (born c. 475 CE) produced the first comprehensive collation of them in Constantinople c. 530 CE. Also, in the sixth century, Eutocius of Ascalon (c. 480–c. 520) wrote commentaries on several of his mathematical works making them available to a wider readership. 

Although his works were translated and known in the medieval period, it was first during the Renaissance that Archimedes’ mathematical works, in particular, On the Equilibrium of Planes and On Floating Bodies, began to have a major influence on the development of physics. Galileo was by no means the first or the only natural philosopher to consciously replace the qualitative physics of Aristotle with the mathematical physics of Archimedes setting in motion the mathematisation of science that is regarded as a key characteristic of the so-called scientific revolution. 

Incorrect casual assumptions

No, she bleedin’ weren’t!

That was my buddy the HISTSCI_HULK expostulating whilst he was indulging in his annoying habit of peering over my shoulder whilst I’m reading.

She never was! That’s simply wrong!

Hulky was getting his nickers in a twist about the following claim:

Hypatia was by all accounts, a fine astronomer and a first rank mathematician…^[1]

He bleedin’ weren’t either! Exploded Hulky as he read further on:

…her father, an equally formidable mathematician…

Hulky is, of course, totally correct. Hypatia’s father was Theon of Alexandria and although such judgements are to a large extent subjective, in the normal run of things nobody would classify Theon as a formidable mathematician or Hypatia a fine astronomer and a first rank mathematician.

We start with Theon from whom Hypatia appears to have learnt and inherited everything. Theon was the head of a (note, not ‘the’) Neoplatonic school in Alexandria where he taught philosophy, mathematics and astronomy. The latter two being part of a basic Neoplatonic curriculum. Here Theon is a teacher of astronomy and mathematics not in any way a formidable mathematician. 

Theon is most well known in the history of mathematics as the editor and commentator of an edition of the Euclid’s Elements. In fact, the only known Greek edition until a different one was found in the nineteenth century. He also produced commentaries on Euclid’s Data, his Optics and Ptolemaios’ Mathēmatikē Syntaxis. All of these are works of elucidation for students and it is more correct to call Theon a textbook editor. 

Theon of Alexandria is best known for having edited the existing text of Euclid’s Elements, shown here in a ninth-century manuscript Vatican Library via Wikimedia Commons

Turing to Hypatia, she appears to have studied under her father and then went on to take over his position as head of his school, also teaching Neoplatonic philosophy with astronomy and mathematics as subsidiaries. Once again, a teacher not a fine astronomer and a first rank mathematician. Unlike Theon there are no known surviving publications by Hypatia. 

The Suda, a tenth-century Byzantine encyclopaedia of the ancient Mediterranean world list three mathematical works for her, which it states have all been lost. The Suda credits her with commentaries on the Conic Sections of the third-century BCE Apollonius of Perga, the “Astronomical Table” and the Arithemica of the second- and third-century CE Diophantus of Alexandria. Alan Cameron, however, argues convincingly that she in fact edited the surviving text of Ptolemaeus’ Handy Tables, (the second item on the Suda list) normally attributed to her father Theon as well as a large part of the text of the Almagest her father used for his commentary.  Only six of the thirteen books of Apollonius’ Conic Sections exist in Greek; historians argue that the additional four books that exist in Arabic are from Hypatia, a plausible assumption^[2]. So once again, what we have is that Hypatia was like her father a textbook editor.

The MacTutor article on Theon contains the following judgement:

Theon was a competent but unoriginal mathematician.

Although we have no direct evidence in her case, the same can almost certainly be said about his daughter, Hypatia. Both of them are Neoplatonic philosophy teachers, a philosophical direction that includes a basic amount of astronomy and mathematics. They both produced textbooks for students by editing existing standard texts and adding commentaries to aid understanding. There is absolutely no evidence that their mathematical competence went beyond this pedagogical level.

Because they both feature fairly prominently in the history of mathematics, people, and unfortunately, not just the quoted author make the lazy, unfounded assumptions that they are “a fine astronomer and a first rank mathematician” and “an equally formidable mathematician.” Assumptions that have absolutely no foundation in the known historical facts. Theon is famous because of his edition of Euclid’s Elements and Hypatia because she was brutally murdered, and not for their mathematical abilities.

I will, however, add, as a sort of footnote, that textbook authors and editors play a very important role in the history of a scientific discipline, a role that unfortunately, all too often, simply gets ignored in the standard accounts of the history of science. 

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^[1] I’m not going to mention the source on this occasion because the assumption made here turns up time and again and has somehow become gospel. I will however be reviewing the book in question in due course.

^[2] This paragraph is borrowed from an early blog post about Hypatia that I wrote.

From τὰ φυσικά (ta physika) to physics – VI

It was not only the philosophers in antiquity who laid down theories that would reappear in the Early Modern Period, as that which we now term physics began to be created, several Ancient Greek mathematicians made contributions that would go on to play a significant role in that creation; most notable are Eudoxus of Cnidus (c. 408–c. 355 BCE), Euclid (fl. [maybe?] 300 BCE), Archimedes (c. 287–c. 212 BCE), Apollonius of Perga (c. 240–c. 190 BCE), Hipparchus  (c. 190–c. 120 BCE), and Ptolemaios (c. 100–c. 170 CE). In what follows I shall be taking a brief look at the work of Euclid and Eudoxus, taking Euclid first, although Eudoxus predates him and will deal with the others in later episodes. 

