In the comments on my recent post on books on the history of maths Fernando Q. Gouvêa jumped in to draw attention to the book Math Through the Ages: A Gentle History for Teachers and Others, which he coauthored with William P. Berlinghoff. I had not come across this book before, as I noted in my post I gave up reading general histories of maths long ago, so, I went looking and came across the following glowing recommendation:
“Math Through the Ages is a treasure, one of the best history of math books at its level ever written. Somehow, it manages to stay true to a surprisingly sophisticated story, while respecting the needs of its audience. Its overview of the subject captures most of what one needs to know, and the 30 sketches are small gems of exposition that stimulate further exploration.” — Glen Van Brummelen, Quest University
Glen Van Brummelen is an excellent historian of maths and regular readers will already know that I’m a very big fan of his books on the history of trigonometry, which I reviewed here. Intrigued I decided to acquire a copy and see if it lives up to Glen’s description. The book that I acquired is the paperback Dover reprint from 2019 of the second edition from Oxton House Publishers, Farmington, Maine, it has a recommended price in the USA of $18, which is certainly affordable for its intended audience, school, college and university students. The cover blurb says, “Designed for students just beginning their study of the discipline…”
The book distinguishes itself by not being a long continuous narrative covering the chronological progress of the historical development of mathematics, the usual format of one volume histories. Instead, it opens with a comparatively brief such survey, just fifty-eight pages entitled, The History of Mathematics in a Large Nutshell, before presenting the thirty sketches mentioned by Van Brummelen in his recommendation. These are brief expositions of single topics from the history of maths, each one conceived and presented as a complete, independent pedagogical unit. The sketches however contain references to other units where a term or concept used in the current unit is explained more fully. Normally when reviewing books, I usually leave the discussion of the bibliography till the end of my review but in this case the bibliography is an integral part of the unit presentation. The bibliography contains one hundred and eighty numbered titles listed alphabetically by the authors’ names. At the end of each sketch there is a list of numbers of the titles where the reader can find more information on the topic of the sketch. Within the sketches themselves this recommendation system is also applied to single themes. The interaction between sketches and bibliography is a very central aspect of the books pedagogical structure.
Given its very obvious limitations, my own far from complete survey of the history of mathematics now occupies more than two metres of bookshelf space and that is not counting the numerous books on the histories of astronomy, physics, navigation, cartography, technology and so forth, which I own, that contain elements on the history of mathematics, the Large Nutshell is actually reasonably good. It sketches the development of mathematics in Mesopotamia, Egypt, China, Ancient Greece, India, and the Islamic Empires, before leaping to Europe in the fifteenth and sixteenth centuries and from there is fairly large steps down to the present day. Naturally, a lot gets left out that other authors might have considered important but it still manages to give a general feeling of the depth and breadth of the history of mathematics. The real work begins with the Sketches.
Sketch No.1 is naturally Keeping Count: Writing Whole Numbers. Our authors deliver competent but brief accounts of the Egyptian, Mesopotamian, Mayan, Roman and Hindu-Arabic number systems but don’t consider the Greek system worth mentioning. They of course emphasise the difficulty of calculating with Roman numerals, as everybody does, but at least mention the calculation was actually done with a counting board or abacus, without seeming to realise that abacus is the Greek name for a counting board, an error that leads them to a stupid statement in another sketch. They also don’t mention that everybody, Mesopotamians, Greeks, Egyptians etc used counting boards to do calculations. They of course mention this in order to emphasise the advantages of the Hindu-Arabic numerals for doing calculations on paper, which leads to another error later.
Sketch No.2 Reading and Writing Arithmetic: The Basic Symbols now makes a big leap to the introduction of symbolic arithmetic in the Renaissance to replace the previous rhetorical arithmetic. This precedes chronologically through the various innovations, including, unfortunately, the myth that Robert Recorde was the first to introduce the equal’s sign. Is OK, but they admit that their brief sketch skips over many, many symbols that were used from time to time.
