In the last episode I outlined those aspects of Aristotle’s philosophy that would go on to play a significant role of the history of physics in later centuries. Because Aristotelian philosophy came to play such a central role in medieval thought in the High Middle Ages, there is a strong tendency to think that it also dominated the philosophical scene throughout antiquity. However, this was not the case. Although his school, the Lyceum, survived his death in 332 BCE and under the leadership of Theophrastus (c. 371–c. 287 BCE) enjoyed a good reputation. Theophrastus produced some good science, but it was natural history–biology, botany–not physics as indeed the majority of Aristotle’s scientific work had been. After Theophrastus died the Lyceum went into decline. However, the school was in no way dominant, the main philosophy of Ancient Greek following Aristotle being Stoicism and Epicureanism. Both of these philosophical directions “shared the aim of attaining tranquillity or freedom from worry (ataraxia), yet their explanations of the world were often radically different, even opposed.”[1]
Epicurus (341–270) had a rather negative attitude towards natural philosophy:
Epicurus was content for his followers to have an ‘in principle’ approach to understanding the world. He explained that it is not necessary to become too caught up in details. In fact, he argues that less is more, because knowing less can prevent confusion and worry. In Epicurus’ view, there is a distinct possibility of being blinded by science, and this is to be avoided. While Epicurus indicates that he had himself worked out the details of his views more fully, he advocates that a summary is sufficient. A physical theory that can withstand objections and leads to peace of mind should be accepted.[2]
Portrait of Epicurus, founder of the Epicurean school. Roman copy after a lost Hellenistic original. Source: Wikimedia Commons
Epicurean natural philosophy was retrograde, it rejected the developments made by Empedocles, Plato, Eudoxus, and Aristotle and returned to the views of the Atomists. There existed only bodies and space and the bodies were composed of atoms. Because space, the void was infinite and had no centre they rejected Aristotle’s arguments for a spherical earth at the centre of a spherical cosmos and supported a flat earth floating in the void. As with so many of these figures, Epicurus supposedly wrote a large number of books of which only a few short works survive. However, the Roman poet and philosopher, Lucretius (c. 99–c. 55 BCE) presented the Epicurean philosophy and physical theory in his poem, De rerum natura (On the Nature ofThings), This would go on to play a role in the introduction of a particle theory of matter in the Early Modern Period.
Opening of Pope Sixtus IV’s 1483 manuscript of De rerum natura, scribed by Girolamo di Matteo de Tauris Source:Wikimedia Commons
Unlike Epicureanism, Stoicism did not have a single direction determining founder, although Zeno of Citium (c. 334–c. 226) BCE, is regarded as the first Stoic.
Zeno of Citium. Bust in the Farnese collection, Naples. Photo by Paolo Monti, 1969. Source: Wikimedia Commons
He developed his ideas out of the philosophy of the Cynics being, for a time, a pupil of Crates of Thebes (c. 365–c. 285 BCE). However, equally important in the early phase of Stoicism were, Zeno’s pupil Cleanthes of Assos (c. 330–c. 230 BCE) and his pupil Chrysippus of Soli (C. 279–c. 206 BCE).
Chrysippos of Soli, third founder of Stoicism. Marble, Roman copy after a lost Hellenistic original of the late 3rd century BC. Source: Wikimedia Commons
As with almost all major Greek philosophers in this period, according to hearsay, they all wrote lots of books but only fragments of their actual writings have survived. Most of the reports on what they actually believed and practiced come from writers active centuries after they lived. As opposed to Epicurus and Epicureanism, Stoic philosophy is thought to be the result of a collective of writers rather than one dominant individual.
The Stoic philosophy has three major pillars logic, physics (natural philosophy), and ethics. As this series is about the history of the evolution of physics I shall ignore the ethics, although it was the ethics that made, and still makes, Stoicism attractive to many people. The Stoics, like Aristotle, placed a strong emphasis on logic but whereas Aristotle laid his emphasis on logical deductive reasoning using the syllogism, which is a logic of classes, the Stoics are credited with developing the earliest European logic of proposition or predicative logic. This development is usually attributed to Chrysippus but we have little or nothing of his original texts and rely on later reports by Diogenes Laëtius (fl. 3rd century CE), Sextus Empiricus (2nd century CE), Galen (129–c. 216 CE), Aulus Gellius (c. 125–after 180 CE), Alexander of Aphrodisias (fl. 200 CE), and Cicero (106–43 BCE).
Stoic physic is interesting both for its similarities with and differences to Aristotle. Unlike Epicurus, the Stoics accepted the basic Platonic-Aristotelian model of the cosmos as a sphere with a spherical Earth at its centre. They were in fact mainly responsible for the acceptance of this model by the Roman. They also largely accepted the existing astronomical model of the planets revolving around the Earth on circular orbits. However, their cosmology had a major difference to that of Aristotle that would become highly significant when Stoicism was revived in the Early Modern Period.
The Stoics were pandeists, that is they believed that god was the cosmos and everything in it and the cosmos and everything in it was god. As a result, their matter theory was radically different to that of both Plato and Aristotle. To quote Wikipedia:
According to the Stoics, the Universe is a material reasoning substance (logos), which was divided into two classes: the active and the passive. The passive substance is matter, which “lies sluggish, a substance ready for any use, but sure to remain unemployed if no one sets it in motion.” The active substance is an intelligent aether or primordial fire (pneuma) which acts on the passive matter:
The universe itself is God and the universal outpouring of its soul; it is this same world’s guiding principle, operating in mind and reason, together with the common nature of things and the totality that embraces all existence; then the foreordained might and necessity of the future; then fire and the principle of aether; then those elements whose natural state is one of flux and transition, such as water, earth, and air; then the sun, the moon, the stars; and the universal existence in which all things are contained.
