Passages from the Life of a Philosopher

By Charles Babbage.

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“I’m a philosopher. Confound them all⁠—
Birds, beasts, and men; but no, not womankind.”

Don Juan

“I now gave my mind to philosophy: the great object of my ambition was to make out a complete system of the universe, including and comprehending the origin, causes, consequences, and termination of all things. Instead of countenance, encouragement, and applause, which I should have received from everyone who has the true dignity of an oyster at heart, I was exposed to calumny and misrepresentation. While engaged in my great work on the universe, some even went so far as to accuse me of infidelity;⁠—such is the malignity of oysters.”

Autobiography of an Oyster deciphered by the aid of photography in the shell of a philosopher of that race⁠—recently scalloped

Sire,

In dedicating this volume to your Majesty, I am also doing an act of justice to the memory of your illustrious father.

In 1840, the King, Charles Albert, invited the learned of Italy to assemble in his capital. At the request of her most gifted Analyst, I brought with me the drawings and explanations of the Analytical Engine. These were thoroughly examined and their truth acknowledged by Italy’s choicest sons.

To the King, your father, I am indebted for the first public and official acknowledgment of this invention.

I am happy in thus expressing my deep sense of that obligation to his son, the Sovereign of united Italy, the country of Archimedes and of Galileo.

Preface

Some men write their lives to save themselves from ennui, careless of the amount they inflict on their readers.

Others write their personal history, lest some kind friend should survive them, and, in showing off his own talent, unwittingly show them up.

Others, again, write their own life from a different motive⁠—from fear that the vampires of literature might make it their prey.

I have frequently had applications to write my life, both from my countrymen and from foreigners. Some caterers for the public offered to pay me for it. Others required that I should pay them for its insertion; others offered to insert it without charge. One proposed to give me a quarter of a column gratis, and as many additional lines of eloge as I chose to write and pay for at ten-pence per line. To many of these I sent a list of my works, with the remark that they formed the best life of an author; but nobody cared to insert them.

I have no desire to write my own biography, as long as I have strength and means to do better work.

The remarkable circumstances attending those Calculating Machines, on which I have spent so large a portion of my life, make me wish to place on record some account of their past history. As, however, such a work would be utterly uninteresting to the greater part of my countrymen, I thought it might be rendered less unpalatable by relating some of my experience amongst various classes of society, widely differing from each other, in which I have occasionally mixed.

This volume does not aspire to the name of an autobiography. It relates a variety of isolated circumstances in which I have taken part⁠—some of them arranged in the order of time, and others grouped together in separate chapters, from similarity of subject.

The selection has been made in some cases from the importance of the matter. In others, from the celebrity of the persons concerned; whilst several of them furnish interesting illustrations of human character.

Passages from the Life of a Philosopher

What is there in a name? It is merely an empty basket, until you put something into it. My earliest visit to the Continent taught me the value of such a basket, filled with the name of my venerable friend the first Herschel, ere yet my younger friend his son, had adorned his distinguished patronymic with the additional laurels of his own well-earned fame.

The inheritance of a celebrated name is not, however, without its disadvantages. This truth I never found more fully appreciated, nor more admirably expressed, than in a conversation with the son of Filangieri, the author of the celebrated Treatise on Legislation, with whom I became acquainted at Naples, and in whose company I visited several of the most interesting institutions of that capital.

In the course of one of our drives, I alluded to the advantages of inheriting a distinguished name, as in the case of the second Herschel. His remark was, “For my own part, I think it a great disadvantage. Such a man must feel in the position of one inheriting a vast estate, so deeply mortgaged that he can never hope, by any efforts of his own, to redeem it.”

Without reverting to the philosophic, but unromantic, views of our origin taken by Darwin, I shall pass over the long history of our progress from a monad up to man, and commence tracing my ancestry as the world generally do: namely, as soon as there is the slightest ground for conjecture. Although I have contended for the Mosaic date of the creation of man as long as I decently could, and have even endeavoured to explain away1 some of the facts relied upon to prove man’s long anterior origin; yet I must admit that the continual accumulation of evidence probably will, at last, compel me to acknowledge that, in this single instance, the writings of Moses may have been misapprehended.

Let us, therefore, take for granted that man and certain extinct races of animals lived together, thousands of years before Adam. We find, at that period, a race who formed knives, and hammers, and arrowheads out of flint. Now, considering my own inveterate habit of contriving tools, it is more probable that I should derive my passion by hereditary transmission from these original toolmakers, than from any other inferior race existing at that period.

Many years ago I met a very agreeable party at Mr. Rogers’ table. Somebody introduced the subject of ancestry. I remarked that most people are reluctant to acknowledge as their father or grandfather, any person who had committed a dishonest action or a crime. But that no one ever scrupled to be proud of a remote ancestor, even though he might have been a thief or a murderer. Various remarks were made, and reasons assigned, for this tendency of the educated mind. I then turned to my next neighbour, Sir Robert H. Inglis, and asked him what he would do, supposing he possessed undoubted documents, that he was lineally descended from Cain.