Almost predictably, Sketch No.3 is Nothing Becomes a Number: The Story of Zero. We start with the Babylonians, who had a place value number system, and after a long time without one, introduced a place holder zero. We then spring to India and from them straight to the Arabs and the logistic origins of the word zero before being served up the first major error in the book. Our authors tell us:
By the 9th century A.D., the Indians had made a conceptual leap that ranks as one of the most important mathematical events of all time. They had begun to recognise sunya, the absence of quantity, as a quantity in its own right! That is, they had begun to treat zero as a number. For instance, the mathematician Mahāvīra wrote that a number multiplied by zero results in zero, and that zero subtracted from a number leaves the number unchanged. He also claimed that a number divided by zero remain unchanged. A couple of centuries later. Bhāskara declared, a number divided by zero to be an infinite quantity.
This is quite frankly bizarre! Already in the 628 CE, Brahmagupta (c. 598–c. 668) had in Chater 12 of his Brāhma-sphuṭa-siddhānta given the rules for addition, subtraction, multiplication, and division with positive and negative numbers and zero in the same way as they can be found in any elementary arithmetic textbook today, with the exception that he defined division by zero. His work was well known in India. In fact, Bhāskara II, mentioned by our authors, based his account on that of Brahmagupta. His work was also translated into Arabic in the eight century and provided the basis for al-Khwārimī’s account, which has not survived in Arabic but was translated into Latin in the twelfth century as Dixit Algorizmi, which was the first introduction of the Hindu-Arabic number system in Europe, as is noted by our authors. Our authors then over emphasise the reluctance of some European mathematicians to accept zero as a number, whilst introducing Thomas Harriot’s method for solving polynomials and zero as a defining property of rings and fields.
Sketch No.4 Broken Numbers: Writing Fractions is a quite good account of their history considering its brevity.
Sketch No.5 Less than Nothing: Negative Numbers is also a reasonably good account of the very problematic history of accepting the existence of negative numbers. Somewhat bizarrely, having totally ignored him on the topic of zero, our authors now state that:
A prominent Indian mathematician, Brahmagupta, recognised and worked with negative quantities to some extent as early as the 7th century. He treated positive numbers as possessions and negative numbers as debts, and also stated rules for adding, subtracting, multiplying, and dividing with negative numbers.
It’s the same section of his book where he deals with zero as a number!
Sketch No. 6 By Ten and Tenths: Metric Measurement is a good brief account of the French introduction of the metric system and its subsequent modernisation. I would have appreciated a brief nod for the English, polymath John Wilkins (1614–1672), who advocated for a metric system already in the seventeenth century.
Sketch No. 7 Measuring the Circle: The Story of π a simple brief account. An interesting addition is a table showing the size of the error in calculating the circumference of a circular lake with a diameter of one kilometre using different historical values for π.
Sketch No. 8 The Cossic Art: Writing Algebra with Symbols is a competent sketch of the gradual but zig zag transition from rhetorical to symbolic algebra.
Sketch No. 9 Linear Thinking: Solving First Degree Equations starts with the Egyptians mentions the Chinese but completely ignores the Babylonians. It then wanders off into an account of the false proposition method of solution.
Sketch No. 10 A Square and Things: Quadratic Equations after one dimension we move onto two dimensions. This is devoted to al- Khwārimī’s discussion of methods of solving quadratic equations completely ignoring the fact that the Babylonians had the general solution of the quadratic equation a thousand years earlier, albeit in a different form to the one we teach schoolchildren and, of course ignoring negative solutions. Every worse ignoring the fact that Brahmagupta had the general solution in the form we use today including negative solutions!
Sketch No. 11 Intrigue in Renaissance Italy: Solving Cubic Equations and now onto three dimensions. We start with the Ancient Geek problem of trisecting an angle and then surprisingly, considering that this is supposed to be “for students just beginning their study of the subject,” we suddenly get a piece of fairly advanced trigonometry blasted into the text:
cos(3𝛂) = 4 cos3(𝛂) – 3 cos(𝛂)
to point out that the solution can be found with a cubic equation!
