Babbage invited Ada to see the miraculous Difference Engine a few weeks after they met. Ada’s impression of the Difference Engine was recalled by Sophia Frend more than fifty years later: “Miss Byron, young as she was, understood its working, and saw the great beauty of the invention.” An excerpt from Lady Byron’s letter to Dr King on 21 June gives us more information about the first time she went with Ada to see the Difference Engine. Lady Byron called the Difference Engine a “thinking machine,” a mistake that still persists. It was the most advanced calculating machine ever built and though Babbage had visions of it performing metaphysical feats, which he described in his book “The Ninth Bridgewater Treatise,” it was just a massive calculator.
Lady Byron’s calling the Difference Engine a “thinking machine” was not an accurate description of its ability but a common name given to such machines at the time. It was a calculating engine. Numbers were put on successive carriages consisting of toothed wheels that had ten digits marked on the edge. Whenever any wheel, in performing addition, passed from nine to zero, the projecting tooth pushed over a certain lever. Babbage described how a calculation could be done in nine seconds, quickly and accurately.
Babbage’s description must have touched Ada, since it was more than just an explanation of the mechanical manipulation of numbers. To both Ada and Babbage the manipulation of numbers was not just a practical exercise but a path to mathematical and metaphysical understanding.
Metaphysics was not an area that interested Lady Byron. She regarded Babbage’s metaphysical views as “the whim of the moment,” and preferred that Ada be grounded in what she considered the facts. She was always watching for signs of Lord Byron’s influence—his passionate nature—in Ada’s personality. Lady Byron enlisted the sober Dr King to help Ada not only with mathematics, but to insure that Ada’s passions were directed along a proper path. Dr King wrote Ada sermons, and when she asked him concrete questions about mathematics, he replied that he was puzzled. He explained that as a student at Cambridge University he seldom read a book that was not assigned.
Despite Dr King’s moralizing, when Ada met Babbage, learning more about mathematics, science, and technology became not a duty to her but a joy. Ada stepped back from asking Dr King any more questions about mathematics and attempted to find solutions by using her own imagination, visualizing mathematical problems and solutions. She would not destroy her imagination but use it in her own way. She could not understand the rainbow and directed her questions not to Dr King but to her mother’s old tutor, Dr Frend. She asked Dr Frend “why a rainbow always appears to the spectator to be an arc of a circle. Why is it a curve at all, and why a circle rather than any other curve? I believe I clearly understand how it is that the colours are separated, and the different angles which the different colours must make with the original incident ray. I am not sure that I entirely understand the secondary rainbow. –
Is the spectator’s eye supposed to be in the centre of the circle of which the arc of the rainbow forms a portion? –”
She ended the letter by asking Dr Frend whether he had read Mrs Somerville’s latest book. Mrs. Somerville was another new friend who had a profound influence on Ada’s life and helped her to put mathematics and science in a human context.
Poetical Science
Ada was curious at this time about the secondary rainbow. Today she could search books, the Internet, or go to Wikipedia and find this explanation:
Frequently, a dim secondary rainbow is seen outside the primary bow. Secondary rainbows are caused by a double reflection of sunlight inside the raindrops, and appear at an angle of 50°–53°. As a result of the second reflection, the colours of a secondary rainbow are inverted compared to the primary bow, with blue on the outside and red on the inside. The dark area of unlit sky lying between the primary and secondary bows is called Alexander’s band, after Alexander of Aphrodisias, who first described it.
There are many sites that describe the secondary rainbow. This is my favorite: http://www.webexhibits.org/causesofcolor/13.html
What is yours?
5
Make It Part of Your Mind, Solving Unsolvable Equations,
The Universal Machine
[1834-1835]
Even though Dr Frend was helpful in trying to answer Ada’s questions, he was growing old. He encouraged her to become friendly with Mrs Somerville, a prominent scientist, whose Connection of the Physical Sciences had just been published. Today there is a college named in her honor at Oxford University.
Mary Somerville was essentially self-taught. Reading mathematical puzzles in a sewing magazine sparked her interest in mathematics. After the death of her first husband, she studied mathematics on her own, reading Newton’s works in Latin. George Peacock (Babbage’s friend) used her translation from French of LaPlace’s work as a textbook at Cambridge University. In addition to respecting her as a scientist, everyone regarded Mrs Somerville as a gentle, humble, and kind human being.
From Mrs Somerville’s letters to Ada, it is apparent that she was anxious not only to help Ada with her mathematics but also to put that passion in a proper perspective. As a result Mrs Somerville encouraged Ada in all pursuits from knitting caps to riding on the downs. Ada became a frequent visitor (sometimes at her own request) to the Somerville home at the Royal Hospital in the Chelsea section of London, where Dr Somerville (Mrs Somerville’s second husband) was a physician.
This was Ada’s first involvement with a traditional family. She became close friends with the whole family: Martha and Mary, Mrs Somerville’s daughters, and Woronzow Greig, her son from her first marriage. Greig became Ada’s friend, confidant, and attorney. She spent many happy moments playing music, riding horses, and going to Babbage’s home with members of the Somerville family.