To begin I will deal with the maybe above in the life dates of Euclid, which doesn’t refer to the dates but to the man himself. For a man, who supposedly wrote the most successful textbook of any sort in the history of the world, he is remarkably elusive. There are four books attributed to him–the Elements, Optics, Data, Phaenomena–and that is quite literally all that we know about the man: 

In 1966, French mathematician Jean Itard, introducing a reissue of François Peyrard’s translation of the Elements, asserted that the book was the work not of an individual but of a group, a school, for which ‘Euclid’ was a collective name. (As it happens, a group of French mathematicians had been publishing under the name ‘Bourbaki’ since the 1930s, providing Itard with the relevant model for his claim about Euclid.) In the twenty-first century it is still sometimes said that the Elements is an accretive text for which there is no need to name an author.^[1]

The Elements is a sort of geometrical encyclopaedia that incorporates and systemises the work of several others and as such, one can understand Itard’s thought process, but I shall simply assume that there really was a man, who lived around 300 BCE in ancient Greece, was named Euclid and was the author of the book we know of as the Elements.

In what follows I shall only be dealing with the Elements. I shall cover the Optics in a separate episode over Greek optics. To quote Wikipedia: “Data (Greek: Δεδομένα, Dedomena) is a work by Euclid. It deals with the nature and implications of “given” information in geometrical problems.” The subject matter is closely related to the first four books of Euclid’s Elements. As such it need not concern us here, which is also true of Phaenomena, which is a non-technical introduction to astronomy.

It is almost impossible to overemphasise the significance and importance of the Elements as a textbook in European science over a period of more than two thousand years, only giving way to other texts in the nineteenth century, much to the annoyance of Lewis Carrol, a mathematics lecturer at Oxford University, who wrote a superb play, Euclid and His Modern Rivals^[2], criticising this development. 

Source: My private copy Details see footnote 2

Euclid’s wasn’t the first Elements (Στοιχεῖα, Stoicheia) to be written by an ancient Greek mathematician. The earliest known was written by Hippocrates of Chios (c. 470–410 BCE) of which only one fragment is known to exist embedded in the work of Simplicius of Cilicia (c. 480–c. 560 CE). Other Elements were supposedly written by Leon (fl. 370 BCE), Theudius (4^th C BCE) and Hermotimus of Colophon (4^th C BCE) all three mentioned by Proclus (412–485 CE). It can be assumed that just as Ptolemaios’ Mathēmatikē Syntaxisbecause it was so superior to all previous astronomy textbooks made them obsolete and they simply disappeared. Now known only through later hearsay or because Ptolemaios mentions them, so Euclid’s Elements made all previous geometry textbooks obsolete, and they too disappeared. 

Because it was the only major mathematics textbook to survive antiquity. The opening books formed the geometry part of the quadrivium, the basic mathematical education on the medieval Latin schools and later the undergraduate curriculum on the medieval universities.  This meant that, at least within science, if, over the centuries, scholars were doing mathematics they were mostly doing Euclidean geometry. This applies to John Philoponus in the sixth century, to the Oxford Calculatores and the Paris physicists in the fourteenth century, to Tartaglia, Benedetti, Copernicus, and Tyco Brahe in the sixteenth century, and even to Kepler and Galileo in the seventeenth century. The Euclidean element of Newton’s Principia is even greater, but I’ll deal with that in more detail later. 

Title page of the first printed edition of Euclid’s Elements printed and Published by Erhard Ratdolt in 1482 Source with all of the pages online at the Library of Congress

I would need a whole blog series to explain Euclid’s Elements in detail. It has thirteen books and Thomas Heath’s three volume, annotated, English translation runs to more than fourteen hundred pages. However, I will briefly cover some of the salient points that help to explain its longevity.

The first aspect of the Elements is its logical structure. In general, the mathematical propositions within it have proofs. This is new. If you look at the earlier mathematics of Mesopotamia, Egypt, or India there are no proofs. The propositions in these various early forms of mathematics are demonstrated and explained by worked examples. When you have a problem of this type, then you solve it in this way. To use a modern term the mathematics was algorithmic. This changes with Euclid, here mathematical propositions have proofs utilising deductive logic. Perhaps the most well-known example being his proof of the so-called Pythagorean Theorem, which Euclid does not attribute to Pythagoras.

One of the oldest surviving fragments of Euclid’s Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5 Source: Wikimedia Commons

Book II Proposition 5 in the first printed edition of The Elements printed and published by Erhard Ratdolt in 1482

Book II Proposition 5 in Thomas Heath’s translation of The Elements

More significant is the logical structure of the entire book. Book 1 opens with a set of basic self-evidently true concepts that require no proof on which the whole structure of the book is erected using step by step logical deduction, adding new definitions where necessary. The entire book has a coherent logical structure. If you accept the basics, then everything else follows logically and must therefore also be accepted. Caveat, critical examination in the nineteenth century, in particular by Moritz Pash (1843–1930) and David Hilbert (1862–1943), showed that many of Euclid’s definitions were inadequate and some of his logic suffered from lacuna but as the historian of mathematics W. W. Rouse Ball pointed out “the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose.”^[3]

Of interest, is the fact that the Elements is the most obvious work from antiquity that fulfils Aristotle’s fundament epistemological concept: Starting from self-evident premises that require no proof one uses a chain of deductive logic until one arrives at empirically observed facts. This is ironical as Aristotle was fundamentally anti-mathematics because did not consider mathematical object as related in anyway to the real world. Some Aristotle fans argue that he was not anti-mathematics exactly because his epistemology was borrowed from mathematics. 

Returning to Newton, not only is his Principia written in Euclidean geometry and not as often falsely assumed in the then new analysis that he helped substantially to create but the whole structure of the work mirrors the same epistemology as used by Euclid in the Elements. This form of epistemology, generally known today as the axiomatic method, is now well-known and widely used in many branches of science and mathematics. Two well known examples in the basics of mathematics are the Peano Axioms for the natural numbers, and the Zermelo-Fraenkel Axioms for set theory. 

A short overview of the contents of the thirteen books:

Book I starts with five postulates, including the infamous parallel postulate, and five common notions continuing to cover basics of plane geometry ending with his proof of proposition 47, the Pythagorean theorem and proposition 48, its corollary.