We move on to Omar Khayyám and his solution of cubic equations using conic sections. We then get a mangled version of the Antonio Fiore/Tartaglia challenge. Out authors claim that Tartaglia bragged that he could solve cubic equations and so Fiore, who had learnt the solution from his teacher Scipione del Ferro, challenged him to a mathematical contest. In fact, Tartaglia didn’t know how to solve them when challenged by Fiore and realising he was onto a hiding to nothing sat down and discovered how to solve them. End result victory for Tartaglia. They get the rest of the story right. How Cardano persuaded Tartaglia to reveal his solution after promising not to publish it before Tartaglia did. Then discovering that Scipione del Ferro had discovered the solution before Tartaglia, and having extended Tartaglia’s solution, went on and published anyway, although giving full credit to Tartaglia for his work. You can read the whole story here. Our authors, however, contradict themselves. In their Large Nutshell they gave a brief version of the story and wrote:
Once he knew Tartaglia’s method for solving some cubic equations. Cardano was able to generalise it to a way of solving any cubic equation. Felling he had made a contribution of his own, Cardano decided he was no longer bound by his promise of secrecy.
Now in Sketch 11 they write:
At this point, Cardano knew that he had made a real contribution to mathematics. But how could he publish it without breaking his promise? He found a way. He discovered that del Ferro had found the solution of a crucial case before Tartaglia had. Since he had the not promised to keep del Ferro’s solution secret, he felt he could publish it, even though it was identical to the one he had learnt from Tartaglia.
We then get Cardano’s discovery of conjugate pairs of complex numbers sometimes in the solutions of cubic equations and a somewhat confused explanation of his reaction to them. Our authors tell us that Cardano wrote to Tartaglia and asked him about the problem and Tartaglia simply suggest that Cardano had not understood how to solve such problems. This is true, quoting MacTutor:
One of the first problems that Cardan hit was that the formula sometimes involved square roots of negative numbers even though the answer was a ‘proper’ number. On 4 August 1539 Cardan wrote to Tartaglia:-
I have sent to enquire after the solution to various problems for which you have given me no answer, one of which concerns the cube equal to an unknown plus a number. I have certainly grasped this rule, but when the cube of one-third of the coefficient of the unknown is greater in value than the square of one-half of the number, then, it appears, I cannot make it fit into the equation.
Indeed, Cardan gives precisely the conditions here for the formula to involve square roots of negative numbers. Tartaglia, by this time, greatly regretted telling Cardan the method and tried to confuse him with his reply (although in fact Tartaglia, like Cardan, would not have understood the complex numbers now entering into mathematics):-
… and thus, I say in reply that you have not mastered the true way of solving problems of this kind, and indeed I would say that your methods are totally false.
Our authors simply move on to state, It fell to Rafael Bombelli to resolve the issue.”
This is selling Cardano short, firstly he was well aware that if one multiplies complex conjugate pairs together the imaginary term disappears as he demonstrated in Ars Magna, he simply thought that it didn’t make sense, mathematically.
Dismissing mental tortures, and multiplying 5 + √-15 by 5 – √-15, we obtain 25 – (–15). Therefore the product is 40. …. and thus far does arithmetical subtlety go, of which this, the extreme, is, as I have said, so subtle that it is useless. (MacTutor)
Secondly, Bombelli used Cardano’s work as the starting point for his own work.
Sketch No. 12 A Cheerful Fact: The Pythagorean Theorem This is a reasonable overview of some of the history of what is probably the most well-known of all mathematical theorems. The first half of the title is a Gilbert and Sullivan quote!
Sketch No. 13 A Marvellous Proof: Fermat’s Last Theorem Once again a reasonably synopsis of the history of another possible candidate for the most well-known of all mathematical theorems.
Sketch No. 14 On Beauty Bare: Euclid’s Plane Geometry Another reasonably synopsis of the world’s most famous and biggest selling maths textbook.
Sketch No. 15 In Perfect Shape: The Platonic Solids A brief, compact and largely accurate introduction to the history of the Platonic regular solids and the Archimedean semi-regular solids. I would have wished for more detail on the Renaissance rediscovery of the Archimedean solids, which was actually carried out by artists rather than mathematicians, a fact that our authors seem not to be aware of, as part of the explorations into the then new linear perspective.