Sir Robert said he was at that moment proposing to himself the very same question. After some consideration, he said he should burn them; and then inquired what I should do in the same circumstances. My reply was, that I should preserve them: but simply because I thought the preservation of any fact might ultimately be useful.

I possess no evidence that I am descended from Cain. If any herald suppose that there may be such a presumption, I think it must arise from his confounding Cain with Tubal Cain, who was a great worker in iron. Still, however he might argue that, the probabilities are in favour of his opinion: for I, too, work in iron. But a friend of mine, to whose kind criticisms I am much indebted, suggests that as Tubal Cain invented the Organ, this probability is opposed to the former one.

The next step in my pedigree is to determine whence the origin of my modern family name.

Some have supposed it to be derived from the cry of sheep. If so, that would point to a descent from the Shepherd Kings. Others have supposed it is derived from the name of a place called Bab or Babb, as we have, in the West of England, Bab Tor, Babbacombe, etc. But this is evidently erroneous; for, when a people took possession of a desert country, its various localities could possess no names; consequently, the colonists could not take names from the country to which they migrated, but would very naturally give their own names to the several lands they appropriated: “mais revenons à nos moutons.”

How my blood was transmitted to me through more modern races, is quite immaterial, seeing the admitted antiquity of the flint-workers.

In recent times, that is, since the Conquest, my knowledge of the history of my family is limited by the unfortunate omission of my name from the roll of William’s followers. Those who are curious about the subject, and are idlers, may, if they think it worth while, search all the parish registers in the West of England and elsewhere.

The light I can throw upon it is not great, and rests on a few documents, and on family tradition. During the past four generations I have no surviving collateral relatives of my own name.

The name of Babbage is not uncommon in the West of England. One day during my boyhood, I observed it over a small grocer’s shop, whilst riding through the town of Chudley. I dismounted, went into the shop, purchased some figs, and found a very old man of whom I made inquiry as to his family. He had not a good memory himself, but his wife told me that his name was Babb when she married him, and that it was only during the last twenty years he had adopted the name of Babbage, which, the old man thought, sounded better. Of course I told his wife that I entirely agreed with her husband, and thought him a very sensible fellow.

The craft most frequently practised by my ancestors seems to have been that of a goldsmith, although several are believed to have practised less dignified trades.

In the time of Henry the Eighth one of my ancestors, together with a hundred men, were taken prisoners at the siege of Calais.

When William the Third landed in Torbay, another ancestor of mine, a yeoman possessing some small estate, undertook to distribute his proclamations. For this bit of high treason he was rewarded with a silver medal, which I well remember seeing, when I was a boy. It had descended to a very venerable and truthful old lady, an unmarried aunt, the historian of our family, on whose authority the identity of the medal I saw with that given by King William must rest.

Another ancestor married one of two daughters, the only children of a wealthy physician, Dr. Burthogge, an intimate friend and correspondent of John Locke.

Somewhere about 1700 a member of my family, one Richard Babbage, who appears to have been a very wild fellow, having tried his hand at various trades, and given them all up, offended a wealthy relative.

To punish this idleness, his relative entailed all his large estates upon eleven different people, after whom he gave it to this Richard Babbage, who, had there been no entail, would have taken them as heir-at-law.

Ten of these lives had dropped, and the eleventh was in a consumption, when Richard Babbage took it into his head to go off to America with Bamfylde Moore Carew, the King of the Beggars.

The last only of the eleven lives existed when he embarked, and that life expired within twelve months after Richard Babbage sailed. The estates remained in possession of the representatives of the eleventh in the entail.

If it could have been proved that Richard Babbage had survived twelve months after his voyage to America, these estates would have remained in my own branch of the family.

I possess a letter from Richard Babbage, dated on board the ship in which he sailed for America.

In the year 1773 it became necessary to sell a portion of this property, for the purpose of building a church at Ashbrenton. A private Act of Parliament was passed for that purpose, in which the rights of the true heir were reserved.

From my earliest years I had a great desire to inquire into the causes of all those little things and events which astonish the childish mind. At a later period I commenced the still more important inquiry into those laws of thought and those aids which assist the human mind in passing from received knowledge to that other knowledge then unknown to our race. I now think it fit to record some of those views to which, at various periods of my life, my reasoning has led me. Truth only has been the object of my search, and I am not conscious of ever having turned aside in my inquiries from any fear of the conclusions to which they might lead.

As it may be interesting to some of those who will hereafter read these lines, I shall briefly mention a few events of my earliest, and even of my childish years. My parents being born at a certain period of history, and in a certain latitude and longitude, of course followed the religion of their country. They brought me up in the Protestant form of the Christian faith. My excellent mother taught me the usual forms of my daily and nightly prayer; and neither in my father nor my mother was there any mixture of bigotry and intolerance on the one hand, nor on the other of that unbecoming and familiar mode of addressing the Almighty which afterwards so much disgusted me in my youthful years.

My invariable question on receiving any new toy, was “Mamma, what is inside of it?” Until this information was obtained those around me had no repose, and the toy itself, I have been told, was generally broken open if the answer did not satisfy my own little ideas of the “fitness of things.”