Book II introduces the so-called geometric algebra i.e., presenting and solving algebraic propositions geometrically. It covers a lot of propositions about squares and rectangles, including the general solution of the quadratic equation. Euclid in the reason we use geometrical names quadratic, cubic etc. to refer to algebraic terms. 

Book III moves onto circles.

Book IV construction of incircles and circumcircles of triangles and the construction of regular polygons with 4, 5, 6, and 15 sides.

Book V changes gear and we learn about the theory of proportion of magnitudes. 

Book VI extends the theory of proportions to plane geometry and the construction of similar figures.

Book VII Elementary number theory, including the Euclidean algorithm for finding the greatest common divider and the lowest common multiple.  

Book VIII deals with geometric series 

Book IX is the application of Books VII and VIII and includes Proposition 20, Euclid’s elegant proof of the infinity of primes.

Book X deal with incommensurable magnitudes i.e., lines that cannot be used to measure each other, a geometrical presentation of irrational numbers. For example, the diagonal of a unit square, which has length √2, cannot be measured by the side. 

Book XI applies Book VI to solid figures.

Book XII determines the volumes of cones, pyramids and cylinders using the method of exhaustion, showing, for example, that the volume of a cone is a third of the volume of the corresponding cylinder, and the volume of a sphere is proportional to the cube of its radius. 

Book XIII Constructs and discusses the properties of the five regular Platonic Solids. 

Having sketched the Elements, we can now turn our attention to Eudoxus of Cnidus (c. 408–c. 355 BCE) and the reason why we deal with him after Euclid, although he lived earlier. As we saw earlier in the episodes about Plato and Aristotle Eudoxus was the creator of the homocentric spheres model for modelling the orbits of the planets as viewed from the Earth, including a mechanism to recreate retrograde motion. However, Eudoxus was not just an astronomer, he was also an excellent mathematician, who can be counted alongside Archimedes as one of the best Ancient Greek mathematicians. Starting with Book V, much of the latter part of the Elements rests on mathematics created by him.

The theory of proportions, a powerful tool in the geometrical presentation of the Elements, and its subsequent application in other books is due to Eudoxus. Eudoxus’ theory of proportions provides a rigorous definition for the real numbers; a definition that inspired Richard Dedekind in developing his theory of Dedkind cuts. Eudoxus introduced the concept of incommensurable lengths as a replacement for working with irrational numbers. 

The mathematics most closely associated with Eudoxus is the method of exhaustion, an early form of integration. It is claimed that the method was first propagated by Antiphon who used inscribed polygons with increasing numbers of sides to approximate the area of a circle. As is so often the case we don’t actually known who exactly this Antiphon was. He is mostly identified with Antiphon of Rhamnus (480–411 BCE) an orator but the identification is highly questionable. Eudoxus took over the theory and made it rigorous. In his hands and later those of Archimedes it proved a powerful tool, in particular for determining areas and volumes. The method of exhaustion was used to prove the following proposition in Book XII of the Elements:

Proposition 2: The area of circles is proportional to the square of their diameters. 

Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. 

Proposition 10: The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. 

Proposition 11: The volume of a cone (or cylinder) of the same height is proportional to the area of the base. 

Proposition 12: The volume of a cone (or cylinder) that is similar to another is proportional to the cube of the ratio of the diameters of the bases. 

Proposition 18: The volume of a sphere is proportional to the cube of its diameter. 

Today, the names Euclid and The Elements are for most people just something to do with Ancient Greek mathematics, but, as already emphasised above, no other book in the history of humanity has had such a powerful influence on the discipline of mathematics and also those disciplines that used mathematics such as physics. 

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^[1] Benjamin Wardhaugh, The Book of Wonders: The Many Lives of Euclid’s Elements, William Collins, 2020 ppb. p. 304 Wardhaugh’s book is an excellent guide to the more than 2000-year history of the Elements, you can read my review here.

^[2] Lewis Carroll, Euclid and His Modern Rivals, Dover Books, NY, 1973

^[3] W. W. Rouse Ball, A short Account of the History of Mathematics, MacMillan, 6^th ed., 1915, p. 55

The equestrian country gentleman, who turned his hand to navigation.

The last third of the sixteenth century and the first third of the seventeenth century saw the emergence of published handbooks on the art of navigation in England. This trend started with the publication of Richard Eden’s translation into English of the Breve compendio de la sphere y de la arte de navegar (Seville, 1551) by Cortés de Albacar (1510–1582), as The Arte of Navigation in 1561. The first handbook on the art of navigation written and published by an Englishman was A Regiment for the Sea published by William Bourne (c. 1535–1582) in 1574. Beginning in 1585, John Blagrave (d. 1611) began the publication of a series of manuals on mathematical instruments beginning with his universal astrolabe, The Mathematical Jewel designed to replace a whole range of navigational instruments. John Davis (c. 1550–1605) became the first active seaman and professional navigator to add to the handbooks on the art of navigation with his The Seaman’s Secrets published in 1594. Although Thomas Hood (1556– 1620), England’s first publicly appointed lecturer for mathematics centred on navigation, published several books on the use of diverse instruments, he never wrote a comprehensive handbook on the art of navigation but in 1592 he edited a new edition of Bourne’s A Regiment for the Sea. Edward Wright (c. 1520–1576) added his contribution to this growing literature, his Certaine Errors in Navigation in 1599. In 1623, Edmund Gunter published his guide to the use of navigation instruments Description and Use of the Sector, the Crosse-staffe and other Instruments. 

All of these books went through several editions, showing that there was an eager and expanding market for vernacular literature on navigation in the period. A market that was also exploited by the gentlemanly humanist scholar Thomas Blundeville (c. 1522–c. 1606), probably writing for a different, more popular, readership than the others.