Sketch No. 16 Shapes by the Numbers: Coordinate Geometry Another good synopsis covering both Fermat and Descartes and includes the important fact that the Cartesian system was actually popularised by the expanded Latin edition of La Géométrie put together by Frans van Schooten jr. Although our authors miss the junior. Having acknowledge his contribution they then miss one of his biggest contributions. Our authors write:
The two-axis rectangular coordinate system that we commonly attribute to Descarte seems instead to have evolved gradually during the century and a half after he published La Géométrie.
Rectangular Cartesian coordinates were introduced by Frans van Schooten jr. in his Latin edition.
Sketch No. 17 Impossible, Imaginary, Useful: Complex Numbers Having, in their discussion of cubic equations, failed to acknowledge the fact that Cardano actually showed how to eliminate conjugate pairs of complex numbers in his Ars Magna, even if doing so offended his feelings for mathematics, our authors now bring the relevant quote to this effect. Having opened with Cardano we then move onto Bombelli’s acceptance and Descartes’ rejection before our authors attribute an implicit anticipation of Euler’s Formula in the work of Abraham De Moivre, whilst ignoring the explicit formulation in the work of Roger Cotes. Dealing comparatively extensively with Euler our authors then move onto Argand and Gauss and the geometrical representation of complex numbers, whilst completely ignoring Casper Wessel.
Sketch No. 18 Half is Better: Sine and Cosine We now get introduced to the history of trigonometry. Once again a reasonable short presentation of the main points of the history down to the seventeenth century, with an all too brief comment about the development of the trigonometrical ratios as functions.
Sketch No. 19 Strange New Worlds: The Non-Euclidian Geometries Once again nothing here to criticise.
Sketch No. 20 In the Eye of the Beholder: Projective Geometry Starts with some brief comments about the discovery of linear perspective during the Renaissance, then moves on to the development of projective geometry out of that and ends with Pascal’s Theorem. Once again no complaints on my part.
Sketch No. 21 What’s in a Game?: The Start of Probability Theory Having had the teenage Pascal on conics we now have the mature Pascal on divvying up the stakes in an interrupted game of chance. This is followed by standard story of the evolution of probability theory.
Sketch No. 22 Making Sense of Data: Statistics Becomes a Science Once again presents a standard account without any significant errors.
Sketch No. 23 Machines that Think?: Electronic Computers Having presented twenty-two sketches with only occasional historical error, our authors now come completely of the rails: To start:
Some would say that the story begins 5000 years ago with the abacus, a calculating device of beads and rods that is still used today.
The Greek term abacus refers to a counting board. The calculating device of beads and rods, which is also referred to as an abacus, was developed in Asia and didn’t become known in Europe until the end of the seventeenth beginning of the eighteenth centuries, via Russia, well after the counting board had ceased to be used in Europe.
We get no mention of the astrolabe, which is universally described as an analogue computer,
We then get Napier’s Bones and Oughtred’s slide rule as calculating devices implying that they are somehow related. They aren’t the slide rule is based on logarithms and although Napier invented logarithms his Bones aren’t.
Next up is the Pascaline with, quotes correctly the information that they were difficult to manufacture and so were a flop, although they don’t put it that bluntly. No mention of Wilhelm Schickard, whose calculator was earlier that the Pascaline. Of course, if we have Pascal’s calculator then we must have Leibniz’s, and we get the following hammer statement:
Leibniz’s machine, the Stepped Reckoner, represented a major theoretical advance over the Pascaline in that its calculations were done in binary (base-two) arithmetic (my emphasis), the basis of all modern computer architecture.
When I read that I had a genuine what the fuck moment. There are three possibilities, either our authors are using a truly crappy source on the history of reckoning machines, or they are simply making shit up, or they created a truly bad syllogistic argument.
1) Leibniz was one of the first European mathematicians to develop binary arithmetic
2) Leibniz created a reckoning machine
Therefore: Leibniz’s reckoning machine must have used binary arithmetic
Of course, the conclusion does not follow from the premises and Leibniz’s Stepped Reckoner used decimal not binary numbers.
A fourth possibility is a truly cataclysmic typo but our authors repeat the claim at the beginning of the next sketch of that is definitively not the case.
We get a brief respite with the description of a successful stepped reckoner in the nineteenth century, then we move onto Charles Babbage and a total train wreck. Our authors tell us:
Early in the 19th century, Cambridge mathematics professor Charles Babbage began work on a machine for generating accurate logarithmic and astronomical tables.