During my boyhood my mother took me to several exhibitions of machinery. I well remember one of them in Hanover Square, by a man who called himself Merlin. I was so greatly interested in it, that the Exhibitor remarked the circumstance, and after explaining some of the objects to which the public had access, proposed to my mother to take me up to his workshop, where I should see still more wonderful automata. We accordingly ascended to the attic. There were two uncovered female figures of silver, about twelve inches high.

One of these walked or rather glided along a space of about four feet, when she turned round and went back to her original place. She used an eyeglass occasionally, and bowed frequently, as if recognizing her acquaintances. The motions of her limbs were singularly graceful.

The other silver figure was an admirable danseuse, with a bird on the fore finger of her right hand, which wagged its tail, flapped its wings, and opened its beak. This lady attitudinized in a most fascinating manner. Her eyes were full of imagination, and irresistible.

These silver figures were the chef-d’œuvres of the artist: they had cost him years of unwearied labour, and were not even then finished.

After I left Devonshire I was placed at a school in the neighbourhood of London, in which there were about thirty boys.

My first experience was unfortunate, and probably gave an unfavourable turn to my whole career during my residence of three years.

After I had been at school a few weeks, I went with one of my companions into the playground in the dusk of the evening. We heard a noise, as of people talking in an orchard at some distance, which belonged to our master. As the orchard had recently been robbed, we thought that thieves were again at work. We accordingly climbed over the boundary wall, ran across the field, and saw in the orchard beyond a couple of fellows evidently running away. We pursued as fast as our legs could carry us, and just got up to the supposed thieves at the ditch on the opposite side of the orchard.

A roar of laughter then greeted us from two of our own companions, who had entered the orchard for the purpose of getting some manure for their flowers out of a rotten mulberry-tree. These boys were aware of our mistake, and had humoured it.

We now returned all together towards the playground, when we met our master, who immediately pronounced that we were each fined one shilling for being out of bounds. We two boys who had gone out of bounds to protect our master’s property, and who if thieves had really been there would probably have been half-killed by them, attempted to remonstrate and explain the case; but all remonstrance was vain, and we were accordingly fined. I never forgot that injustice.

The schoolroom adjoined the house, but was not directly connected with it. It contained a library of about three hundred volumes on various subjects, generally very well selected; it also contained one or two works on subjects which do not usually attract at that period of life. I derived much advantage from this library; and I now mention it because I think it of great importance that a library should exist in every schoolroom.

Amongst the books was a treatise on Algebra, called Ward’s Young Mathematician’s Guide. I was always partial to my arithmetical lessons, but this book attracted my particular attention. After I had been at this school for about a twelvemonth, I proposed to one of my schoolfellows, who was of a studious habit, that we should get up every morning at three o’clock, light a fire in the schoolroom, and work until five or half-past five. We accomplished this pretty regularly for several months. Our plan had, however, become partially known to a few of our companions. One of these, a tall boy, bigger than ourselves, having heard of it, asked me to allow him to get up with us, urging that his sole object was to study, and that it would be of great importance to him in afterlife. I had the cruelty to refuse this very reasonable request. The subject has often recurred to my memory, but never without regret.

Another of my young companions, Frederick Marryat,3 made the same request, but not with the same motive. I told him we got up in order to work; that he would only play, and that we should then be found out. After some time, having exhausted all his arguments, Marryat told me he was determined to get up, and would do it whether I liked it or not.

Marryat slept in the same room as myself: it contained five beds. Our room opened upon a landing, and its door was exactly opposite that of the master. A flight of stairs led up to a passage just over the room in which the master and mistress slept. Passing along this passage, another flight of stairs led down, on the other side of the master’s bedroom, to another landing, from which another flight of stairs led down to the external door of the house, leading by a long passage to the schoolroom.

Through this devious course I had cautiously threaded my way, calling up my companion in his room at the top of the last flight of stairs, almost every night for several months.

One night on trying to open the door of my own bedroom, I found Marryat’s bed projecting a little before the door, so that I could not open it. I perceived that this was done purposely, in order that I might awaken him. I therefore cautiously, and by degrees, pushed his bed back without awaking him, and went as usual to my work. This occurred two or three nights successively.

One night, however, I found a piece of packthread tied to the door lock, which I traced to Marryat’s bed, and concluded it was tied to his arm or hand. I merely untied the cord from the lock, and passed on.

A few nights after I found it impossible to untie the cord, so I cut it with my pocketknife. The cord then became thicker and thicker for several nights, but still my penknife did its work.

One night I found a small chain fixed to the lock, and passing thence into Marryat’s bed. This defeated my efforts for that night, and I retired to my own bed. The next night I was provided with a pair of plyers, and unbent one of the links, leaving the two portions attached to Marryat’s arm and to the lock of the door. This occurred several times, varying by stouter chains, and by having a padlock which I could not pick in the dark.

At last one morning I found a chain too strong for the tools I possessed; so I retired to my own bed, defeated. The next night, however, I provided myself with a ball of packthread. As soon as I heard by his breathing that Marryat was asleep, I crept over to the door, drew one end of my ball of packthread through a link of the too-powerful chain, and bringing it back with me to bed, gave it a sudden jerk by pulling both ends of the packthread passing through the link of the chain.