Thomas was born in the manor house of Newton Flotman in Norfolk, a small village about 13 km south of Norwich. He was the eldest of four sons of Edward Blundevill (1492–1568) and Elizabeth Godsalve. He had one sister and two half-brothers from his father’s second marriage to Barbara Drake. Unfortunately, as is all too often the case, that is all we know about his background, his upbringing, or his education. 

The authors of Athenae Cantabrigienses claim that he studied at Cambridge but there are no details of his having studied there. He is said to have been in Cambridge at the same time as John Dee (1527–c. 1608) but there is no corroboration of this, although they were friends in later life.  However, based on his publications Blundeville does appear to have obtained a good education somewhere, somehow. Blundeville seems to have lived in London for some time before returning to live in Newton Flotman Manor, which he inherited, when his father died in 1568. Much of his writing also seems to indicate that he spent some time in Italy.

Blundeville was well connected, along with his acquaintances with John Dee, Edward Wright, and Edmund Gunter he was also friends with Henry Briggs (1561–1630). Elizabeth I’s favourite Robert Dudley, 1^st Earl of Leicester, who took a great interest in the expanding field of exploration and maritime trade, investing in many companies and endeavours, was one of his patrons. He was also, for a time, mathematics tutor to Elizabeth Bacon, daughter of Sir William Bacon (1510–1579, Lord Keeper of the Great Seal, and elder half sister of Francis Bacon (1561–1626), 1^st Viscount St Alban. He was also mathematics tutor in the household of the judge Francis Wyndham (d. 1592) of Norwich. We will return to his tutorship later.

Blundeville only turned to writing on mathematics, astronomy, and navigation late in life having previously published books on a wide range of topics. 

Blundeville’s first publication, 1561, was a partial verse translation of Plutarch’s Moralia, entitled Three Moral Treatises, which was to mark the accession of Elizabeth I to the throne and one of which was dedicated to her: 

‘Three Morall Treatises, no less pleasant than necessary for all men to read, whereof the one is called the Learned Prince, the other the Fruites of Foes, the thyrde the Porte of Rest,’ The first two pieces are in verse, the third in prose; the first is dedicated to the queen. Prefixed to the second piece are three four-line stanzas by Roger Ascham.

About the same time, he published The arte of ryding and breakinge greate horses, an abridged and adapted translation of Gli ordini di cavalcare by Federico Grisone a Neapolitan nobleman and an early master of dressage.

Grisone’s book was the first book on equitation published in early modern Europe and Blundeville’s translation the first in English. Blundeville followed this in 1565/6 with The fower chiefyst offices belonging to Horsemanshippe, which included a revised translation of Grisone together with other treatises. 

In 1570, under the title A very briefe and profitable Treatise, declaring howe many Counsels and what manner of Counselers a Prince that will governe well ought to have. he translated into English, Alfonso d’Woa’s Italian translation of a Spanish treatise by Federigo Furio Ceriól. He now followed up with historiography, his True Order and Methode (1574) was a loose translation and summery of historiographical works by the Italians Jacopo Aconcio (c. 1520–c. 1566) and Francesco Patrizzi (1529–1597). The first work emphasised the importance of historiography as a prerequisite for a counsellor. Both volumes were dedicated to the Earl of Leicester. 

In 1575 he wrote Arte of Logike, which was first published in 1599. Strongly Ramist it displays the influences Galen (129–216 CE), De Methodo (1558) of Jacopo Aconcio (c. 1520–c. 1566), Philip Melanchthon (1497–1560), and Thomas Wilson (1524–1581). 

Arte of Logike Plainely taught in the English tongue, according to the best approved authors. Very necessary for all students in any profession, how to defend any argument against all subtill sophisters, and cauelling schismatikes, and how to confute their false syllogismes, and captious arguments. By M. Blundevile.  

It contains a section on fallacies and examples of Aristotelian and Copernican arguments on the motion of the Earth.

This is very typical of Blundeville’s publications. He is rather more a synthesist of the works of others than an original thinker. This is very clear in his mathematical and geographical works. Blunderville published three mathematical works covering a wide range including cartography, studies in magnetism, astronomy, and navigation. The first of these works was his A Briefe Description of Universal Mappes and Cardes

This contains the following interesting passage:

For mine owne part, having to seek out, in these latter Maps, the way by sea or land to any place I would use none other instrument by direction then half a Circle divided with lines like a Mariner’s Flie [compass rose] [my emphasis]. Truly, I do thinke the use of this flie a more easie and speedy way of direction, then the manifold tracing of the Maps or Mariners Cards, with such crosse lines as commonly are drawn therein…  

What Blundeville is describing here is the humble geometrical protractor, which we all used at school to draw or measure angles. This is the earliest known reference to a protractor, and he is credited with its invention. 

Blundeville’s second mathematical work, is the most important of all his publications, M. Bludeville His exercises… or to give it its full title:

M. BLVNDEVILE 

His Exercises, containing sixe Treatises, the titles wherof are set down 

in the next printed page: which Treatises are verie necessarie to be read and learned of all yoong Gentlemen that haue not bene exercised in such disciplines, and yet are desirous to haue knowledge as well in Cosmographie, Astronomie, and Geographie, as also in the Arte of Navigation, in which Arte it is impossible to profite without the helpe of these, or such like instructions. To the furtherance of which Arte of Navigation, the said M. Blundevile speciallie wrote the said Treatises and of meere good will doth dedicate the same to all the young Gentlemen of this Realme.

This is a fat quarto volume of 350 pages, which covers a lot of territory. Blundeville is not aiming for originality but has read and synthesised the works of Martín Cortés de Albacar (1510–1582), Pedro de Medina (1493–1567), William Bourne (c. 1535–1582), Robert Norman (before 1560–after 1596), William Borough (1536–1599), Michel Coignet (1549–1623), and Thomas Hood (1556–1620) and is very much up to date on the latest developments.