[…]
By 1822, Babbage was literally cranking out tables with six-figure accuracy on a small machine he called a Difference Engine.
I think our authors live in an alternative universe, only this could explain how Babbage was “cranking out tables with six-figure accuracy” on a machine that was never built! They appear to be confusing the small working model he produced to demonstrate the principles on which the engine was to function with the full engine which was never constructed. They even include a picture of that small working model labelled, Babbage’s Difference Engine.
In 1832, Babbage and Joseph Clement produced a small working model (one-seventh of the plan), which operated on 6-digit numbers by second-order differences. Lady Byron described seeing the working prototype in 1833: “We both went to see the thinking machine (or so it seems) last Monday. It raised several Nos. to the 2nd and 3rd powers, and extracted the root of a Quadratic equation.” Work on the larger engine was suspended in 1833. (Wikipedia)
1822 was the date that Babbage began work on the Difference Engine. As to being small:
This first difference engine would have been composed of around 25,000 parts, weighed fifteen short tons (13,600 kg), and would have been 8 ft (2.4 m) tall. (Wikipedia)
Not exactly a desktop computer.
Our authors continue with the ahistorical fantasies:
In 1801, Joseph-Marie Jacquard had designed a loom that wove complex patterns guided by a series of cards with holes punched in them. Its success in turning out “pre-programmed” patterns led Babbage to try to make a calculating machine that would accept instructions and data from punch cards. He called the proposed device an Analytical Engine.
In 1801 Jacquard exhibited an earlier loom at the Exposition des produits de l’industrie française and was awarded a bronze medal. He started developing the punch card loom in 1804. The claim about Babbage is truly putting the cart before the horse. Following the death of his wife in 1827, Babbage undertook an extended journey throughout continental Europe, studying and researching all sorts of industrial plant to study and analyse their uses of mechanisation and automation. Something he had already done in the 1820s in Britain. He published the results of this research in his On the Economy of Machinery and Manufactures in 1832. To quote myself, “It would be safe to say that in 1832 Babbage knew more about mechanisation and automation that almost anybody else on the entire planet and what it was capable of doing and which activities could be mechanised and/or automated. It was in this situation that Babbage decided to transfer his main interest from the Difference Engine to developing the concept of the Analytical Engine conceived from the very beginning as a general-purpose computer capable of carrying out everything that could be accomplished by such a machine, far more than just a super number cruncher.” The Jacquard loom was just one of the machines he had studied on his odyssey and he incorporated its concept of punch card programming into the plans for his new computer.
Of course, our authors can’t resit repeating the Ada Lovelace myths and hagiography:
Babbage’s assistant in this undertaking was Augusta Ada Lovelace […] Lovelace translated, clarified and extended a French description of Babbage’s project, adding a large amount of original commentary. She expanded on the idea of “programming” the machine with punched-card instructions and wrote what is considered to be the first significant computer program…
Ada was in no way ever Babbage’s assistant. The notes added to the French description of Babbage’s project were cowritten with Babbage. They did not expand on the idea of “programming” the machine with punched-card instructions. The computer program included in Note G was written by Babbage and not Lovelace and wasn’t the first significant computer program. Babbage had already written several before the Lovelace translation.
We now turn to George Boole and the advent of Boolean algebraic logic, which is dealt with extremely briefly but contains the following rather strange comment:
Although Boole reportedly thought his system would never have any practical applications…
I spent several years at university researching the life and work of George Boole and never came across any such claim. In fact, Boole himself applied his logical algebra to probability theory in Laws of Thought, as is stated quite clearly in its full title An Investigation of the Laws of Thought: on Which are Founded the Mathematical Theories of Logic and Probabilities.
After a very brief account of Herman Hollerith’s use of punch cards to accelerate the counting of 1880 US census, we take a big leap to Claude Shannon.
The pieces started to come together in 1937, when Claude Shannon, in his master’s thesis at M.I.T., combined Boolean algebra with electrical relays and switching circuits to show how machines can “do” mathematical logic.