— Chrysippus, in Cicero, De Natura Deorum, i. 39
For their cosmology, a major consequence of this philosophy was that the Stoics rejected Aristotle’s division of the cosmos into the supralunar and sublunar spheres, they were both the same for the Stoics, and they considered comets, which as I said in the last episode played a major role in the emergence of modern astronomy, to be a supralunar phenomenon as opposed to Aristotle who regarded them as a meteorological or atmospheric phenomenon.
Both Epicureanism and Stoicism were very popular amongst educated Romans, such as Cicero and Seneca (c. 54 BCE–c. 39 CE), and the later Greek philosopher Plutarch (c. 46–119 BCE), who quoted and discussed Epicurean and Stoic philosophers in their writings. They remained so until about 300 CE when they in turn faded into the background and were replaced by the Neoplatonist. However, those writers who paid the most attention to them were those Latin stylists who were most popular amongst the Humanist philosophers of the Renaissance and so their ideas experienced a revival in the Early Modern Period and there had an impact on the evolution of science.
[1] Liba Taub, Ancient Greek and Roman Science: A Very Short Introduction, OUP, 2023 p.
There is very little doubt that Aristotle (384–§22 BCE) is the predominant figure in the narrative of the history of European science in the twenty-two centuries from 400 BCE to 1800 CE, and even after that he remains a central figure in the discourse. The general literature on Aristotle as a philosopher would fill a sizable library and the specific literature on his contributions to the history of science would fill a whole wing of that library. In this post, I will be limiting myself to a brief description of those aspects of his philosophy that had an impact on the history of physics.
As already explained in the first episode of this series, although Aristotle gave us the word physics in his book title τὰ φυσικά (ta physika), what he means with this is very different from what is meant by the modern use of the term physics. As this series is actually about the evolution of modern physics in the early modern period, I shall here only deal with various aspects of Aristotle’s philosophy that relate to that evolution and they are by no means confined to his τὰ φυσικά (ta physika).
Before I start, a brief biographical note on Aristotle. Born in Stagira in Northern Greece, the son of the prominent physician Nicomachus (fl. c. 375 BCE), who died whilst he was still young, he was brought up by a guardian. In his late teens he joined Plato’s Academy in Athens, where he remained until the death of Plato in 347/348 BCE. Around 343 BCE he left Athens at the request of Philip II of Macedon to become the tutor to his son Alexander (356–323 BCE), the future Alexander III of Macedon, better known as Alexander the Great. Around 340 BCE he returned to Athens and set up his own school of philosophy, the Lyceum. Having, for so long, been a pupil of Plato much of his philosophy was formed either by acceptance or rejection of Plato’s teaching.
Bust of Aristotle. Marble, Roman copy after a Greek bronze original by Lysippos from 330 BC; the alabaster mantle is a modern addition. Source: Wikimedia Commons
One major area of acceptance, that would have consequences up to the seventeenth century, was his adoption of the cosmological and astronomical theories of Plato and Eudoxus, together with the four element theory, originally expounded by Empedocles. Aristotle took over the basic two sphere model from Plato. The cosmos is a sphere, which would become the sphere of the stars in astronomy, and the Earth is a sphere at its centre. Unlike Plato, Aristotle explains why the Earth is a sphere and also gives empirical evidence to show that it is truly a sphere. He argued that if something with gravitas falls then it falls towards the centre of the cosmos until it stops. The natural consequence of everything falling towards a centre from all direction is a sphere. This fall was not due to a force but to the tendency for objects to return to their natural or proper place. So, objects that have gravitas i.e., those predominantly consisting of earth or water, return to the Earth. Those with levitas, consisting predominantly of air or fire rise up into the atmosphere.
Of course, he has to explain why the celestial objects don’t fall to the Earth. Aristotle divides the cosmos into two zones, everything below the orbit of the Moon i.e., sublunar and everything above the orbit of the Moon i.e., supralunar. The sublunar zone consists of the four elements and is subject to change and corruption, whereas the supralunar zone consists of a fifth element, the aether or quintessence, which is unchanging and incorruptible. Natural motion in the supralunar zone is, once again following Plato and Empedocles, uniform circular motion.
Unlike his predecessors, Aristotle provides empirical arguments to demonstrate that the Earth is actually a sphere, to quote James Hannam:
Aristotle concluded his discission by showing how the theory of the Globe explained observations that might seem otherwise inexplicable. In the first place, he said that when there is a lunar eclipse the shadow of the Earth on the Moon is always curved. This corroborates what he had already shown from his first principles. The umbra during a lunar eclipse follows from the shape of the Earth. If it is a ball, its shadow must always be an arc.
His second piece of empirical evidence is the way the visible stars change as we travel north or south. He noted that some stars, which are visible in Egypt and Cyprus, can’t be seen in the north. He is almost certainly referring to Eudoxus’ observations of Canopus. It is bright enough to be hard to miss in Egypt, albeit usually low in the sky. Its absence from view in Athens would have been obvious to anyone who had seen it further south. This is only explicable if the Earth is rounded, if it were a flat plane, everyone would see the same stars. Since it is spherical, it’s inevitable that our view of the heavens will change with latitude.[1]
Hannam closes his section on Aristotle with the following:
By any conventional standards he [Aristotle] knew the Earth was a sphere, and he was probably the first person who did. On that basis, he discovered the theory of the Globe. As we will see in the remainder of this book, everyone today who knows the Earth is round indirectly learnt it from Aristotle. This makes the Globe the greatest scientific achievement of antiquity. It’s only because we take it as obvious that we don’t give Aristotle the credit he deserves.[2]
For the planets Aristotle takes over the concentric or homocentric spheres of Eudoxus and Callippus but adds more spheres filling out the spaces between the planets making a complete set of spheres within spheres from the Moon to the stars. All motion within the heavens is driven by a sort of friction drive by the outer most sphere. This in turn is driven by an unmoved mover, a concept that appealed to the Church in the medieval period, who simply assumed that the unmoved mover was God. Although more scientific in his explanations than any of his predecessors, Aristotle can, at times, also be totally metaphysical. What motivates the unmoved mover? The spheres have souls, and it is the love of those souls for the unmoved mover that motivates it. The origin of the phrase, “love makes the world go round.”