Ada refers in her letters to Dionysius Lardner, who was a friend of Babbage’s. Lardner was a popular scientific personality who gave lectures about the Difference Engine at the Mechanics Institute, which Ada attended. She used the mathematics textbooks he wrote and was particularly impressed by his article about the Difference Engine published in the Edinburgh Review. By June 1834 Lady Byron wrote that Ada regarded “the Difference Engine as a friend.”
Ada continued to attend Babbage’s Saturday night soirées. He sent her an invitation enticing her to come to see the “Silver Lady,” one of the two automatic dolls he had. In Passages he described one of the dolls, “whose eyes were filled with imagination . . . ”; however, from Ada’s letters to Mrs Somerville, it appears that she was more captivated by the Difference Engine than by the “Silver Lady.”
Jacquard Loom and punched cards
During the summer of 1834 Ada had a chance to see the Industrial Revolution in action. She went with her mother on a trip to the north of England and visited many of the new factories blossoming throughout England. They visited printers and ribbon factories in Coventry. The machinery reminded her of Babbage’s “gem of all machinery.” Lady Byron drew a picture of a punched card used to instruct the loom. After several weeks of factory tours in the Midlands, they settled down with one of Lady Byron’s friends, Lady Gosford, and her two daughters Annabella (named after Lady Byron) and Olivia (Livy) Acheson. While the mothers were busy with a health cure, Ada decided to make good use of her time and teach Annabella and Livy mathematics.
As a teacher Ada revealed how she approached a subject that normally does not evoke passion, especially in young women. The quality of her understanding is evidenced by her use of imagination and quantitative skills. Ada’s letters were written on Cambridge quire paper, a parchment paper about four times the size of regular paper. She used color and what were then considered vulgar instruments—compasses and protractors. She scolded her students for using indirect proofs when a direct proof should be used. As did her father, she believed that mathematical explanations and allusions are reinforced by the apt use of metaphor.
She used Lardner’s text, wrote out a proposition, calling it “neat, tidy
and pretty” and tried to develop an esprit de corps. Ada used every skill she could muster to get her message across. She used both reason and imagination. Like her mother, she insisted on excellent performance, but her method did not involve the use of tickets. She defined mathematical terms verbally and visually. She developed an esprit de corp, a joint mission, by signing a letter, “A Sentimental Mathematical Correspondence carried on for years between two young ladies of rank, to be hereafter published no doubt for the edification of mankind, or womankind.”
Ada rotated between understanding mathematics using imagination and insisting on conceptual understanding. She wrote that she did not consider that she knew a proposition, “until I can imagine to myself a figure in the air, and go through the construction & demonstration without any book or assistance whatever.” Even though Ada was busy with mathematics and Babbage’s Difference Engine, she did not forget her music. She was still playing the guitar and at this time started harp lessons.
Varying moods and language emerged in Ada’s growing correspondence. She began to show different aspects of her personality depending on whom she was writing to. She continued to write to Dr King, explaining her method of understanding mathematics; to his wife, explaining the state of her moral development (perhaps more to please than to give the true state of her thoughts); and to Mrs Somerville, explaining more truthfully, without fear of a sermon, what she was doing, thinking, and feeling.
Ada continued her correspondence with Dr King, assuring him that she was studying Euclid’s theorems using Dionysius Lardner’s textbook. Her question was “Can it be proved by means of propositions & deductions from the 1st book only, that equilateral triangles being constructed on the sides of a right angled triangle, and also on the hypotenuse, the sum of the triangles on the sides is equal to the triangle on the hypotenuse?”
The letters flew back and forth to her students. Ada taught concepts in geometry using figures to illustrate the propositions as well as color to make the subject more interesting. She always admonish her students “to make the subject part of their mind.” Sometimes she realized she was overbearing and apologized stating how she often became so eager she forgot about everything else.
Back in London she wrote Mary Somerville on 8 November to ask her to thank Babbage for a copy of the blueprints for the machine and noted that she was copying the Steam Engine paper which she “carried off from Mr Babbage’s the other day.” Her letters to Mary Somerville were not only about Mary’s daughters but contained questions about Babbage. Mrs Somerville often took Ada to his house at One Dorset Street.
One Dorset Street
The autumn of 1834 was an exciting period in Charles Babbage’s life, and Ada was witness to the birth of an idea that has changed all our lives. Ada did not write letters about this exciting period; however, she did leave an account of several evenings during which the fate of the Difference Engine was discussed and Babbage revealed his conception for a new calculating engine. What Ada heard at this time no doubt formed the foundation of her understanding of the Analytical Engine, which is now regarded by many as the conceptual birth of the computer revolution.
Lady Byron accompanied Ada on a few occasions to the dinner parties, but when she was ill, Ada went on her own to dine with Mrs Somerville and Babbage. When she returned home she recounted to her mother what had been discussed during the evening. These discussions give us a glimpse of Babbage’s creative mind. During this time he discussed the importance of color and how he had experimented with the printing of logarithm tables using different inks and colored paper.