We get no mention of the fact that Shannon was working on the circuitry of Vannevar Bush’s Differential Analyser, an analogue computer. The Differential Analyser played a highly significant role in the history of the computer as all three, US, war time computers, the ABC, the Harvard Mark I, and the ENIAC, all mentioned by our authors, were all projects set in motion to create an improved version of the Differential Analyser.
We move onto WWII:
Alan Turing, the mathematician who spearheaded the successful British attempt to break the German U-boat command’s so-called “Enigma code,” designed several electronic machines to help in the crypto analysis.
Turing designed one machine the Bombe based on the existing Polish Bomba, his design was improved by Gordon Welchman and the machine was actually designed and constructed by Harold Keen. We get a brief not totally accurate account of Max Newman, Tommy Flowers and the Colossus, before moving to Germany and Konrad Zuse. Our authors tell us:
Meanwhile in Germany, Konrad Zuse had also built a programmable electronic computer. His work begun in the late 1930s, resulted in a functioning electro-mechanical machine by sometime in the early 1940s, giving him some historical claim to the title of inventory of electronic computers.
This contains a central contradiction an electro-mechanical computer is not an electronic computer. Also, we have a precise time line for the development of Zuse’s computers. His Z1, a purely mechanical computer was finished in 1938, his Z2, an electro-mechanical computer in 1939. The Z3, which our authors are talking about, another electro-mechanical computer, an improvement on the Z2, was finished in 1941. Our authors erroneously claimed that Leibniz’s steeped reckoner used binary arithmetic but didn’t think it necessary to point out that Zuse was the first to use binary rather than decimal in his computers beginning with the Z1. They also state:
However, wartime secrecy kept his work hidden as well.
Zuse already set up his computer company during the war and went straight into development and production after the war. Although he couldn’t complete construction of his Z4, begun in 1944, until 1949, delivering the finished product to the ETH in Zurich in 1950.
The Z4 was arguably the world’s first commercial digital computer, and is the oldest surviving programmable computer. (Wikipedia).
Our authors now deliver surprisingly brief accounts of the ABC, the Harvard Mark I, and ENIAC, before delivering the John von Neumann myth.
John von Neuman is generally credited with devising a way to store programs inside a computer.
The stored program computer in question is the EDVAC devised by John Mauchly and J. Presper Eckert, the inventors of ENIAC. Von Neuman merely described their design, without attribution, in his First Draft of a Report on the EDVAC, thereby stealing the credit.
The sketch ends with a very rapid gallop through the first stored program computers, EDSAC in Britain and UNIVAC I in the US, the introduction of first transistors and then integrated circuitry.
There is so much good, detailed history of the development of computers that I simply don’t understand how our authors could get so much wrong.
Sketch No. 24 The Arithmetic of Reasoning: Boolean Algebra A brief nod to Aristotelian logic and then the repeat of the false claim that Leibniz’s steeped reckoner used binary numbers leads us into Leibniz’s attempts to create a calculus of logic, which however our authors admit remained unpublished and unknown until the twentieth century. All of this is merely a lead in to Augustus De Morgan and George Boole. We get brief, pathetic biographies of both of them emphasising their struggles in life. This leads into a presentation of Boole’s logic, presented in the form of truth tables which didn’t exist till much later! We then get an extraordinary ahistorical statement:
De Morgan, too, was an influential, persuasive proponent of the algebraic treatment of logic. His publications helped to refine, extend, and popularise the system started by Boole.
De Morgan was an outspoken supporter of Boole’s work but , his publications did not help to refine, extend, and popularise the system started by Boole. We do however get De Morgan’s Laws and his logic of relations. The latter gives our authors the chance to wax lyrical about Charles Saunders Pierce.
Towards the end of this very short sketch, we get the following, “But it was the work of Boole, De Morgan, C. S. Pierce and others…” (my emphasis) That “and others” covers a multitude of omissions, probably the most important being the work of William Stanley Jevons (1835–1882). An oft repeated truisms is that Boole’s algebraic logic is not Boolean algebra! It was first Jevons, who modifying Boole system turned it into the Boolean algebra used by computers today.
