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• "…I was sitting in the rooms of the Analytical Society, at Cambridge, my head leaning forward on the table in a kind of dreamy mood, with a table of logarithms lying open before me. Another member, coming into the room, and seeing me half asleep, called out, 'Well, Babbage, what are you dreaming about?' to which I replied 'I am thinking that all these tables' (pointing to the logarithms) 'might be calculated by machinery.'"
                                                                               Charles Babbage

  "The distinctive characteristic of the Analytical Engine, and that which has rendered it possible to endow mechanism with such extensive faculties as bid fair to make this engine the executive right-hand of abstract algebra, is the introduction into it of the principle which Jacquard devised for regulating, by means of punched cards, the most complicated patterns in the fabrication of brocaded stuffs. It is in this that the distinction between the two engines lies. Nothing of the sort exists in the Difference Engine. We may say most aptly that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves."
                                                                  Augusta Ada King Lovelace

• The nineteenth century continues and reinforces the trends surveyed in the previous lecture. Many areas of mathematics are explored in depth, with a growing preoccupation for their logical foundations. The widespread belief that most sciences can ultimately be reduced to physics and mathematics reaches a peak in this century. Mathematicians begin to play in earnest with non-Euclidean geometries, which in turns leads them to ask fundamental questions about the nature of mathematics. The proliferation of mathematical fields creates a need to systematize them and to connect them together in one organic whole. Some of such attempts will prove crucial in the development of modern computers.
  Although substantially a singular phenomenon, this century witnesses the invention of a computer which, in in an essential way, represents the first modern computer: Babbage's Analytical Engine.
• The nineteenth century opens with a sort of test of the power of the computational techniques developed by Newton and many others in the past couple of centuries. On January 1st, 1801, the Italian astronomer Giuseppe Piazzi (1746-1826) discovered Ceres, the first and largest (about 914km in diameter) asteroid. See 1801-2001: The Bicentenary of Ceres' Discovery and Bode's Law and the Discovery of Ceres.
• In 1772 a German astronomer, Johann Ehlert Bode published what became known as Bode's Law. In fact, the credit should go to a German mathematician, Johann Titius. Titius and Bode, and many other astronomers of the time, were interested in understanding why the planets are where they are in the solar system. Why should the Earth orbit at about 150,000,000 km from the Sun? This is a difficult question, and even today we don't really have a satisfactory answer. By the late eighteenth century astronomers had already measured the distances of all the planets then known from the Sun. These were the initial observations Bode started with. When you look at the data,  keep in mind that the farthest planet known at the time was Saturn. Bode tried to find a hypothesis that would fit these data. He came up with a simple formula. He first expressed all the data in so-called astronomical units. One AU is the average distance of the Earth from the Sun. His hypothesis was as follows:
  • Assign the number 0 to Mercury, 3 to Venus, 6 to Earth, 12 to Mars, 24 to Jupiter and 48 to Saturn
  • Add 4 to each of these numbers
  • Divide the results by 10

  If you now compare the numbers so obtained to the actual distances of these planets from the Sun, you find a remarkable match. Of course you may well ask: why 0, 3, 6. etc.? why add 4? why divide by 10? The answer is that initially a hypothesis must only satisfy one criterion: it must fit the data. You may get the idea from a dream, from a genie, from the depths of your unconscious, from wherever. Is it consistent with the observations? If yes, you are in business. Bode was in business. Well,…almost. The Asteroid Belt, an area between Mars and Jupiter where we now know thousands of smaller bodies orbit, had not been discovered yet. So, the predicted distance for Jupiter was too small, and so was that of Saturn (which in fact seemed to correspond to the actual distance of Jupiter). Since however things were fine for Mercury, Venus, Earth and Mars, Bode was reluctant to abandon his hypothesis.

  In 1781 a British astronomer, William Herschel, discovered Uranus. Bode extended his  table,  by assigning 96 to Uranus and applying the formula. Unfortunately the formula predicted the wrong distance for Uranus (but strangely, once again, the number so obtained seemed to correspond to another planet, Saturn).
  Then, on January 1st, 1801, an Italian astronomer, Giuseppe Piazzi, discovered a small planet, Ceres, which was precisely at the (wrong) distance predicted for Jupiter. Now the table seemed to be correct, from Mercury to Uranus. Ceres turned out to be the biggest body in what is now known as the Asteroid Belt.