As is generally well known, having defined fall as natural motion, Aristotle now goes on to elucidate his laws of fall, which, of course, everybody knows were wrong being first brilliantly corrected by Galileo in the seventeenth century. Firstly, Aristotle’s laws of fall are not as wrong as people think, and secondly, they were, as we shall see in later episodes, challenged and corrected much earlier than Galileo.
Aristotle’s laws of fall are actually based on simple everyday empirical observation. If I drop a lead ball from an oak tree it evidently falls to the ground faster than an acorn that I dislodge whilst dropping the ball. In real life not all objects fall at the same speed. It is only in a vacuum that this is the case. People tend to ignore the all-important “vacuum” when praising Galileo’s enthronement of Aristotle’s laws of fall. Naturally if I drop a two lead balls of different weights, they do fall at approximately the same speed but even here the heavier ball will hit the ground a split second earlier than the lighter one.
Aristotle argued that the rate of fall was directly proportional to the weight of the falling object and indirectly proportional to the resistance of the medium through which it falls.
Aristotle’s laws of motion. In On the Heavens he states that objects fall at a speed proportional to their weight and inversely proportional to the density of the fluid they are immersed in. This is a correct approximation for objects in Earth’s gravitational field moving in air or water. Source: Wikimedia Commons
This is a good first approximation for objects on the Earth falling through air or water. Having established this Aristotle then argued that the void (a vacuum) could not exist because in the void a falling object would accelerate to infinity and that was an absurdity. Interestingly he also argues that in a vacuum all objects would fall at the same speed, an absurdity! Galileo anyone? It is quite common to express his laws of fall either symbolically or even mathematically, but Aristotle never did either.
As already said, although Aristotle gave us the word physics, he uses it in a very different way. For Aristotle physics is the study of natural things, which he sees as the study of the general principles of change. Change is for Aristotle universal, plants grow and then die, there is quantitative change with respect to size and number and so forth. Most important from our point of view is that he considers motion to be change of place.
In Aristotle’s theories of motion, having dealt with natural motion he now had to define and deal with unnatural motion. Of course, there was only natural motion in the uncorruptible supralunar area. On Earth beyond natural motion there was voluntary motion and unnatural motion. Voluntary motion is such as animals moving and need not concern us here. Unnatural motion requires a cause, and it is here that Aristotle’s whole theory of motion ran into difficulties.
If I have a horse and cart or I push a wheelbarrow, then the cart only moves if the horse pulls and the wheelbarrow only moves if I push. If the horse stops pulling or if I stop pushing then the motion stops, no real problems here, although it is difficult to fit this type of motion into the laws of motion that applies to the falling object. Aristotle’s real problems start with projectile motion. If I fire an arrow with a bow or throw a ball, why does the arrow after it has left the bow string or the ball after it has left my hand continue to fly through the air? There is now apparently nothing propelling the arrow or ball. Aristotle’s escape from this impasse is, to say the least, dodgy. He argued that the air displaced by the flying object rushed around to the back and pushed it further along its course. This weak point in his theory was exploited comparatively early by his critics, i.e., long before the seventeenth century.
Aristotle rejected atomism arguing there was no limit to how far one could divide something, so no smallest particles, atoms. His own theory of matter was that there is primal material. Objects consist of two things material and form. This is important because it plays a role in his fourfold theory of cause that dominated his whole philosophy of nature.
According to Aristotle everything in nature has four causes:
The Material Cause: The material out of which it is composed.
The Formal Cause: The pattern or form that makes the material into a particular type of thing.
The Efficient Cause: In general that which brings an object about
The Final Cause: The purpose for the existence of the object in question
The four causes also apply to abstract concepts such as motion, each motion has a material, a formal, an efficient, and a final cause.
Aristotle argued by analogy with woodwork that a thing takes its form from four causes: in the case of a table, the wood used (material cause), its design (formal cause), the tools and techniques used (efficient cause), and its decorative or practical purpose (final cause). Source: Wikimedia Commons
Today, we regard the final cause, for which the technical term is teleology, as bizarre. Since at least the nineteenth century it is not thought that most things have an intrinsic purpose for their existence, they just exist. However, in the Middle Ages, the high point of Aristotelian thought in science, it would have chimed with Christian thought, “everything has a place in God’s great plan.
Introducing Aristotle’s four causes takes us along to, perhaps Aristotle’s greatest contribution to the development of science his methodology and his epistemology, i.e., his theory of knowledge. In six works, collectively known as the Organon, he laid out the earliest known introduction to formal logic. How do I argue correctly, so that I transport truth from my premises to my conclusions. Our understanding of logic and the logic that we use have evolved since Aristotle, but logic still lies at the heart of all formal scientific proofs. Stealing from Wikipedia the six Aristotelian works on logic are:
The Categories (Latin: Categoriae) introduces Aristotle’s 10-fold classification of that which exists: substance, quantity, quality, relation, place, time, situation, condition, action, and passion.
On Interpretation (Latin: De Interpretatione) introduces Aristotle’s conception of proposition and judgement, and the various relations between affirmative, negative, universal, and particular propositions.
The Prior Analytics (Latin: Analytica Priora) introduces his syllogistic method argues for its correctness, and discusses inductive inference.