The first treatise:

First, a verie easie Arithmeticke so plainlie written as any man of a mean capacitie may easilie learn the same without the helpe of any teacher.

What cause first mooved the Author to write this Arithmeticke, and with what order it is here taught, which order the contents of the chapters therof hereafter following doe plainly shew. 

I Began this Arithmeticke more than seuen yeares since for a vertuous Gentlewoman, and my verie deare frend M. Elizabeth Bacon, the daughter of Sir Nicholas Bacon Knight, a man of most excellent wit, and of most deepe iudgment, and sometime Lord Keeper of the great Seale of England, and latelie (as shee hath bene manie yeares past) the most loving and faithfull wife of my worshipfull friend M. Iustice Wyndham, not long since deceased, who for his integritie of life, and for his wisedome and iustice daylie shewed in gouernement, and also for his good hospitalitie deserued great commendation. And though at her request I had made this Arithmeticke so plaine and easie as was possible (to my seeming) yet her continuall sicknesse would not suffer her to exercise her selfe therein. And because that diuerse having seene it, and liking my plaine order of teaching therein, were desirous to haue copies thereof, I thought good therefore to print the same, and to augment it with many necessarie rules meet for those that are desirous to studie any part of Cosmographie, Astronomie, or Geographie, and speciallie the Arte of Navigation, in which without Arithmeticke, as I haue said before, they shall hardly profit.

And moreover, I haue thought good to adde vnto mine Arithmeticke, as an appendix depending thereon, the vse of the Tables of the three right lines belonging to a circle, which lines are called Sines, lines tangent, and lines secant, whereby many profitable and necessarie conclusions aswell of Astronomie, as of Geometrie are to be wrought only by the help of Arithmeticke, which Ta∣bles are set downe by Clauius the Iesuite, a most excellent Mathematician, in his booke of demonstrations made vpon the Spherickes of Theodosius, more trulie printed than those of Monte Regio, which booke whilest I read at mine owne house, together with a loving friend of mine, I took such delight therein, as I mind (God willing) if God giue me life, to translate all those propositions, which Clauius himselfe hath set downe of his owne, touching the quantitie of Angles, and of their sides, as well in right line triangles, as in Sphericall triangles: of which matter, a Monte Regio wrote diffusedlie and at large, so Copernicus wrote of the same brieflie, but therewith somewhat obscurelie, as Clauius saith. Moreover, in reading the Geometrie of Albertus Durcrus, that excellent painter, and finding manie of his conclusions verie obscurelie interpreted by his Latine interpreter (for he himselfe wrote in high Dutch) I requested a friend of mine, whome I knewe to haue spent some time in the studie of the Mathematicals, not onelie plainelie to translate the foresaide Durerus into English, but also to adde thereunto manie necessary propositions of his owne, which my request he hath (I thanke him) verie well perfourmed, not onely to my satisfaction, but also to the great commo∣ditie and profite of all those that desire to bee perfect in Architecture, in the Arte of Painting, in free Masons craft, in Ioyners craft, in Carvers craft, or anie such like Arte commodious and serviceable in any common Wealth, and I hope that he will put the same in print ere it be long, his name I conceale at his owne earnest intreatie, although much against my will, but I hope that he will make himselfe known in the publishing of his Arithmeticke, and the great Arte of Algebra, the one being almost finished, and the other to bee vndertaken at his best leasure, as also in the printing of Durerus, vnto whom he hath added many necessary Geometrical conclusions, not heard of heretofore, together with divers other of his workes as wel in Geometrie as as in other of the Mathe∣maticall sciences, if he be not called away from these his studies by other affaires. In the mean time I pray al young Gentlemen and seamen to take these my labours already ended in good part, whereby I seeke neither praise nor glorie, but onely to profite my countrey.

Blundeville obviously prefers the trigonometry of Christoph Clavius over that of Johannes Regiomontanus but is well acquainted with both. More interesting is the fact that he took his geometry from Albertus Durcrus or Durerus, who is obviously Albrecht Dürer and his Underweysung der Messung mit dem Zirkel und Richtscheyt (Instruction in Measurement with Compass and Straightedge, 1525. Blundeville even goes so far as to have an English translation made from the original German (high Dutch!), as he considers the Latin translation defective. 

Title page of Albrecht Dürer’s Underweysung der Messung mit dem Zirkel und Richtscheyt 

The second treatise: 

Item the first principles of Cosmographie, and especi∣ally a plaine treatise of the Spheare, representing the shape of the whole world, together with the chiefest and most necessarie vses of the said Spheare.

The third treatise:

Item a plaine and full description of both the Globes, aswell Terrestriall as Celestiall, and all the chiefest and most necessary vses of the same, in the end whereof are set downe the chiefest vses of the Ephemerides of Iohan∣nes Stadius, and of certaine necessarie Tables therein con∣tained for the better finding out of the true place of the Sunne and Moone, and of all the rest of the Planets vpon the Celestiall Globe.

A plaine description of the two globes of Mercator, that is to say, of the Terrestriall Globe, and of the Celestiall Globe, and of either of them, together with the most necessary vses thereof, and first of the Terrestriall Globe, written by M. Blundeuill. 

This ends with A briefe description of the two great Globes lately set forth first by M. Sanderson, and the by M. Molineux.

The first voyage of Sir Francis Drake by sea vnto the West and East Indies both outward and homeward.

The voyage of M. Candish vntothe West and East Indies, described on the Terrestriall Globe by blew line.

Johannes Stadius’ ephemerides were the first ephemerides based on Copernicus’ De revolutionibus. 

The fourth treatise: 

Item a plaine and full description of Petrus Plancius his vniversall Mappe, lately set forth in the yeare of our Lord 1592. contayning more places newly found, aswell in the East and West Indies, as also towards the North Pole, which no other Map made heretofore hath, whereunto is also added how to find out the true distance betwixt anie two places on the land or sea, their longitudes and la∣titudes being first knowne, and thereby you may correct the skales or Tronkes that be not trulie set downe in anie Map or Carde.