Marryat jumped up, put out his hand to the door, found his chain all right, and then lay down. As soon as he was asleep again, I repeated the operation. Having awakened him for the third time, I let go one end of the string, and drew it back by the other, so that he was unable at daylight to detect the cause.

At last, however, I found it expedient to enter into a treaty of peace, the basis of which was that I should allow Marryat to join the night party; but that nobody else should be admitted. This continued for a short time; but, one by one, three or four other boys, friends of Marryat, joined our party, and, as I had anticipated, no work was done. We all got to play; we let off fireworks in the playground, and were of course discovered.

Our master read us a very grave lecture at breakfast upon the impropriety of this irregular system of turning night into day, and pointed out its injurious effects upon the health. This, he said, was so remarkable that he could distinguish by their pallid countenances those who had taken part in it. Now he certainly did point out every boy who had been up on the night we were detected. But it appeared to me very odd that the same means of judging had not enabled him long before to discover the two boys who had for several months habitually practised this system of turning night into day.

Another of our pranks never received its solution in our master’s mind; indeed I myself scarcely knew its early history. Somehow or other, a Russian young gentleman, who was a parlour-boarder, had I believe, expatiated to Marryat on the virtues of Cognac.

One evening my friend came to me with a quart bottle of what he called excellent stuff. A council was held amongst a few of us boys to decide how we should dispose of this treasure. I did not myself much admire the liquid, but suggested that it might be very good when mixed up with a lot of treacle. This thought was unanimously adopted, and a subscription made to purchase the treacle. Having no vessel sufficiently large to hold the intended mixture, I proposed to take one of our garden-pots, stopping up the hole in its bottom with a cork.

A good big earthen vessel, thus extemporised, was then filled with this wonderful mixture. A spoon or two, an oyster-shell, and various other contrivances delivered it to its numerous consumers, and all the boys got a greater or less share, according to their taste for this extraordinary liqueur.

The feast was over, the garden-pot was restored to its owner, and the treacled lips of the boys had been wiped with their handkerchiefs or on their coat-sleeves, when the bell announced that it was prayer-time. We all knelt in silence at our respective desks. As soon as the prayers were over, one of the oddest scenes occurred.

Many boys rose up from their knees⁠—but some fell down again. Some turned round several times, and then fell. Some turned round so often that they resembled spinning dervishes. Others were only more stupid than usual; some complained of being sick; many were very sleepy; others were sound asleep, and had to be carried to bed; some talked fast and heroically, two attempted psalmody, but none listened.

All investigation at the time was useless: we were sent off to bed as quickly as possible. It was only known that Count Cognac had married the sweet Miss Treacle, whom all the boys knew and loved, and who lodged at the grocer’s, in the neighbouring village. But I believe neither the pedigree of the bridegroom nor his domicile were ever discovered. It is probable that he was of French origin, and dwelt in a cellar.

After I left this school I was for a few years under the care of an excellent clergyman in the neighbourhood of Cambridge. There were only six boys; but I fear I did not derive from it all the advantage that I might have done. I came into frequent contact with the Rev. Charles Simeon, and with many of his enthusiastic disciples. Every Sunday I had to write from memory an abstract of the sermon he preached in our village. Even at that period of my life I had a taste for generalization. Accordingly, having generalized some of Mr. Simeon’s sermons up to a kind of skeleton form, I tried, by way of experiment, to fill up such a form in a sermon of my own composing from the text of “Alexander the coppersmith hath done us much harm.” As well as I remember, there were in my sermon some queer deductions from this text; but then they fulfilled all the usual conditions of our sermons: so thought also two of my companions to whom I communicated in confidence this new manufacture.

By some unexplained circumstance my sermon relating to copper being isomorphous with Simeon’s own productions, got by substitution into the hands of our master as the recollections of one of the other boys. Thereupon arose an awful explosion which I decline to paint.

I did, however, learn something at this school, for I observed a striking illustration of the Economy of Manufactures. Mr. Simeon had the cure of a very wicked parish in Cambridge, whilst my instructor held that of a tolerably decent country village. If each minister had stuck to the instruction of his own parish, it would have necessitated the manufacture of four sermons per week, whilst, by this beneficial interchange of duties, only two were required.

Each congregation enjoyed also another advantage from this arrangement⁠—the advantage of variety, which, when moderately indulged in, excites the appetite.

My father, with a view of acquiring some information which might be of use to me at Cambridge, had consulted a tutor of one of the colleges, who was passing his long vacation at the neighbouring watering-place, Teignmouth. He dined with us frequently. The advice of the Rev. Doctor was quite sound, but very limited. It might be summed up in one short sentence: “Advise your son not to purchase his wine in Cambridge.”

Previously to my entrance at Trinity College, Cambridge, I resided for a time at Totnes, under the guidance of an Oxford tutor, who undertook to superintend my classical studies only.