The Posterior Analytics (Latin: Analytica Posteriora) deals with definition, demonstration, inductive reasoning, and scientific knowledge.
The Topics (Latin: Topica) treats issues in constructing valid arguments, and inference that is probable, rather than certain. It is in this treatise that Aristotle mentions the Predicables, later discussed by Porphy and the scholastic logicians.
The On Sophistical Refutations (Latin: De Sophisticis Elenchis) gives a treatment of logical fallacies, and provides a key link to Aristotle’s tractate on rhetoric.
Added 17 August
Although not really clearly spelt out, Aristotle propagated an axiomatic deductive system for securing knowledge. Starting from self-evident premises that require no proof one uses a chain of deductive logic until one arrives at empirically observed facts. Although we would regard his premise that the Earth is a sphere because all falling objects fall to the centre of the universe as self-evident, this is the form of argument, sketched above, he uses to demonstrate that the Earth is really a sphere.
It is important to note, for the evolution of scientific thought in Europe throughout the centuries after Aristotle, that when applied to nature he didn’t regard mathematical proofs as valid. He argued that the objects of mathematics were not natural and so could not be applied to nature. He did however allow mathematics in what were termed the mixed sciences, astronomy, statics, and optics. For Aristotle mathematical astronomy merely delivered empirical information on the position of the celestial bodies. Their true nature was, however, delivered by non-mathematical cosmology. I shall deal with statics and optics separately.
In recent times, various voices have claimed that the adherence to Aristotle’s vision of science hindered the evolution of the discipline. It is a similar claim to that of the gnu atheists that Christianity blocked the evolution of science. In the case of Aristotle, I think we should bear in mind that in antiquity his popularity waned fairly quickly after his death, and he was superceded by the Stoics and then the Epicureans as the flavour of the century in philosophy and although this included the period of the greatest Greek mathematicians Archimedes (c. 287–c.212 BCE) Apollonius of Perga (c. 240–c. 190 BCE), who both made significant advances in both pure and applied mathematics, it cannot be said that the world advanced significantly towards modern science.
[1] James Hannam, The Globe: How the Earth Became Round, Reaktion Books, London, 2023 p. 93
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, 1st 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), 1st 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.
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.
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 Ageof 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 Roadfrom Leibniz to Turing (3rded. CRC Press, 2018). Given the subject of his own research, Binder really should know that Martin Davis was one of the 20th 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
As I have mentioned more than once on this blog, I served my apprenticeship as a historian of science working for ten years in a major research project into the external history of formal or mathematical logic. During the semester, we held a weekly research seminar in which one or more of the members of the project would present a talk on the current state of their research. These seminars were held early evening and afterwards we would all go for a meal at a local Italian restaurant. I think, I probably learnt more through the discussions during those meals than through any other part of my life as a student.
From time to time those research seminars would be graced by a guest lecture by visiting historians. Over the years I got to hear lectures by many of the world’s leading historians of logic and mathematics. More important was being able to talk informally with these luminaries of the discipline during those post seminar meals.
Martin Davis
One of those guest lecturers was Martin Davis, not only one of the best historians of twentieth century meta-logic but also a world class logician in his own right, who died 1 January. He held an excellent lecture on the American logician, Emil Post (1897–1954), who published a paper in 1936 giving an almost identical solution to the Entscheidungsproblem as Turing.
It was the meal after the lecture that we will forever remain in my memory. Martin was travelling through Europe with his wife and the two of them were incredibly friendly and delightful diner companions.
Martin & Virginia Davis
Two things are particularly present in my mind. The first is being in full flow in my inimitable style answering a question that Martin’s wife Virginia had posed about something in logic, when I suddenly realised that I, a mere student, albeit a mature one, was sitting between two of the world’s leading historians of logic, who were listening intently to what I was saying. Feeling somewhat more than flustered, I somehow managed to finish what I was saying. Nobody said anything negative.
The other was an incredible display of generosity from Martin. Amongst his publications, was a book that he edited called The Undecidable, a collection of the original papers on the topic from Post, Turing, Gödel et al. During the course of the meal, I asked Matin if there were plans to republish it as it was out of print, and I couldn’t get hold of a copy. He asked what I was doing after the meal and if I had time to accompany him back to his hotel. I said yes. When we got there, he gave me his personal copy of The Undecidable. I was mind blown.
The book has now been republished
Many years later I gave it to my professor, Christian Thiel, who has a very impressive collection of logic books. I was sad when I read of Martin’s death yesterday, remembering a very kind and friendly man, who once did a student a very generous favour.
The is an old saying in English, “if you’ve got an itch scratch it!” A medically more correct piece of advice is offered, usually by mothers in a loud stern voice, “Don’t scratch!” I have had an itch since the start of December and have been manfully trying to heed the wise words of mother but have finally cracked and am going to have a bloody good scratch.
I actually don’t wish to dump on Lady Science, which I regard as a usually excellent website promoting the role of women in science, particularly in the history of science but the essay,Before Lovelace, that they posted 3 December 2020 is so full of errors concerning Ada Lovelace and Charles Babbage that I simply cannot ignore it. In and of itself the main point that the concept of the algorithm exists in many fields and did so long before the invention of the computer is interesting and of course correct. In fact, it is a trivial point, that is trivial in the sense of simple and obvious. An algorithm is just a finite, step by step procedure to complete a task or solve a problem, a recipe!