Sketch No. 25 Beyond Counting: Infinity and the Theory of Sets Enter stage right George Cantor. We get a reasonable introduction to Cantorian set theory, Kronecker’s objections and the problem of set theory paradoxes. We then get a long digression on nineteenth century neo-Thomism, metaphysics, infinite sets and the Mind of God. Sorry but this has no place in a very short, supposedly elementary introduction to the history of mathematics, “designed for students just beginning their study of the discipline.”
Sketch No. 26 Out of the Shadows: The Tangent Function is a reasonable account of the gradual inclusion of the tangent function into the trigonometrical canon. My only criticism is although they correctly state that Regiomontanus introduced the tangent function in his Tabulae directionum profectionumque written in 1467. Our authors only give the first two words of the title and translate it as Table of Directions. This is not incorrect but left so probably leads to misconceptions. Directions here does not refer to finding ones way geographical but to a method used in astrology to determine from a birth horoscope the major events, including death, in the subject life. This require complex spherical trigonometrical calculation transferring points from one system of celestial coordinates to another system. Hence the need for tangents.
Sketch No. 27 Counting Ratios: Logarithms A detailed introduction to the history of the invention of logarithms. In general, OK but a bit off the rails on the story of Jost Bürgi.
As Napier and Briggs worked in Scotland, the Thirty Years’ War was spreading misery on the European continent. One of its side effects was the loss of most copies of a 1620 publication by Joost Bürgi, a Swiss clockmaker. Bürgi had discovered the basic principles of logarithms while assisting the astronomer Johannes Kepler in Prague in 1588, some years before Napier, but his book of tables was not published until six years after Napier’s Descriptio appeared. When most of the copies disappeared, Bürgi’s work faded into obscurity.
The failure of Bürgi’s work to make an impact almost certainly had to do with the facts that only a very small number were ever printed and it only consisted of a table of what we now call anti-logarithms with no explanation of how they were created or how to use them. Also, in 1588 Kepler was still living and working in Graz and Bürgi was living and working in Kassel. Kepler first moved to Prague in 1600 and Bürgi in 1604. The reference to 1588 in Bürgi’s work is almost certainly to prosthaphaeresis, a method of using trigonometrical formulars to turn difficult multiplications and divisions into addition and subtractions which Bürgi had learnt from Paul Wittich, and not to logarithms. His work on logarithms almost certainly started later than that of Napier but was independent. Interestingly it is thought that Napier was led to logarithms after being taught prosthaphaeresis by John Craig, who had also learnt it from Wittich
Sketch No. 28 Anyway You Slice It: Conic Sections What is actually quite a good section on the history of conic sections is spoilt by one minor error and a cluster fuck concerning Kepler. The small error concerns Witelo (c.1230–after 1280), our authors write:
There is some evidence that Apollonius’s Conics was known in Europe as far back as the 113th century, when Erazmus Witelo used conics in his book on optics and perspective.
Witelo’s book is titled De Perspectiva and is purely a book on optics not perspective. Perspectivist optics is a school of optical theory derived from the optics of Ibn al-Haytham, whose book Kitāb al-Manāẓir is titled De Perspectiva in Latin translation. On to Kepler:
Sorry about the scans but I couldn’t be arsed to type out the whole section.
The camera obscura that Kepler assembled on the market place in Graz in the summer of 1600 was a small tent and not “a large wooden structure.” It was a pin hole camera and not a kind of large-size one.
The pin hole camera problem, the image created is larger than it should be, had been known since the Middle Ages, was widely discussed in the literature, and Kepler had been expecting it before he began his observations. His attention had been drawn to the problem by Tycho when he was in Prague at the beginning of 1600 to negotiate his move there to work with Tycho. Tycho had also drawn his attention to the problem of atmospheric refraction in the astronomy. These were the triggers that motivated his studies in optics. As noted he moved to Prague to work with Tycho but there was no Imperial Observatory in Prague.
Kepler inherited Tycho’s position of Imperial Mathematicus but not his observation, which were inherited by his daughter. This led to several years of difficult negotiations before he obtained permission to publish his work based on those observations. Tycho had set Kepler on the problem of calculating the orbit of Mars, the work that led to his first two planetary laws, before his death.