During my residence at this place I accidentally heard, for the first time, of an idea of forming a universal language. I was much fascinated by it, and, soon after, proceeded to write a kind of grammar, and then to devise a dictionary. Some trace of the former, I think, I still possess: but I was stopped in my idea of making a universal dictionary by the apparent impossibility of arranging signs in any consecutive order, so as to find, as in a dictionary, the meaning of each when wanted. It was only after I had been some time at Cambridge that I became acquainted with the work of Bishop Wilkins on Universal Language.

Being passionately fond of algebra, I had instructed myself by means of Ward’s Young Mathematician’s Guide, which had casually fallen into my hands at school. I now employed all my leisure in studying such mathematical works as accident brought to my knowledge. Amongst these were Humphrey Ditton’s Fluxions, of which I could make nothing; Madame Agnesi’s Analytical Institutions, from which I acquired some knowledge; Woodhouse’s Principles of Analytical Calculation, from which I learned the notation of Leibnitz; and Lagrange’s Théorie des Fonctions. I possessed also the Fluxions of Maclaurin and of Simpson.

Thus it happened that when I went to Cambridge I could work out such questions as the very moderate amount of mathematics which I then possessed admitted, with equal facility, in the dots of Newton, the d’s of Leibnitz, or the dashes of Lagrange. I had, however, met with many difficulties, and looked forward with intense delight to the certainty of having them all removed on my arrival at Cambridge. I had in my imagination formed a plan for the institution amongst my future friends of a chess club, and also of another club for the discussion of mathematical subjects.

In 1811, during the war, it was very difficult to procure foreign books. I had heard of the great work of Lacroix, on the Differential and Integral Calculus, which I longed to possess, and being misinformed that its price was two guineas, I resolved to purchase it in London on my passage to Cambridge. As soon as I arrived I went to the French bookseller, Dulau, and to my great surprise found that the price of the book was seven guineas. After much thought I made the costly purchase, went on immediately to Cambridge, saw my tutor, Hudson, got lodgings, and then spent the greater part of the night in turning over the pages of my newly-acquired purchase. After a few days, I went to my public tutor Hudson, to ask the explanation of one of my mathematical difficulties. He listened to my question, said it would not be asked in the Senate House, and was of no sort of consequence, and advised me to get up the earlier subjects of the university studies.

After some little while I went to ask the explanation of another difficulty from one of the lecturers. He treated the question just in the same way. I made a third effort to be enlightened about what was really a doubtful question, and felt satisfied that the person I addressed knew nothing of the matter, although he took some pains to disguise his ignorance.

I thus acquired a distaste for the routine of the studies of the place, and devoured the papers of Euler and other mathematicians, scattered through innumerable volumes of the academies of Petersburgh, Berlin, and Paris, which the libraries I had recourse to contained.

Under these circumstances it was not surprising that I should perceive and be penetrated with the superior power of the notation of Leibnitz.

At an early period, probably at the commencement of the second year of my residence at Cambridge, a friend of mine, Michael Slegg, of Trinity, was taking wine with me, discussing mathematical subjects, to which he also was enthusiastically attached. Hearing the chapel bell ring, he took leave of me, promising to return for a cup of coffee.

At this period Cambridge was agitated by a fierce controversy. Societies had been formed for printing and circulating the Bible. One party proposed to circulate it with notes, in order to make it intelligible; whilst the other scornfully rejected all explanations of the word of God as profane attempts to mend that which was perfect.

The walls of the town were placarded with broadsides, and posters were sent from house to house. One of the latter form of advertisement was lying upon my table when Slegg left me. Taking up the paper, and looking through it, I thought it, from its exaggerated tone, a good subject for a parody.

I then drew up the sketch of a society to be instituted for translating the small work of Lacroix on the Differential and Integral Lacroix. It proposed that we should have periodical meetings for the propagation of d’s; and consigned to perdition all who supported the heresy of dots. It maintained that the work of Lacroix was so perfect that any comment was unnecessary.

On Slegg’s return from chapel I put the parody into his hands. My friend enjoyed the joke heartily, and at parting asked my permission to show the parody to a mathematical friend of his, Mr. Bromhead.4

The next day Slegg called on me, and said that he had put the joke into the hand of his friend, who, after laughing heartily, remarked that it was too good a joke to be lost, and proposed seriously that we should form a society for the cultivation of mathematics.

The next day Bromhead called on me. We talked the subject over, and agreed to hold a meeting at his lodgings for the purpose of forming a society for the promotion of analysis.

At that meeting, besides the projectors, there were present Herschel, Peacock, D’Arblay,5 Ryan,6 Robinson,7 Frederick Maule,8 and several others. We constituted ourselves “The Analytical Society;” hired a meeting-room, open daily; held meetings, read papers, and discussed them. Of course we were much ridiculed by the Dons; and, not being put down, it was darkly hinted that we were young infidels, and that no good would come of us.

In the meantime we quietly pursued our course, and at last resolved to publish a volume of our Transactions. Owing to the illness of one of the number, and to various other circumstances, the volume which was published was entirely contributed by Herschel and myself.