On the evening of 15 November, Babbage was concerned about the future of the Difference Engine since the British government was not going to financially support the completion of the engine. According to the Times, the week of 15 November was filled with stormy weather and political turmoil. On the 15th there were rumors that Lord Melbourne (Lady Byron’s first cousin) had resigned as prime minister and that Lord Wellington would head a caretaker government. A few days later, on the 18th, those rumors were confirmed. Since the duke was impressed with the Difference Engine, Babbage hoped that this change in government would mean that he could get the necessary funds to complete the Difference Engine.
Ada went with Mrs Somerville to see Babbage on 28 November, and the discussion that evening was lively. According to Lady Byron’s diaries, Ada listened as Babbage and Mrs Somerville argued about the reasons for the lack of continued support of Babbage’s plans. Mrs Somerville suggested that perhaps the world was not ready for such an engine. Babbage replied that if he did not develop it, someone else would. It appeared that Sir Robert Peel, no friend of the Difference Engine, would become the new prime minister.
The discussion then turned to miracles and mathematics, Babbage alleged that the engine could show that miracles were not only possible but probable. Ada listened to his ideas and was captivated by the thought of where mathematics might lead. She looked forward to her next evening with Babbage and Mrs Somerville.
On 15 December Ada once again spent the evening with Babbage and Mrs Somerville. This time Babbage explained his theory of the process of discovery. He said: “. . . the difference of opinion amongst mankind could be traced to the different degrees of power to individualize.” Babbage continued by explaining that a person would have a general idea of an objective and then would proceed, for example, in designing a machine, to the particular part of the mechanism that could achieve that objective. He concluded that between these two types of minds there are various grades: “. . . the rarest mind is that which unites both powers in considerable degree.” In order to understand anything we ought first, said Babbage, to place ourselves at a distance from it and then approach gradually to investigate the details. Astronomy, therefore, was the most perfect science since we were forced to view it at a distance.
Babbage then described the strange sensation that the first glimpse of his discovery aroused in his mind, “. . . when the possibility of throwing a bridge from the known to the unknown was first apprehended.” Babbage used an example to clarify his point. A man was standing on a mountain with mountains surrounding him on all sides. The man felt closed in by the mountains. As the man watched, the mist in the valley below began to disperse and he caught a glimpse of a river whose course he could not follow, yet he felt sure that there must be an egress.
Babbage recalled in Passages how he conceived the idea of the Analytical Engine: “. . . it occurred to me that it might be possible to teach a mechanism to accomplish another mental process, namely – to foresee. The idea occurred to me in October, 1834. It cost me much thought, but the principle was arrived at in a short time. As soon as that was attained, the next step was to teach the mechanism which could foresee to act on that foresight.”
Lady Byron wrote in her journal the evening of 15 December that Babbage reported he had made a discovery “. . . in the highest department of mathematics. – I understand it to include the means of solving equations that hitherto had been considered insolvable.” Lady Byron discounted Babbage’s views as “unsound and paradoxical.” Ada, on the other hand, stated to her mother that she was touched by the “universality” of Babbage’s ideas: to Ada it was the threshold of a new world.
Poetical Science
What follows are two activities suggested by Scott Kim which will help you to understand Babbage’s Difference Engine.
Activity 1: Differences
Babbage’s Difference Engine is based on a neat mathematical shortcut called difference sequences. Here’s how it works. A square number is a number times itself, like 25 (5×5) or 49 (7×7). Here are the first few square numbers.
Now write the difference between each pair of numbers below and between each pair of numbers, like this.
See the pattern in the second row? If you write the differences of the differences, you get a third row of numbers, this time all 2s. What do you get if you take the differences one more time?
Now it’s your turn. Write th
e differences for the following sequences of numbers. What do you get?
Differences are useful in computers because they let you compute sequences of numbers by using just addition instead of more complex operations like multiplication and exponents.
Activity 2: Build a Difference Engine out of Lego Pieces.
http://acarol.woz.org
Babbage built his Difference Engine out of precisely milled metal parts. But you don’t need a machine shop to build a working computer. The difference engine (pictured above) was built entirely out of standard Lego pieces.
6
A Royal Road to Love,
Marriage, Establishing Three Households,
The Birth of Byron
[1835-1836]
Though Ada continued to correspond with Annabella and Livy in 1835, her social life centered on the Somerville family and London activities. It took several hours to go by carriage from Fordhook to London, and Ada often spent the night at the Somerville home in Chelsea.
Woronzow Greig wrote a short biography of Ada after her death, about twenty handwritten pages, which is found in the Somerville collection. He recalled that when he first met Ada, she still suffered lingering side effects from her bout with the measles—giddiness while riding horses and moments of nervousness and fatigue. However, Ada’s busy schedule—her continued interest in mathematics as well as her harp, riding, and active social life—filled every moment. In a letter it appears that Mrs Somerville was concerned that Ada was overdoing all these activities. Ada’s reaction, there was no stopping her.
Ada, the Enchantress of Numbers:Poetical Science Page 4