This map was published under the title, Nova et exacta Terrarum Orbis Tabula geographica ac hydrographica. 

Petrus Plancius’ world map from 1594

The fifth treatise: 

Item, A briefe and plaine description of M. Blagraue his Astrolabe, otherwise called the Mathematicall Iewel, shewing the most necessary vses thereof, and meetest for sea men to know.

I wrote about Blagrave and his Mathematical Jewel here

Title Page Source Note the title page illustration is an  armillary sphere and not the Mathematical Jewel

The sixth treatise:

Item the first & chiefest principles of Navigation more plainlie and more orderly taught than they haue bene heretofore by some that haue written thereof, lately col∣lected out of the best modern writers, and treaters of that Arte.

Towards the end of this section, we find the first published account of Edward Wright’s mathematical solution of the construction of the Mercator chart

in the meane time to reforme the saide faults, Mercator hath in his vniuersal carde or Mappe made the spaces of the Parallels of latitude to bée wider euerie one than other from the E∣quinoctiall towards either of the Poles, by what rule I knowe not, vnlesse it be by such a Table, as my friende M. Wright of Caius colledge in Cambridge at my request sent me (I thanke him) not long since for that purpose, which Table with his consent, I haue here plainlie set downe together with the vse thereof as followeth.

The Table followeth on the other side of the leafe.

The first edition was published in 1594 and was obviously a success with a second edition in 1597, a third in 1606, and a fourth in 1613. The eighth and final edition appeared in 1638. Beginning with the second edition two extra treatises were added. The first was his A Briefe Description of Universal Mappes and Cardes. The second, the true order of making Ptolomie his Tables. 

Blundeville’s Exercises contains almost everything that was actual at the end of the sixteenth century in mathematics, cartography, and navigation. 

Blundeville’s final book was The Theoriques of the Seuen Planets written with some assistance from Lancelot Browne (c. 1545–1605) a friend of William Gilbert (c. 1544–1603), and like Gilbert a royal physician, published in 1602:

THE Theoriques of the seuen Planets, shewing all their diuerse motions, and all other Accidents, cal∣led Passions, thereunto belonging. Now more plainly set forth in our mother tongue by M. Blundeuile, than euer they haue been heretofore in any other tongue whatsoeuer, and that with such pleasant demonstratiue figures, as eue∣ry man that hath any skill in Arithmeticke, may easily vnderstand the same. A Booke most necessarie for all Gentlemen that are desirous to be skil∣full in Astronomie, and for all Pilots and Sea-men, or any others that loue to serue the Prince on the Sea, or by the Sea to trauell into forraine Countries.

Whereunto is added by the said Master Blundeuile, a breefe Extract by him made, of Maginus his Theoriques, for the better vnderstanding of the Prutenicall Tables, to calculate thereby the diuerse mo∣tions of the seuen Planets.

There is also hereto added, The making, description, and vse, of two most ingenious and necessarie Instruments for Sea-men, to find out thereby the latitude of any Place vpon the Sea or Land, in the darkest night that is, without the helpe of Sunne, Moone, or Starr. First inuented by M. Doctor Gilbert, a most excellent Philosopher, and one of the ordinarie Physicians to her Maiestie: and now here plainely set downe in our mother tongue by Master Blundeuile.

LONDON, Printed by Adam Islip. 1602.

A short Appendix annexed to the former Treatise by Edward Wright, at the motion of the right Worshipfull M. Doctor Gilbert. 

To the Reader.

Being aduertised by diuers of my good friends, how fauorably it hath pleased the Gentlemen, both of the Court and Country, and specially the Gentlemen of the Innes of Court, to accept of my poore Pamphlets, entituled Blundeuiles Exercises; yea, and that many haue earnestly studied the same, because they plainly teach the first Principles, as well of Geographie as of Astronomie: I thought I could not shew my selfe any way more thankfull vnto them, than by setting forth the Theoriques of the Planets, vvhich I haue collected, partly out of Ptolomey, and partly out of Purbachius, and of his Commentator Reinholdus, also out of Copernicus, but most out of Mestelyn, whom I haue cheefely followed, because his method and order of writing greatly contenteth my humor. I haue also in many things followed Maginus, a later vvriter, vvho came not vnto my hands, before that I had almost ended the first part of my booke, neither should I haue had him at all, if my good friend M. Doctor Browne, one of the ordinarie Physicians to her Maiestie, had not gotten him for me…

It is interesting to note the sources that Blundeville consulted to write what is basically an astronomy-astrology* textbook. He names Ptolemy, Georg von Peuerbach’s Theoricae novae planetarum and Erasmus Reinhold’s commentary on it, Copernicus, but names Michael Mästlin as his primary source. Although Copernicus is a named source, the book is, as one would expect at the juncture, solidly geocentric. *Blundeville never mentions the word astrology in any of his astronomy texts, but it is clear from the contents of his books that they were also written for and expected to be used by astrologers. 

The Theoriques contains an appendix on the use of magnetic declination to determine the height of the pole very much state of the art research.

Because the making and vsing of the foresaid Instrument, for finding the latitude by the declination of the Mag∣neticall Needle, will bee too troublesome for the most part of Sea-men, being notwithstanding a thing most worthie to be put in daily practise, especially by such as vndertake long voyages: it was thought meet by my worshipfull friend M. Doctor Gilbert, that (ac∣cording to M. Blundeuiles earnest request) this Table following should be hereunto adioined; which M. Henry Brigs (professor of Geometrie in Gre∣sham Colledge at London) calculated and made out of the doctrine and ta∣bles of Triangles, according to the Geometricall grounds and reason of this Instrument, appearing in the 7 and 8 Chapter of M. Doctor Gilberts fift booke of the Loadstone. By helpe of which Table, the Magneticall declination being giuen, the height of the Pole may most easily be found, after this manner.