At last our work was printed, and it became necessary to decide upon a title. Recalling the slight imputation which had been made upon our faith, I suggested that the most appropriate title would be⁠—

The Principles of pure D-ism in opposition to the Dot-age of the University.9

In thus reviving this wicked pun, I ought at the same time to record an instance of forgiveness unparalleled in history. Fourteen years after, being then at Rome, I accidentally read in Galignani’s newspaper the following paragraph, dated Cambridge:⁠—“Yesterday the bells of St. Mary rang on the election of Mr. Babbage as Lucasian Professor of Mathematics.”

If this event had happened during the lifetime of my father, it would have been most gratifying to myself, because, whilst it would have given him much pleasure, it would then also have afforded intense delight to my mother.

I concluded that the next post would bring me the official confirmation of this report, and after some consideration I sketched the draft of a letter, in which I proposed to thank the University sincerely for the honour they had done me, but to decline it.

This sketch of a letter was hardly dry when two of my intimate friends, the Rev. Mr. Lunn and Mr. Beilby Thompson,10 who resided close to me in the Piazza del Populo, came over to congratulate me on the appointment. I showed them my proposed reply, against which they earnestly protested. Their first, and as they believed their strongest, reason was that it would give so much pleasure to my mother. To this I answered that my mother’s opinion of her son had been confirmed by the reception he had met with in every foreign country he had visited, and that this, in her estimation, would add but little to it. To their next argument I had no satisfactory answer. It was that this election could not have occurred unless some friends of mine in England had taken active measures to promote it; that some of these might have been personal friends, but that many others might have exerted themselves entirely upon principle, and that it would be harsh to disappoint such friends, and reject such a compliment.

My own feelings were of a mixed nature. I saw the vast field that the Difference Engine had opened out; for, before I left England in the previous year, I had extended its mechanism to the tabulation of functions having no constant difference, and more particularly I had arrived at the knowledge of the entire command it would have over the computation of the most important classes of tables, those of astronomy and of navigation. I was also most anxious to give my whole time to the completion of the mechanism of the Difference Engine No. 1 which I had then in hand. Small as the admitted duties of the Lucasian Chair were, I felt that they would absorb time which I thought better devoted to the completion of the Difference Engine. If I had then been aware that the lapse of a few years would have thrown upon me the enormous labour which the Analytical Engine absorbed, no motive short of absolute necessity would have induced me to accept any office which might, in the slightest degree, withdraw my attention from its contrivance.

The result of this consultation with my two friends was, that I determined to accept the Chair of Newton, and to hold it for a few years. In 1839 the demands of the Analytical Engine upon my attention had become so incessant and so exhausting, that even the few duties of the Lucasian Chair had a sensible effect in impairing my bodily strength. I therefore sent in my resignation.

In January, 1829, I visited Cambridge, to fulfil one of the first duties of my new office, the examination for Dr. Smith’s prizes.

These two prizes, of twenty-five pounds each, exercise a very curious and important influence. Usually three or four hundred young men are examined previously to taking their degree. The University officers examine and place them in the order of their mathematical merit. The class called Wranglers is the highest; of these the first is called the senior wrangler, the others the second and third, etc., wranglers.

All the young men who have just taken their degree, whether with or without honours, are qualified to compete for the Smith’s prizes by sending in notice to the electors, who consist of the three Professors of Geometry, Astronomy, and Physics, assisted occasionally by two official electors, the Vice-Chancellor and the Master of Trinity College. However, in point of fact, generally three, and rarely above six young men compete.

It is manifest that the University officers, who examine several hundred young men, cannot bestow the same minute attention upon each as those who, at the utmost, only examine six. Nor is this of any importance, except to the few first wranglers, who usually are candidates for these prizes. The consequence is that the examiners of the Smith’s prizes constitute, as it were, a court of appeal from the decision of the University officers. The decision of the latter is thus therefore, necessarily appealed against upon every occasion. Perhaps in one out of five or six cases the second or third wrangler obtains the first Smith’s prize. I may add that in the few cases known to me previously to my becoming an examiner, the public opinion of the University always approved those decisions, without implying any censure on the officers of the University.

In forming my set of questions, I consulted the late Dean of Ely and another friend, in order that I might not suddenly deviate too much from the usual style of examinations.

After having examined the young men, I sat up the whole night, carefully weighing the relative merits of their answers. I found, with some mortification, that, according to my marks, the second wrangler ought to have the first prize. I therefore put aside the papers until the day before the decision. I then took an unmarked copy of my questions, and put new numbers for their respective values. After very carefully going over the whole of the examination-papers again, I arrived almost exactly at my former conclusion.

On our meeting at the Vice-Chancellor’s, that functionary asked me, as the senior professor, what was my decision as to the two prizes. I stated that the result of my examination obliged me to award the first prize to the second wrangler. Professor Airy was then asked the same question. He made the same reply. Professor Lax being then asked, said he had arrived at the same conclusion as his two colleagues.

The Vice-Chancellor remarked that when we altered the arrangement of the University Examiners, it was very satisfactory that we should be unanimous. Professor Airy observed that this satisfaction was enhanced by the fact of the remarkable difference in the tastes of the three examiners.