The book that Kepler used for his optical studies and on which he based his own work was Friedrich Risner’s “Opticae thesaurus: Alhazeni Arabis libri septem, nuncprimum editi; Eiusdem liber De Crepusculis et nubium ascensionibus, Item Vitellonis Thuringopoloni libri X” (Optical Treasure: Seven books of Alhazen the Arab, published for the first time; His book On Twilight and the Rising of Clouds, Also of Vitello Thuringopoloni book X), which was published in 1572.
“Of course he thought of the conic sections!” Actually, when Kepler first realised that the orbit of Mars was some sort of oval, he didn’t think of the conic section at all. Something for which he criticised himself in his Astronomia Nova, when he finally realised, after much more wasted effort, that the orbit was actually an ellipse. A realisation not based on more measurements. The Astronomia Nova from 1609 only contains the first two planetary laws. The third was first published in his Harmonice Mundi in 1619.
Sketch No. 29 Beyond the Pale: Irrational Numbers Once again an acceptable historical account.
Sketch No. 30 Barely Touching: From Tangents to Derivatives Not perfect but more than reasonable
The book ends with a What to Read Next which begins with The Reference Shelf. Here they recommend several one volume histories of mathematics starting with Victor J Katz and then moving onto Howard Eves’s An Introduction to the History of Mathematics and David M . Burton’s The History of Mathematics which they admit are both showing their age. They move on to lot more recommendations but studiously ignore, what was certainly the market leader before Katz, Carl B. Boyer, A History of Mathematics in the third edition edited by Uta Merzbach. I seriously wonder what they have against Boyer whose books on the history of mathematics are excellent.
Having covered a fairly wide range of reference books, including a lot of Grattan-Guinness we now move onto Twelve Historical Books You Ought to Read. Of course, any such recommendation is subjective but some of their choices are more that questionable. They start, for example, with Tobias Dantzig’s Number the Language of Science. This was first published in 1930 and is definitively severely dated.
Our authors also recommend Reviel Netz & William Noel, The Archimedes Codex (2007). This book was the object of an incredible media hype when it first appeared. Whilst the imaging techniques used to expose the Archimedean manuscript on the palimpsest are truly fascinating the claims made by the authors about the newly won knowledge about Archimedes works are extremely hyperbolic and contain more than a little bullshit. Should this really be on a list of twelve history of maths books one ought to read?
I did a double take when I saw that they recommend Eric Templer Bell’s Men of Mathematics. They themselves say:
The book has lost some of its original popularity, not (or at least not primarily) because of its politically incorrect, but rather because Bell takes too many liberties with his sources. (Some critics would say “because he makes things up.”) The book is fun to read, but don’t rely solely on Bell for facts.
My take don’t waste your time reading this crap book. I read it when I was sixteen and spent at least twenty years unlearning all the straight forward lies that Bell spews out!
Another disaster recommendation is Dava Sobel’s Longitude, which our authors say, “provides a good picture of the interactions among mathematics, astronomy, and navigation in the 18th century.” Sobel’s much hyped book give an extremely distorted “picture of the interactions among mathematics, astronomy, and navigation in the 18th century,” which at times is closer to a fantasy novel than a serios history book. If you really want to learn the true facts about those interactions then read Richard Dunn & Rebekah Higgitt, Finding Longitude: How ships, clocks and stars helped solve the longitude problem (Collins, 2014) and Katy Barrett, Looking for Longitude: A Cultural History (Liverpool University Press), both volumes are written by professional historians, who spent several years researching the topic in a major research project.
The section closes with History Online, which can’t be any good because it doesn’t include The Renaissance Mathematicus! 🙃
The apparatus begins with a useful When They Lived, which simply lists lots of the most well-known mathematicians alphabetically with their birth and death dates. This is followed by the bibliography, already mentioned above, and a good index. There are numerous, small, black and white illustration scattered throughout the pages.
Despite my negative comment about some points in the book, I would actually recommend it as a reasonably priced, mostly accurate, introduction to the history of mathematics. A small criticism is it does at times display a US American bias, but it was written primarily for American students. A good jumping off point for somebody developing an interest in the discipline. In any case considerably better that my jumping off point, E. T. Bell!