My objections concern the wildly inaccurate claims about the respective roles of Charles Babbage and Ada Lovelace in the story of the Analytical Engine. Let us examine those claims, the essay opens at follows:
Charles Babbage and Ada Lovelace loom large in the history of computing. These famous 19th-century figures are consistently cited as the origin points for the modern day computer: Babbage hailed as the “father of computing” and Lovelace as the “first computer programmer” Babbage was a mathematician, inventor, and engineer, famous for his lavish parties and his curmudgeonly attitude. Lady Augusta Ada King, Countess of Lovelace was a mathematician and scientist, introduced to Babbage when she was a teenager. The two developed a long professional relationship, which included their collaborative work on a machine called the Analytical Engine, a design for the first mechanical, programmable computer.
They might be cited as the origin points of the modern-day computer, but such claims are historically wrong. For all of Babbage’s ingenuity in the design and conception of his mechanical, programmable calculating machines they played absolutely no role in and had no influence on the later development of the computer in the twentieth century. They were and remain an interesting historical anomaly. Regular readers of this blog will know that I reject the use of the expression “the father of” for anything in #histSTM and that for very good reasons. They will also know that I reject Ada Lovelace being called the “first computer programmer” for the very simple reason that she wasn’t. (See addendum below) I am of the opinion that Ada Lovelace was not a mathematician in any meaningful sense of the word, and she was in absolutely no way a scientist. Ada Lovelace and Charles Babbage did not have a long professional relationship and did not collaborate on the design of the Analytical Engine, which was entirely the work of Charles Babbage alone, and in which Ada Lovelace played absolutely no part. Assigning co-authorship and co-development to Ada Lovelace for Babbage’s work is no different to saying that a journalist, who interviews a scientist about his research work and then write a puff piece about it, is the scientist’s co-researcher! The train-wreck continues:
Much of what we know about the Analytical Engine comes from Lovelace’s paper on the machine. In 1842, she published” A Sketch of the Analytical Engine, with notes by the Translator”,” a translation of an earlier article by mathematician Luigi Menabrea. Lovelace’s English translation of Menabrea’s article included her own extended appendix in which she elaborated on the machine’s design and proposed several early computer programs. Her notes were instrumental for Alan Turing’s work on the first modern computer in the 1930s. His work would later provide the basis for the Colossus computer, the world’s first large-scale programmable, electronic, digital computer, developed to assist with cryptography work during World War II. Machines like the Colossus were the precursors to the computers we carry around today in our pockets and our backpacks.
We actually know far more about the Analytical Engine from Babbage’s biography (see footnote 1) and his own extensive papers on it, which were collected and published by his son Henry, Babbage’s Calculating Engines: Being a Collection of Papers Relating to Them; Their History and Construction, Charles Babbage, Edited by Henry P. Babbage, CUP, 1889. The notes to the translation, which the author calls an appendix, we know to have been co-authored by Babbage and Lovelace and not as here stated written by Lovelace alone. There is only one computer program in the notes and that we know to have been written by Babbage and not Lovelace. (See addendum below) Her notes played absolutely no role whatsoever in Turing’s work in the 1930s, which was not on the first modern computer but on a problem in metamathematics, known as the Entscheidungsproblem (English: decision problem). Turing discussed one part of the notes in his paper on artificial intelligence, Computing Machinery and Intelligence, (Mind, October 1950). Turing’s 1930s work had nothing to do with the design of the Colossus, although his work on the use of probability in cryptoanalysis did. Colossus was designed and built by Tommy Flowers, who generally gets far too little credit for his pioneering work in computers. The Colossus played no role in the future development of computers because the British government dismantled or hid all of the Colossus computers from Bletchley Park after the war and closed access to the information on the Colossus for thirty years under the official secret act. We are not done yet:
With Babbage and Lovelace’s work as the foundation and the Turing Machine as the next step toward what we now think of as computers…
Babbage’s work, not Babbage’s and Lovelace’s, was not, as already stated above, the foundation and the Turing Machine was very definitely not the next step towards what we now think of as the computer. I really do wish that people would take the trouble to find out what a Turing Machine really is. It’s an abstract metamathematical concept that is useful for describing, on an abstract level, how a computer works and for defining the computing power or capabilities of a given computer. It played no role in the development of real computers in the 1940s and wasn’t even referenced in the computer industry before the 1950s at the very earliest. Small tip for future authors, if you are going to write about the history of the computer, it pays to learn something about that history before you start. We are approaching the finish line:
One part of the history of computing that is much less familiar is the role the textile industry played in Babbage and Lovelace’s plans for the Analytical Engine. In a key line from Lovelace’s publication, she observes, “we may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard loom weaves flowers and leaves.” The Jacquard Loom was a mechanical weaving system controlled by a chain of punched cards. The punched cards were fed into the weaving loom and dictated which threads were activated as the machine wove each row. The result was an intricate textile pattern that had been “programmed” by the punch cards.
Impressed by the ingenuity of this automation system, Babbage and Lovelace used punched cards as the processing input for the Analytical Engine. The punched cards, Lovelace explains in her notes, contain “the impress of whatever special function we may desire to develop or to tabulate” using the machine.
Why is it that so many authors use ‘less familiar’ or ‘less well known’ about things that are very well known to those, who take an interest in the given topic? For those, who take an interest in Babbage and his computers, the fact that he borrowed the punch card concept from Jacquard’s mechanical, silk weaving loom is very well known. Once again, I must emphasise, Babbage and not Babbage and Lovelace. He adopted the idea of using punch cards to program the Analytical Engine entirely alone, Ada Lovelace was not in anyway involved in this decision.
Itch now successfully scratched! As, I said at the beginning the rest of the essay makes some interesting points and is well worth a read, but I really do wish she had done some real research before writing the totally crap introduction.