It is very clear that Thomas Blundeville was a very well connected and integral part of the scientific scene in England at the end of the sixteenth century. An obviously erudite scholar he distilled a wide range of the actual literature on astronomy, cartography, and navigation in popular form into his books making it available to a wide readership. In this endeavour he was obviously very successful as the numerous editions of The Exercises show.

When is an algorithm not an algorithm?

A word previously well known to mathematicians but probably not to the general public, algorithm had begun to seep into the general awareness during the early years of the computer age. As the computer age mutated into the information age, algorithm became one of the buzz words, echoing around the world and seeming to transmute from a piece of vocabulary into a sentient being. Social media became littered with talk of sexist algorithms, racists algorithms, blind algorithms… With the supposed rise of AI, the much vaunted and eagerly sort after, but at the same time feared, artificial intelligence, talk turned to the search for the elusive intelligent algorithm. In little more than the seventy years since the Second World War the word algorithm has come to occupy a dominant position in much of the public discourse. 

But what exactly does the word algorithm mean? Where did it come from? What is an algorithm? The word algorithm has an almost thousand-year history and over the centuries its meaning has mutated and evolved and the computer algorithms of today are not the same as the algebraic algorithms of medieval mathematics. Jeffrey M Binder, who describes himself as a programmer, historian, and writer, has written a book, Language and the Rise of the Algorithm,^[1] which tracks those mutations and the evolution of the current meaning of the word algorithm over the time since it first appeared in the early thirteenth century. 

I will start my review by saying that Binder’s book is excellent and if you have any interest in the topic at all then you should definitely read it. It certainly has the potential to become a classic in the tangled field of the histories of mathematics, language, logic, and computer science. 

As Binder points out early in the introduction algebra was introduced into Europe by the Latin translation of Muḥammad ibn Mūsā al-Khwārizmī’s al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah in the twelfth century; its title gave us the word algebra and the mangled transliteration of his name into Latin the term algorithm. Because algebra is the practice of doing mathematics with symbols a large part of Binder’s book is a review over the centuries of how algebra was perceived, understood, and accepted or not. Part of that perception involved the question whether symbolic algebra was a language, so the book also traces the thoughts on the nature of language over the same period. 

Central to Binder’s narrative is his systematic debunking of the commonly held belief that the computer age was heralded by Leibniz with his calculating machine and his attempts to create a calculus ratiocinator to resolve differences of opinion, expressed in the famous quote:

The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right.

To a certain extent the book is divided into three sections pre-Leibniz, Leibniz, post-Leibniz. A finer division is presented in the fact that the book takes the reader chronologically through the history of symbolic mathematics, and the evolution of symbolic logic out of it from the sixteenth century down to the present. Throughout this journey Binder shows how the various actors used and or defined the concept of the algorithm and how the term took on differently meanings in different contexts. He shows how the term algorithm, that today non-experts seem to consider has always meant roughly the same, has actually been a linguistic chameleon taking on many different meanings over the last eight hundred years.

Binder packs far too much detailed information and analysis into each chapter of his book for me to attempt a detailed chapter by chapter review. To do so I would probably end up writing a review as long as the book itself. I can’t see anything of real relevance that Binder has left out of his account despite the fact that his book is hardly more that two hundred pages long. I will, however, give brief outlines of the five chapters and coda.

The opening chapter is a whirlwind tour of the introduction of both the Hindu-Arabic number system and algebra in the medieval and early modern periods, starting with Brahmagupta in the sixth century and ending with Descartes’ unification of algebra in the seventeenth century. The Hindu-Arabic number system because as Binder correctly points out algorithm, usually then spelt algorism, was the name for the rules governing the use of this new arithmetic. Despite its brevity this tour is excellently done. 

The second chapter starts in the seventeenth century, enter Leibniz and his attempt to create a universal symbolic language that translates natural language. This chapter looks not only at Leibniz’s thoughts on language, both symbolic and natural, but at this of other protagonists of the so-called scientific revolution, in particular John Wilkins but also George Delgarno, John Ray, Descartes, Locke et al. It also covers the discussion amongst the mathematicians of the use of symbolism in the newly created calculus of Leibniz and Newton. 

Moving into the eighteenth century, the third chapter centres on the thoughts on language, algebra, and symbolism of Marie Jean Antoine Nicolas Caritat, Marquis de Condercet. As in the previous chapter there is a list of significant contributors to the debates on these topics such as Locke, Euler, Jean Le Rond d’Alembert, and the Abbé de Condillac. A central question that is discussed by these participants, is algebra a language? Once more Binder covers a complex of thoughts and ideas briefly but comprehensively and clearly. There follows an account of the thoughts of the English mathematician, Francis Maseres, who firmly rejected the modern, continental thoughts on the relationship between algebra and language. I found particularly interesting that at this late date Maseres still had problems with the acceptance of negative numbers. For me surprisingly, this view was shared by his associate the political radical, Willian Frend, after all Frend’s eldest daughter Sophia Elizabeth married Augustus De Morgan. This is followed by a highly informative essay on an English ally of Condorcet, Charles Mohon, third Earl of Stanhope, the creator of the Stanhope Demonstrator, a logic machine. The chapter closes with rumination on Immanuel Kant and how he fits or doesn’t into these ongoing debates. 

Throughout these chapters Binder draws the readers attention to the varying and various ways that the proponents in the diverse debates used and defined both implicitly and explicitly the term algorithm. 

In terms of symbolic systems, language, and logic, the nineteenth century saw the dawning of a new age that Binder takes us through in his fourth chapter. At the centre of this new perception is George Boole and his algebraic logic. Boole divorces his logical symbols from natural language. The symbols are no longer defined in terms of an interpretation in natural language but instead through the rules of the system for their use.  They don’t not have a single fixed linguistic interpretation but can be used to stand for many different things. Binder’s presentation of Boole’s logic and his motivations for creating it is excellent. Although Boole separated logic and natural language Binder points out that this development ran parallel to the new theories on the genesis of language developed by the Romantics. 