   

  

  Ceres (Southwest Research Institute)

   

  Bode died long before new planets were discovered. In 1846, the German astronomer Johann Galle discovered Neptune, and in 1930 the British astronomer Clyde Tombaugh discovered Pluto. Bode would have been rather disturbed by these discoveries, as you can see in the table. What is the problem with Bode's Law? Here is one important observation: Bode's formula never stops. You can apply it over and over again, obtaining distances which do not correspond to anything. Thus, even if the known planets were to fit it perfectly, Bode's law is intrinsically flawed, as it predicts infinitely more bodies than it was asked to.
• Now, almost as soon as Ceres was discovered, it was lost. Fortunately Piazzi had measured the position of the asteroid "on a total of 24 nights between 1 January and 11 February, though some positions were marked as 'doubtful' or even 'very uncertain'." [ see The Night Piazzi Discovered Ceres ] Fortunately Johann Carl Friedrich Gauss (1777 - 1855) proceeded to compute its orbit from the few observations that had been made by Piazzi, and predicted the position of Ceres, which was rediscovered almost exactly where Gauss had calculated it should be.
  It is interesting to compare the theory proposed by Bode with the calculations performed by Gauss. Bode's Law is an ad hoc hypothesis, which works—to some extent—only with a particular set of data, and is not based on any general set of theoretical principles. In a word, it is not really a physical law. Gauss, instead, based his calculations on the by-then well-tested theory of Newton, and given the empirical data collected by Piazzi, successfully predicted the orbit of Ceres.
• Many other adavances are made in this century in 'applied' mathematics [see Chronology of the History of Computing and A Mathematical Chronology ]. Here are just a few examples:
  • in 1804 "Bessel publishes a paper on the orbit of Halley's comet using data from Harriot's observations 200 years earlier,"
  • in 1806 "Legendre develops the method of least squares to find best approximations to a set of observed data,"
  • in 1807 "Fourier discovers his method of representing continuous functions by the sum of a series of trigonometric functions and uses the method in his paper On the Propagation of Heat in Solid Bodies which he submits to the Paris Academy"
  • in 1810 "Gergonne publishes the first volume of his new mathematics journal Annales de Mathématique Pures et Appliquées, where, probably for the first time, a distinction is drawn between 'pure' and 'applied' mathematics)"
  • in 1818, "inspired by the work of Laplace, Adrain publishes Investigation of the Figure of the Earth and of the Gravity in Different Latitudes"
  • in 1821 "Navier gives the well known 'Navier-Stokes equations' for an incompressible fluid"
  • in 1824 "Sadi Carnot publishes Réflexions sur la Puissance Motrice du Feu et sur les Machines Propres à Développer cette Puissance (Thoughts on the Motive Power of Fire, and on Machines Suitable for Developing that Power). A book on steam engines, it will be of fundamental importance in thermodynamics. The 'Carnot cycle' which forms the basis of the second law of thermodynamics also appears in the book"
  • in 1835, "Coriolis publishes Sur les Équations du Mouvement Relatif des Systèmes de Corps. He introduces the 'Coriolis force' and shows that the laws of motion can be used in a rotating frame of reference if an extra force called the 'Coriolis acceleration' is added to the equations of motion. In the same year Coriolis publishes a work on a mathematical theory of billiards"
  • in 1836 "Poncelet publishes Cours de Mécanique Appliquée aux Machines (A Course in Mechanics Applied to Machines). It is the first to propose the use of mathematics in machine design."
  • in 1837 "Poisson publishes Recherches sur la Probabilité des Jugements (Researches on the Probabilities of Opinions). In this work he establishes the rules of probability, gives 'Poisson's law of large numbers' and describes the 'Poisson distribution' for a discrete random variable which is a limiting case of the binomial distribution"
  • in 1838 "Bessel measures the parallax of the star 61 Cygni, the first star for which this is calculated"
  • in 1838 "Cournot publishes Recherches sur les Principes Mathématiques de la Théorie des Richesses in which he discusses mathematical economics, in particular supply-and-demand functions"
  • in 1873 " James Clerk Maxwell (1831 - 1879) publishes Electricity and Magnetism. This work contains the four partial differential equations, now known as 'Maxwell's equations.'" The stature of Maxwell's work in electricity and magnetism can be compared with that of Newton's in mechanics. It is one of the greatest works of all times.
  • in 1876 "Gibbs publishes On the Equilibrium of Heterogeneous Substances which represents a major application of mathematics to chemistry"
  • in 1883 "Reynolds publishes An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water in Parallel Channels Shall Be Direct or Sinuous and of the Law of Resistance in Parallel Channels. The 'Reynolds number' (as it is now called) used in modelling fluid flow appears in this work"
  • in 1886 "Reynolds formulates a theory of lubrication"
  • in 1891 Fedorov and Schönflies independently classify crystallographic space groups showing that there are 230 of them"
  The introduction of the method of least squares is particularly important, since it is one of the fundamental tools of statistics.