Addendum:
I have pointed out on numerous occasions that it was Babbage, who wrote the program for the Analytical Engine to calculate the Bernoulli numbers, as presented in Note G of the Lovelace memoir. He tells us this himself in his autobiography[1]. I have been called a liar for stating this and also challenged to provide evidence by people to lazy to check for themselves, so here are his own words in black and white (16-bit grayscale actually)
[1] Charles Babbage, Passages from the Life of a Philosopher, Longman, Green, Longman, Roberts, Green, London, 1864, p. 136
One of the things that I have been reading recently is a very interesting paper by John N. Crossley, the Anglo-Australian logician and historian of mathematics, about the reception and adoption of the Hindu-Arabic numbers in medieval Europe.[1]Here I came across this wonderful footnote:[2]
[…]
It is interesting to note that Richard Lemay in his entry “Arabic Numerals,” in Joseph Reese Strayer, ed., Dictionary of the Middle Ages(New York, 1982–89) 1:382–98, at 398 reports that in the University of Padua in the mid-fifteenth century, prices of books should be marked “non per cifras sed per literas claras.” He gives a reference to George Gibson Neill Wright, The Writing of Arabic Numerals(London, 1952), 126. Neill Wright in turn gives a reference to a footnote of Susan Cunnigton, The Story of Arithmetic: A Short History of Its Origin and Development(London, 1904), 42, n. 2. She refers to Rouse Ball’s Short History of Mathematics, in fact this work is: Walter William Rouse Ball, A Short Account of the History of Mathematics, 3rded. (London, 1901), and there one finds on p. 192: “…in 1348 the authorities of the university of Padua directed that a list should be kept of books for sale with the prices marked ‘non per cifras sed per literas claras’ [not by cyphers but by clear letters].” I am yet to find an exact reference for this prohibition. (There is none in Rouse Ball.) Chrisomalis Numerical Notations, p. 124, cites J. Lennart Berggren, “Medieval Arithmetic: Arabic Texts and European Motivations,” in Word, Image, Number: Communication in the Middle Ages, ed. John J. Contreni and Santa Casciani (Florence, 2002), 351–65, at 361, who does not give a reference.
Here we have Crossley the historian following a trail of quotes, references and footnotes; his hunt doesn’t so much terminate in a dead-end as fizzle out in the void, leaving the reader unsure whether the university of Padua really did insist on its book prices being written in Roman numerals rather than Hindu-Arabic ones or not. What we have here is a succession of authors writing up something from a secondary, tertiary, quaternary source with out bothering to check if the claim it makes is actually true or correct by looking for and going back to the original source, which in this case would have been difficult as the trail peters out by Rouse Ball, who doesn’t give a source at all.
This habit of writing up without checking original sources is unfortunately not confined to this wonderful example investigated by John Crossley but is seemingly a widespread bad habit under historians and others who write historical texts.
I have often commented that I served my apprenticeship as a historian of science in a DFG[3]financed research project on Case Studies into a Social History of Formal Logic under the direction of Professor Christian Thiel. Christian Thiel was inspired to launch this research project by a similar story to the one described by Crossley above.
Christian Thiel’s doctoral thesis was Sinn und Bedeutung in der Logik Gottlob Freges(Sense and Reference in Gottlob Frege’s Logic); a work that lifted him into the elite circle of Frege experts and led him to devote his academic life largely to the study of logic and its history. One of those who corresponded with Frege, and thus attracted Thiel interest, was the German meta-logician Leopold Löwenheim, known to students of logic and meta-logic through the Löwenheim-Skolem theorem or paradox. (Don’t ask!) Being a thorough German scholar, one might even say being pedantic, Thiel wished to know Löwenheim’s dates of birth and death. His date of birth was no problem but his date of death turned out to be less simple. In an encyclopaedia article Thiel came across a reference to c.1940; the assumption being that Löwenheim, being a quarter Jewish and as a result having been dismissed from his position as a school teacher in 1933, had somehow perished during the holocaust. In another encyclopaedia article obviously copied from the first the ‘circa 1940’ had become a ‘died 1940’.
Thiel, being the man he is, was not satisfied with this uncertainty and invested a lot of effort in trying to get more precise details of the cause and date of Löwenheim’s death. The Red Cross information service set up after the Second World War in Germany to help trace people who had died or gone missing during the war proved to be a dead end with no information on Löwenheim. Thiel, however, kept on digging and was very surprised when he finally discovered that Löwenheim had not perished in the holocaust after all but had survived the war and had even gone back to teaching in Berlin in the 1950s, where he died 5. May 1957 almost eighty years old. Thiel then did the same as Crossley, tracing back who had written up from whom and was able to show that Löwenheim’s death had already been assumed to have fallen during WWII, as he was still alive and kicking in Berlin in the early 1950s!
This episode convinced Thiel to set up his research project Case Studies into a Social History of Formal Logic in order, in the first instance to provide solid, verified biographical information on all of the logicians listed in Church’s bibliography of logic volume of the Journal of Symbolic Logic, which we then proceeded to do; a lot of very hard work in the pre-Internet age. Our project, however, was not confined to this biographical work, we also undertook other research into the history of formal logic.
As I said above this habit of writing ‘facts’ up from non-primary sources is unfortunately very widespread in #histSTM, particularly in popular books, which of course sell much better and are much more widely read than academic volumes, although academics are themselves not immune to this bad habit. This is, of course, the primary reason for the continued propagation of the myths of science that notoriously bring out the HISTSCI_HULK in yours truly. For example I’ve lost count of the number of times I’ve read that Galileo’s telescopic discoveries proved the truth of Copernicus’ heliocentric hypothesis. People are basically to lazy to do the legwork and check their claims and facts and are much too prepared to follow the maxim: if X said it and it’s in print, then it must be true!
[1]John N. Crossley, Old-fashioned versus newfangled: Reading and writing numbers, 1200–1500, Studies in medieval and Renaissance History, Vol. 10, 2013, pp.79–109
[2]Crossley p. 92 n. 42
[3]DFG = Deutsche Forschungsgemeinschaft = German Research Foundation
Charles Babbage wrote a sort of autobiography, Passages From The Life of a Philosopher.