The introduction to Boole is followed by an essay on the calculating wonder Zerah Colburn and the question as to whether the methods he used could be generalised as algorithms for others to learn. This is followed by the work on symbolic mathematics produced by the Cambridge Analytical Society, in particular the algebra of George Peacock, leading into a wide ranging examination of the thoughts of many other nineteenth century thinkers including Józef Wroński, John Venn, William Wordsworth, Maria Edgeworth, Mary Everest Boole, Samuel Taylor Coleridge, Augustus De Morgan, William Stanley Jevons, Ernst Schröder, Gottlob Frege, and others through which Binder weaves the thoughts and concepts of Boole. 

Chapter five takes us into the twentieth century and finally into the age of the computer. Before the arrival of the computer, we have the meta-logical/meta-mathematical theories of Church, Post, and Turing setting the formal limits on what can and what cannot be calculated or computed. It is nice to see Post getting the credit that is due to him for once, he so often gets overlooked. Of course, Gödel gets a look in as do Andrey Markov, and Stephen Kleene. The latter two with varying definitions of the algorithm. There is a long discourse on the meta-logical and philosophical debates in mathematics and logic kicked off by Whitehead and Russell, with their Principia, involving Carnap, Turing, Wittgenstein, Church et all. 

Near the end of this section is my favourite paragraph in the whole book, because of my perpetual war with the “Alan Turing invented the computer blockheads.”

That one must think of computing machines in terms of either of these models is not self-evident. Babbage had already imagined a programmable computer a century before Church and Turing, and the designers of some early computers, such as Konrad Zuse (1936–38) and the IBM Mark I (1939–43), were initially unaware of their work. John Von Neumann, often held up as the designer of the standard computer architecture, knew Turing personally and was deeply familiar with his paper on the decision problem, but it is not clear that Turing’s imaginary machines had any strong influence on his plan. Yet Church and Turing did eventually become common reference points for the discipline of computer science. The most important thing they provided was less a paradigm for the design of actual machines that a theoretical framework for reasoning mathematically about what came to be called “algorithm.” ^[2]

The computer has arrived and with it computer programming languages, at the beginning COBOL, FORTRAN, and ALGOL. We meet Grace Hopper, one of the later creators of COBOL, and her invention of the compiler–a program that automatically translates a human-readable sequence of instruction into a machine-executable form^[3]. This presaged a minor program language war in the early days that Binder outlines. Some wanted to make programming languages more symbolic, mathematical, and strictly formal to avoid the pitfalls made obvious by the meta-logical results of Church, Turing et al. Other aware of the potential market for computer use by non-mathematicians and non-logicians wanted to write programs in more normal languages to make them accessible to these potential users. Binder points to the emergence of the Apple computer as a vindication for the user friendly party in this dispute. 

Binder now considers the story of ALGOL, usually explained as algorithmic language although the original name was International Algebraic Language, which he sees “as a major factor in securing the widespread adoption of algorithm as a general term for computational procedures.”^[4] The aim of ALGOL was it seems to produce a universal language for describing algorithms. The aims and failures of the ALGOL program are discussed in quite a lot of detail in comparison to other approaches to programming, leading into a wider discussion of approaches to creating computer algorithms.

Binder opens his introduction with the following paragraph:

In May 2020, as much of the world focused on the COVID-19 pandemic and as racial justice protests took place across the United States, a technical development sparked excitement and fear in narrow circles. A computer program called GPT-3, developed by the OpenAI company, produced some of the best computer-generated imitations of human writing yet seen: fake news articles that were, according to the authors, able to fool human readers nearly half the time, and poems in the style of Wallace Stevens.^[5]

His journey through the history of the algorithm ends with a twenty-one-page coda, The Age of Arbitrariness, which deals with the newly emerging age of machine learning and the associated change in the meaning of algorithm. 

If classical algorithms are divided from human understanding, they are also divided from data.

[…]

Machine learning (ML) changes this. The “algorithms” are no longer designed by engineers but instead tuned by machines based on large amounts of data.^[6]

Binder closes out his book ruminating on this difference.

Binder has obviously invested an enormous amount of research in his book, a fact that is reflected in the 991 endnotes from just 225 pages of text, most of which refer to the 35-page bibliography of books and papers he consulted. The book also has a good index.

I have only two very minor negative comments on this excellent book. At one point Binder refers to “abacus and counting board” as if they were two separate things. The counting board is an abacus, Binder obviously doesn’t know that the wire and bead frame abacus that people now think of when they read the word abacus didn’t enter Europe until the early eighteenth century well after the use of the counting board had ceased to be used in everyday calculating. My other problem is that he twice refers to Martin Davis as a popular science writer when referencing one of Davis’ popular books, The Universal Computer: The Road from Leibniz to Turing (3^rded. CRC Press, 2018). Given the subject of his own research, Binder really should know that Martin Davis was one of the 20^th centuries leading meta-mathematicians/meta-logicians!

In his journey from al-Khwārizmī to GPT-3 Binder covers an incredible amount of complex material in a comparatively small number of pages. However, his writing is never cluttered or in anyway incomprehensible, it is always clear, lucid, and easy to follow and somewhat surprisingly, given to topic, actually a pleasure to read. As I said above, it certainly has the potential to become a classic in the tangled field of the histories of mathematics, language, logic, and computer science and I very much think it deserves to do so.

---

^[1] Jeffrey M. Binder, Language and the Rise of the Algorithm, University of Chicago Press, Chicago and London, 2022

^[2] Binder p. 176

^[3] Binder p. 177

^[4] Binder p. 179

^[5] Binder p. 1

^[6] Binder p. 205