The Vice-Chancellor, turning to me, asked whether it might be permitted to inquire the numbers we had respectively assigned to each candidate.

I and my colleagues immediately mentioned our numbers, which Professor Airy at once reduced to a common scale. On this it appeared that the number of marks assigned to each by Professor Airy and myself very nearly agreed, whilst that of Professor Lax differed but little.

On this occasion the first Smith’s prize was assigned to the second wrangler, Mr. Cavendish, now Duke of Devonshire, the present Chancellor of the University.

The result of the whole of my after-experience showed that amongst the highest men the peculiar tastes of the examiners had no effect in disturbing the proper decision.

I held the Chair of Newton for some few years, and still feel deeply grateful for the honour the University conferred upon me⁠—the only honour I ever received in my own country.11

I must now return to my pursuits during my residence at Cambridge, the account of which has been partially interrupted by the history of my appointment to the Chair of Newton.

Whilst I was an undergraduate, I lived probably in a greater variety of sets than any of my young companions. But my chief and choicest consisted of some ten or a dozen friends who usually breakfasted with me every Sunday after chapel; arriving at about nine, and remaining to between twelve and one o’clock. We discussed all knowable and many unknowable things.

At one time we resolved ourselves into a Ghost Club, and proceeded to collect evidence, and entered into a considerable correspondence upon the subject. Some of this was both interesting and instructive.

At another time we resolved ourselves into a Club which we called The Extractors. Its rules were as follows⁠—

1. 1st. Every member shall communicate his address to the Secretary once in six months.

2. 2nd. If this communication is delayed beyond twelve months, it shall be taken for granted that his relatives had shut him up as insane.

3. 3rd. Every effort legal and illegal shall be made to get him out of the madhouse. Hence the name of the club⁠—The Extractors.

4. 4th. Every candidate for admission as a member shall produce six certificates. Three that he is sane and three others that he is insane.

It has often occurred to me to inquire of my legal friends whether, if the sanity of any member of the club had been questioned in afterlife, he would have adduced the fact of membership of the Club of Extractors as an indication of sanity or of insanity.

During the first part of my residence at Cambridge, I played at chess very frequently, often with D’Arblay and with several other good players. There was at that period a fellow-commoner at Trinity named Brande, who devoted almost his whole time to the study of chess. I was invited to meet him one evening at the rooms of a common friend for the purpose of trying our strength.

On arriving at my friend’s rooms, I found a note informing me that he had gone to Newmarket, and had left coffee and the chessmen for us. I was myself tormented by great shyness, and my yet unseen adversary was, I understood, equally diffident. I was sitting before the chessboard when Brande entered. I rose, he advanced, sat down, and took a white and a black pawn from the board, which he held, one in either hand. I pointed with my finger to the left hand and won the move.

The game then commenced; it was rather a long one, and I won it: but not a word was exchanged until the end: when Brande uttered the first word. “Another?” To this I nodded assent.

How that game was decided I do not now remember; but the first sentence pronounced by either of us, was a remark by Brande, that he had lost the first game by a certain move of his white bishop. To this I replied, that I thought he was mistaken, and that the real cause of his losing the game arose from the use I had made of my knight two moves previously to his white bishop’s move.

We then immediately began to replace the men on the board in the positions they occupied at that particular point of the game when the white bishop’s move was made. Each took up any piece indiscriminately, and placed it without hesitation on the exact square on which it had stood. It then became apparent that the effective move to which I had referred was that of my knight.

Brande, during his residence at Cambridge, studied chess regularly several hours each day, and read almost every treatise on the subject. After he left college he travelled abroad, took lessons from every celebrated teacher, and played with all the most eminent players on the Continent.

At intervals of three or four years I occasionally met him in London. After the usual greeting he always proposed that we should play a game of chess.

I found on these occasions, that if I played any of the ordinary openings, such as are found in the books, I was sure to be beaten. The only way in which I had a chance of winning, was by making early in the game a move so bad that it had not been mentioned in any treatise. Brande possessed, and had read, almost every book upon the subject.

Another set which I frequently joined were addicted to sixpenny whist. It consisted of Higman, afterwards Tutor of Trinity; Follet, afterwards Attorney-General; of a learned and accomplished Dean still living, and I have no doubt still playing an excellent rubber, and myself. We not unfrequently sat from chapel-time in the evening until the sound of the morning chapel bell again called us to our religious duties.

I mixed occasionally with a different set of whist players at Jesus College. They played high: guinea points, and five guineas on the rubber. I was always a most welcome visitor, not from my skill at the game; but because I never played more than shilling points and five shillings on the rubber. Consequently my partner had what they considered an advantage: namely, that of playing guinea points with one of our adversaries and pound points with the other.

Totally different in character was another set in which I mixed. I was very fond of boating, not of the manual labour of rowing, but the more intellectual art of sailing. I kept a beautiful light, London-built boat, and occasionally took long voyages down the river, beyond Ely into the fens. To accomplish these trips, it was necessary to have two or three strong fellows to row when the wind failed or was contrary. These were useful friends upon my aquatic expeditions, but not being of exactly the same calibre as my friends of the Ghost Club, were very cruelly and disrespectfully called by them “my Tom fools.”