One of its meandering chapters is devoted to his ideas about and work on his Analytical Engine. In one section he describes explaining to his friend the Irish physicist and mathematician James MacCullagh (1809–1847), who did important work in optics and was awarded the Royal Society’s Copley Medal in 1842,
James MacCullagh artist unknown Source: Wikimedia Commons
how the Analytical Engine could be fed subroutines to evaluate trigonometrical or logarithmic functions, whilst working on algebraic operations. He goes on to explain that three or four days later Carl Gustav Jacob Jacobi (1804–1851) and Friedrich Wilhelm Bessel (1784–1846), two of Germany’s most important 19th century mathematicians, were visiting and discussing the Analytical Engine when MacCullagh returned and he completed his programming explanation. Which historian of 19th century mathematician wouldn’t give their eyeteeth to listen in on that conversation?
Having dealt with the problem of subroutines for the Analytical Engine Babbage moves on to another of his mathematical acquaintances, he tells us:
In 1840 I received from my friend M. Plana a letter pressing me strongly to visit Turin at the then approaching meeting of Italian Philosophers. In that letter M. Plana stated that he had inquired anxiously of many of my countrymen about the power and mechanism of the Analytical Engine.
Plana was Giovanni Antonio Amedeo Plana (1781–1864) mathematician and astronomer, a pupil of the great Joseph-Louis Lagrange (1736–1813), who was appointed to the chair of astronomy in Turin in 1811.
Giovanni Antonio Amedeo Plana Source: Wikimedia Commons
Plana worked in many fields but was most famous for his work on the motions of the moon for which he was awarded the Copley Medal in 1834 and the Gold Medal of the Royal Astronomical Society in 1840. The meeting to which he had invited Babbage took place in the Turin Accademia delle Scienze. This august society was founded in 1757 by Count Angelo Saluzzo di Monesiglio, the physician Gianfrancesco Cigna and Joseph-Louis Lagrange as a private society. In 1759 it founded its own journal the Miscellanea philosophico mathematica Societatis privatae Taurinensis still in print today as the Memorie della Accademia delle Scienze. In 1783 having acquired an excellent international reputation it became the Reale Accademia delle Scienze, first as the Academy of Science of the Kingdom of Sardinia and later of the Kingdom of Italy. In 1874 it lost this status to the newly reconstituted Accademia dei Lincei in Rome. It still exists as a private academy today.
Rooms of the Turin Accademia delle Scienze
The meeting to which Babbage had been invited to explain his Analytical Engine was the second congress of Italian scientists. Babbage’s invitation in 1840 was thus recognition of his work at the highest international levels within the scientific community.
Babbage did not need to be asked twice, packed up his plans, drawings and descriptions of the Analytical Engine and accompanied by MacCullagh set of for Turin.
This was not just your usual conference sixty-minute lecture with time for questions. Babbage spent several days ensconced in his apartments in Turin with the elite of the Turin scientific and engineering community. Babbage writes, “M. Plana had at first planned to make notes, in order to write an outline of the principles of the engine. But his own laborious pursuits induced him to give up this plan, and to transfer this task to a younger friend of his, M. Menabrea, who had already established his reputation as a profound analyst.”
Luigi Federico Menabrea (1809–1896) studied at the University of Turin and was an engineer and mathematician. A professional soldier he was professor at both the military academy and at the university in Turin. Later in life he entered politics first as a diplomat and then later as a politician serving as a government minister. He served as prime minister of Italy from 1867 to 1869.
Luigi Federico Menabrea Source: Wikimedia Commons
After another lengthy explanation of the programming of the Analytical Engine, Babbage writes:
It was during these meetings that my highly valued friend, M. Menabrea, [in reality Babbage had almost certainly never heard of Menabrea before he met him in Turin] collected the materials for that lucid and admirable description which he subsequently published in the Bibli. Uni. de Genève, t. xli. Oct. 1842.
This is of course the famous document that Ada Lovelace would translate from the original French into English and annotate. Babbage writes of the two documents:
These two memoires taken together furnish, for those who are capable of understanding the reasoning, a complete demonstration—That the whole of the developments and operations of analysis are now capable of being executed by machinery. [emphasis in original]
That he was never able to realise his dreams of the Analytical Engine must have been very bitter for Babbage and now that we can execute the whole of the developments and operations of analysis with machinery, which even a Charles Babbage could not have envisaged in the 19th century, we should take a moment to consider just how extraordinary his vision of an Analytical Engine was.
One of the parlour games played by intellectuals and academic, as well as those who like to think of themselves as such, is which famous historical figures would you invite to a cocktail or dinner party and why. One premise for the game being, which historical figure or figures would you most like to meet and converse with. As a historian of mostly Early Modern science I am a bit wary of this question, as many of the people I study or have studied in depth have very unpleasant sides to their characters, as I have commented in the past in more than one blog post. However in my other guise, as a historian of formal or mathematical logic and the history of the computer there is actually one figure, who I would have been more than pleased to have met and that is the mathematician and engineer, Claude Shannon.
A young Claude Shannon Source: Wikimedia Commons
For those who might not know who Claude Shannon was, he was a man who made two very major contributions to the development of the computers on which I am typing this post and on which you are reading it. The first was when he at the age of twenty-one, in what has been described as the most important master’s thesis written in the twentieth century, combined Boolean algebra with electric circuit design thus rationalising the whole process and simplifying the design of complex circuitry beyond measure. The second was sixteen years later when he in his A Mathematical Theory of Communication, building, it should be added, on the work of others, basically laid the foundations of our so-called information age. His work laid out how to transmit digital signals through circuitry without loss of information. He is regarded as the über-guru of information theory, to quote Wikipedia:
Information theory studies the quantification, storage, and communication of information. It was originally proposed by Claude E. Shannon in 1948 to find fundamental limits on signal processing and communication operations such as data compression, in a landmark paper entitled “A Mathematical Theory of Communication”.