  "Field data is often accompanied by noise. Even though all control parameters (independent variables) remain constant, the resultant outcomes (dependent variables) vary. A process of quantitatively estimating the trend of the outcomes, also known as regression or curve fitting, therefore becomes necessary.

  The curve fitting process fits equations of approximating curves to the raw field data. Nevertheless, for a given set of data, the fitting curves of a given type are generally NOT unique. Thus, a curve with a minimal deviation from all data points is desired. This best-fitting curve can be obtained by the method of least squares.

  The method of least squares assumes that the best-fit curve of a given type is the curve that has the minimal sum of the deviations squared (least square error) from a given set of data." [ from Least Square Method ]

• As already noted, the nineteenth century saw a full-scale attack on the problem of the foundations of mathematics. This work perhaps culminated in 1900 with the famous lecture delivered by David Hilbert (1862 - 1943) at the Second International Congress of Mathematicians in Paris, where he posed 23 problems as a challenge for the 20th century. We will discuss this milestone in the next lecture. Even before that, however, there had been important discoveries. For example,
  • in 1823 János Bolyai (1802 - 1860) developed a detailed example of non-Euclidean geometry, and so did, in 1829, Nikolai Ivanovich Lobachevsky (1792 - 1856)
  • in 1854 George Boole (1815 - 1864) "publishes The Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities. He reduces logic to algebra and this algebra of logic is now known as Boolean algebra."
  • in 1854 Georg Friedrich Bernhard Riemann (1826 - 1866) "delivered a lecture, Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry) […] in which posed deep questions about the relationship of geometry to the world we live in […] what the dimension of real space was and what geometry described real space"
  • in 1872 Felix Christian Klein (1849 - 1925) "gives his inaugural address at Erlanger. He defines geometry as the study of the properties of a space that are invariant under a given group of transformations. This became known as the "Erlanger programm" and profoundly influences mathematical development"
  • in 1884 Friedrich Ludwig Gottlob Frege (1848 - 1925) publishes The Foundations of Arithmetic
  • in 1888 Julius Wihelm Richard Dedekind (1831 - 1916) "publishes Was sind und was sollen die Zahlen? (The Nature and Meaning of Numbers) and puts arithmetic on a rigorous foundation giving what were later known as the 'Peano axioms'"
  • in 1889 Giuseppe Peano (1831 - 1916) "publishes Arithmetices Principia, Nova Methodo Exposita (The Principles of Arithmetic) which gives the Peano axioms defining the natural numbers in terms of sets"
  • in 1895 Jules Henri Poincaré (1854 - 1912) "publishes Analysis Situs, his first work on topology, which gives an early systematic treatment of the topic"
  • in 1899 David Hilbert (1862 - 1943) "publishes Grundlagen der Geometrie (Foundations of Geometry) putting geometry in a formal axiomatic setting."

• Finally, let's examine the unique work of Charles Babbage (1791 - 1871) and of Augusta Ada King, Countess of Lovelace (1815 - 1852), who was to play a crucial role in the developemnt of Babbage's ideas.