Given that the period we live in is called both the computer age and the information age, it is somewhat surprising that the first full-length biography of Shannon, A Mind at Play,[1] only appeared this year. Having somewhat foolishly said that I would hold a public lecture in November on Vannevar Bush, who was Shannon’s master’s thesis supervisor, and Shannon, I have been reading Soni’s and Goodman’s Shannon biography, which I have to say I enjoyed immensely.
This is a full length, full width biography that covers both the live of the human being as well as the intellectual achievements of the engineer-mathematician. Shannon couldn’t decide which to study as an undergraduate so he did a double BSc in both engineering and mathematics. This dual course of studies is what led to that extraordinary master’s thesis. Having studied Boolean algebra in his maths courses Shannon realised that he could apply it to rationalise and simplify electrical switching when working, as a postgrad, on the switching circuits for Bush’s analogue computer, the differential analyser. It’s one of those things that seems obvious with hindsight but required the right ‘prepared mind’, Shannon’s, to realise it in the first place. It is a mark of his character that he shrugged off any genius on his part in conceiving the idea, claiming that he had just been lucky.
Shannon’s other great contribution, his treatise on communication and information transmission, came out of his work at Bell Labs as a cryptanalyst during World War II. The analysis of language that he developed in order to break down codes led him to a more general consideration of the transmission of information with languages out of which he then laid down the foundations of his theories on communication and information.
Soni’s and Goodman’s and volume deals well with the algebraic calculus for circuit design and I came away with a much clearer picture of a subject about which I already knew quite a lot. However I found myself working really hard on their explanation of Shannon’s information theory but this is largely because it is not the easiest subject in the world to understand.
The rest of the book contains much of interest about the man and his work and I came away with the impression of a fascinating, very deep thinking, modest man who also possessed a, for me, very personable sense of humour. One aspect that appealed to me was that Shannon was a unicyclist and a juggler, who also loved toys, hence the title of my review. As I said at the beginning Claude Shannon is a man I would have liked to have met for a long chat over a cup of tea.
An elder Claude Shannon Source: Wikimedia Commons
On the whole I found the biography well written and light to read, except for the technical details of Shannon information theory, and it contains a fairly large collection of black and white photos detailing all of Shannon’s life. As far as the notes are concerned we have the worst of all possible solutions, hanging endnotes; that is endnotes, with page numbers, to which there is no link or reference in the text. There is an extensive and comprehensive bibliography as well as a good index. This is a biography that I would whole-heartedly recommend to anybody who might be interested in the man or his area of work or both.
[1] Jimmy Soni & Rob Goodman, A Mind at Play: How Claude Shannon Invented the Information Age, Simon & Shuster, New York etc., 2017
This is something that I wrote this morning as a response on the History of Astronomy mailing list; having written it I have decided to cross post it here.
John Briggs is the second person in two days, who has recommended Eric Temple Bell’s “Men of Mathematics”. I can’t remember who the first one was, as I only registered it in passing, and it might not even have been on this particular mailing list. Immediately after John Briggs recommended it Rudi Lindner endorsed that recommendation. This series of recommendations has led me to say something about the role that book played in my own life and my view of it now.
“Men of Mathematics” was the first book on the history of science and/or mathematics that I ever read. I was deeply passionate fan of maths at school and my father gave me Bell’s book to read when I was sixteen years old. My other great passion was history and I had been reading history books since I taught myself to read at the age of three. Here was a book that magically combined my two great passions. I devoured it. Bell has a fluid narrative style and the book is easy to read and very stimulating.
Bell showed me that the calculus, that I had recently fallen in love with, had been invented/discovered (choose the verb that best fits your philosophy of maths), something I had never even considered before. Not only that but it was done independently by two of the greatest names in the history of science, Newton and Leibniz, and that this led to one of the most embittered priority and plagiarism disputes in intellectual history. He introduced me to George Boole, whom I had never heard of before and whose work and its reception in the 19th century I would seriously study many years later in a long-year research project into the history of formal or mathematical logic, my apprenticeship as a historian of science.
Bell’s tome ignited a burning passion for the history of mathematics in my soul, which rapidly developed into a passion for the whole of the history of science; a passion that is still burning brightly fifty years later. So would I join the chorus of those warmly recommending “Men of Mathematics”? No, actually I wouldn’t.
Why, if as I say Bell’s book played such a decisive role in my own development as a historian of mathematics/science, do I reject it now? Bell’s florid narrative writing style is very seductive but it is unfortunately also very misleading. Bell is always more than prepared to sacrifice truth and historical accuracy for a good story. The result is that his potted biographies are hagiographic, mythologizing and historically inaccurate, often to a painful degree. I spent a lot of time and effort unlearning a lot of what I had learnt from Bell. His is exactly the type of sloppy historiography against which I have taken up my crusade on my blog and in my public lectures in my later life. Sorry but, although it inspired me in my youth, I think Bell’s book should be laid to rest and not recommended to new generations.
If your philosophy of [scientific] history claims that the sequence should have been A→B→C, and it is C→A→B, then your philosophy of history is wrong. You have to take the data of history seriously.
John S. Wilkins 30th August 2009
Culture is part of the unholy trinity—culture, chaos, and cock-up—which roam through our versions of history, substituting for traditional theories of causation. – Filipe Fernández–Armesto “Pathfinders: A Global History of Exploration”
The Renaissance Mathematicus 