   

  

  Addition and Carry Mechanism of the Difference Engine
  (The Powerhouse Museum)

   

  Here are a few additional references to Babbage and Ada King Lovelace:
  • Augusta Ada Byron, Countess of Lovelace
  • Charles Babbage
  • Babbage's Analytical Engine
  • Babbage Difference Engine
  • The Babbage Pages
  • Babbage Printer Finally Runs
  • Babbage Difference Engine #2: How to Initialize the Machine
  As I have stated earlier, not much progress was made in the eighteenth and nineteenth centuries with regard to computational hardware. Yet, the more and more powerful computational techniques introduced in this period always ended up in very laborious calculations. The only aids available were tables, patiently developed by hand. I will just mention here the publication in 1814 of Peter Barlow's New Mathematical Tables which include "factors, squares, cubes, square roots, reciprocals and hyperbolic logs of all numbers from 1 to 10000," and which were "considered so accurate and so useful that [they have] been regularly reprinted ever since."
  In was not until 1819, when he became interested in astronomical instruments, that Charles Babbage "formulated a plan to construct tables using the method of differences by mechanical means. Such a machine would be able to carry out complex operations using only the mechanism for addition. Babbage began to construct a small difference engine in 1819 and had completed it by 1822. He announced his invention in a paper Note on the Application of Machinery to the Computation of Astronomical and Mathematical Tables read to the Royal Astronomical Society on 14 June 1822. […] Babbage illustrated what his small engine was capable of doing by calculating successive terms of the sequence n² + n + 41. The terms of this sequence are 41, 43, 47, 53, 61, … while the differences of the terms are 2, 4, 6, 8, … and the second differences are 2, 2, 2, … The difference engine is given the initial data 2, 0, 41; it constructs the next row 2, (0 + 2), [41 + (0 + 2)], that is 2, 2, 43; then the row 2, (2 + 2), [43 + (2 + 2)], that is 2, 4, 47; then 2, 6, 53; then 2, 8, 61; … Babbage reports that his small difference engine was capable of producing the members of the sequence n² + n + 41 at the rate of about 60 every 5 minutes." [ from MacTutor ]
  To appreciate the importance of such an invention, it is useful to look at what was probably the most massive attempt to produce numerical tables ever undertaken. In 1792 Gaspard Clair François Marie Riche de Prony (1755 - 1839)

  "began a major task of producing logarithmic and trigonometric tables, the Cadastre. With the assistance of Legendre, Carnot and other mathematicians, and between 70 to 80 assistants, the work was undertaken over a period of years, being completed in 1801. The tables were '…vast, with values calculated to between fourteen and twenty-nine decimal places. Each copy consisted of eighteen folio volumes together with another volume of mathematical procedures.' Getting such a massive work published was another matter. Negotiations went on over a number of years until, in 1809, it seemed they would appear. The publisher wrote: 'The present generation would never have witnessed the end of this monumental work if M de Prony had not had the fortunate idea of applying the powerful method of division of labour, conceiving methods to reduce the long and laborious part of the production of the tables to simple additions and subtractions…' However the tables were never published in full and it was near the end of the century before even a part appeared. It was just too expensive to print at a time when France was not in the best of financial states."

  Unfortunately various circumstances, primarily funding problems, led to the termination of the Difference Engine project. Babbage realized that a more powerful machine could be built, and by 1834 the first drawing of the Analytical Engine were ready. The Analytical Engine was essentially the first computer in the modern sense of the term. It had five major components
  • the input device
  • the store
  • the mill
  • the control
  • the ouput device
  The input device was a sort of punched card reader; the mill was the mechanical counterpart of the modern cp; the control employed "a Jacquard loom type device, and it was operated by punched cards and the punched cards contained the program for the particular task;" and the ouput was a printer. Concerning the latter, it has now been built according to the original specifications, and it works (see Babbage Printer Finally Runs). The store was the mechanical counterpart of RAM, and although it had been initially designed "to hold 1000 numbers each of 50 digits […] Babbage designed the analytic engine to effectively have infinite storage. This was done by outputting data to punched cards which could be read in again at a later stage when needed."
  The Analytical Engine was not built until well into this century. Besides the funding problems, the major obstacle was the high mechanical precision and strength required by the components—a feat difficult, if not impossible, in that period.
  "Babbage visited Turin in 1840 and discussed his ideas with mathematicians there including Menabrea. During Babbage's visit, Menabrea collected all the material needed to describe the analytical engine and he published this in October 1842." It is at this point that Lady Ada Lovelace enters the scene. She not only translated Menabrea's article from Italian into English, but she added a large number of notes "considerably more extensive than the original memoir." The translation and the notes were published in 1843. The notes are of great historical importance for two reasons: Ada Lovelace understood the machine perhaps even better than Babbage himself—so much so that she saw the huge potential a machine of that kind had. In addition, the notes included the first computer programs ever written, in particular an algorithm for computing the sequence of Bernoulli numbers. Just to give you an idea of the complexity of such a program, consider the implicit definition of Bernoulli numbers B_n:  x/(e^x - 1) = Σ(B_nxⁿ)/n!  (where  n! = 1×2×3×…×n).
  Finally, here is a summary of the most important 'notes' by Ada Lovelace:

  "Her 'Notes,' appearing under the pseudonym A.A.L. in the prestigious journal Taylor's Scientific Memoirs, were entitled NOTE A, NOTE B, etc. Here are some very brief descriptions of some partial content of the notes.

  • NOTE A told how the planned 'analytic engine' (and any future general computing devices of this type) differed from the already existing difference engine. Her discussion predicted the general purpose computer and went beyond anything that Babbage had envisioned. She showed how the analytic engine could be built to accept various types of cards: 'control cards,' 'data cards' and 'operation cards' and how these would make the computer automatically perform the correct operation on the data as it was entered. In other words, the computer could 'analyze the data.' She also proposed that both numbers and other symbols, such as letters of the alphabet, could be 'coded' as numerical data which the engine could handle and put out as written material. She even contended that the engine could produce music.
  • In NOTE B, Ada discusses the analytic engine's memory capability, called its 'storehouse columns.' She anticipated the idea of memory locations or addresses. She indicated the possibility of inserting comments or memoranda that would not be acted upon by the computer, but simply would let the human reader of the instructions see what was going on. (Computer science students will recognize this as the use of 'documentation' statements.)
  • NOTE C. Here Ada, introduced the concept of 'backing'—making an operation card move back into a position so that it could work on the next data card. She said the reason for doing this was 'to secure the possibility of bringing any particular card or set of cards into use any number of times successively in the solution of one problem.' This idea, which we call 'looping' was hinted at in Menabrea's article but not well developed. It also arose in earlier discussions between Ada and Babbage.
  • NOTE D. In this note Ada explained how to write down a sequence of instructions using the operations cards, the backing process, the storehouses and various control cards to accomplish a specific set of operations.
  • In NOTE E, she explained how the engine could be made to hold trigonometric and other functions. She was, essentialy, introducing the idea of built-in functions in a computer.
  • NOTE F. She showed that by using the backing process (a loop), she could solve a system of linear equations of any size by repeating just a few operations. She wrote out the details for solving a ten by ten system of linear equations. She also speculated about the possibility of generating tables of prime numbers by a simple looping procedure.
  • In NOTE G, Ada warns the readers about the computer's inability to do anything about it if the user entered 'untrue' information. Today we call this concept 'garbage in, garbage out.' Here is the way she said it: 'The Analytical Engine has no pretensions whatever to originate anything. It can do whatever we know how to order it to perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths.'… It was also in Note G that Ada wrote out a program by which the analytic engine could generate a table of the Bernoulli numbers."

   

  

  The Analytical Engine
  (The Merz-Schule)

   

 
Readings, Resources and Questions

• Discuss Bode's Law, elaborating the statement that it is not really a physical law.
• Explore further the puzzling lack of success of Babbage's machines, despite the report on their feasibility issued by the British Association after Babbage's death, which "recorded their opinion that its successful realisation might mark an epoch in the history of computation equally memorable with that of the introduction of logarithms'…"
• A very good reference is Doron Swade's The Difference Engine. Charles Babbage and the Quest to Build the First Computer (Penguin Books, 2000). See also Swade's Redeeming Charles Babbage's Mechanical Computer, an article that appeared in the February 1993 issue of Scientific American. "A successful effort to build a working, three-ton Babbage calculating engine suggests history has misjudged the pioneer of automatic computing."
• Among the many software programs which simulate vintage machines, try Pinwheel. "This type of calculators were designed about 1850 and were in use until 1970."