The plan of our voyage was thus:⁠—I sent my servant to the apothecary for a thing called an ægrotat, which I understood, for I never saw one, meant a certificate that I was indisposed, and that it would be injurious to my health to attend chapel, or hall, or lectures. This was forwarded to the college authorities.

I also directed my servant to order the cook to send me a large well-seasoned meat pie, a couple of fowls, etc. These were packed in a hamper with three or four bottles of wine and one of noyeau. We sailed when the wind was fair, and rowed when there was none. Whittlesea Mere was a very favourite resort for sailing, fishing, and shooting. Sometimes we reached Lynn. After various adventures and five or six days of hard exercise in the open air, we returned with our health more renovated than if the best physician had prescribed for us.

During my residence at Cambridge, Smithson Tennant was the Professor of Chemistry, and I attended his lectures. Having a spare room, I turned it into a kind of laboratory, in which Herschel worked with me, until he set up a rival one of his own. We both occasionally assisted the Professor in preparing his experiments. The science of chemistry had not then assumed the vast development it has now attained. I gave up its practical pursuit soon after I resided in London, but I have never regretted the time I bestowed upon it at the commencement of my career. I had hoped to have long continued to enjoy the friendship of my entertaining and valued instructor, and to have profited by his introducing me to the science of the metropolis, but his tragical fate deprived me of that advantage. Whilst riding with General Bulow across a drawbridge at Boulogne, the bolt having been displaced, Smithson Tennant was precipitated to the bottom, and killed on the spot. The General, having an earlier warning, set spurs to his horse, and just escaped a similar fate.

My views respecting the notation of Leibnitz now (1812) received confirmation from an extensive course of reading. I became convinced that the notation of fluxions must ultimately prove a strong impediment to the progress of English science. But I knew, also, that it was hopeless for any young and unknown author to attempt to introduce the notation of Leibnitz into an elementary work. This opinion naturally suggested to me the idea of translating the smaller work of Lacroix. It is possible, although I have no recollection of it, that the same idea may have occurred to several of my colleagues of the Analytical Society, but most of them were so occupied, first with their degree, and then with their examination for fellowships, that no steps were at that time taken by any of them on that subject.

Unencumbered by these distractions, I commenced the task, but at what period of time I do not exactly recollect. I had finished a portion of the translation, and laid it aside, when, some years afterwards, Peacock called on me in Devonshire Street, and stated that both Herschel and himself were convinced that the change from the dots to the d’s would not be accomplished until some foreign work of eminence should be translated into English. Peacock then proposed that I should either finish the translation which I had commenced, or that Herschel and himself should complete the remainder of my translation. I suggested that we should toss up which alternative to take. It was determined by lot that we should make a joint translation. Some months after, the translation of the small work of Lacroix was published.

For several years after, the progress of the notation of Leibnitz at Cambridge was slow. It is true that the tutors of the two largest colleges had adopted it, but it was taught at none of the other colleges.

It is always difficult to think and reason in a new language, and this difficulty discouraged all but men of energetic minds. I saw, however, that, by making it their interest to do so, the change might be accomplished. I therefore proposed to make a large collection of examples of the differential and integral calculus, consisting merely of the statement of each problem and its final solution. I foresaw that if such a publication existed, all those tutors who did not approve of the change of the Newtonian notation would yet, in order to save their own time and trouble, go to this collection of examples to find problems to set to their pupils. After a short time the use of the new signs would become familiar, and I anticipated their general adoption at Cambridge as a matter of course.

I commenced by copying out a large portion of the work of Hirsch. I then communicated to Peacock and Herschel my view, and proposed that they should each contribute a portion.

Peacock considerably modified my plan by giving the process of solution to a large number of the questions. Herschel prepared the questions in finite differences, and I supplied the examples to the calculus of functions. In a very few years the change was completely established; and thus at last the English cultivators of mathematical science, untrammelled by a limited and imperfect system of signs, entered on equal terms into competition with their continental rivals.

Calculating Machines comprise various pieces of mechanism for assisting the human mind in executing the operations of arithmetic. Some few of these perform the whole operation without any mental attention when once the given numbers have been put into the machine.

Others require a moderate portion of mental attention: these latter are generally of much simpler construction than the former, and it may also be added, are less useful.

The simplest way of deciding to which of these two classes any calculating machine belongs is to ask its maker⁠—Whether, when the numbers on which it is to operate are placed in the instrument, it is capable of arriving at its result by the mere motion of a spring, a descending weight, or any other constant force? If the answer be in the affirmative, the machine is really automatic; if otherwise, it is not self-acting.

Of the various machines I have had occasion to examine, many of those for Addition and Subtraction have been found to be automatic. Of machines for Multiplication and Division, which have fully come under my examination, I cannot at present recall one to my memory as absolutely fulfilling this condition.

The earliest idea that I can trace in my own mind of calculating arithmetical Tables by machinery arose in this manner:⁠—

One evening 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.”

I am indebted to my friend, the Rev. Dr. Robinson, the Master of the Temple, for this anecdote. The event must have happened either in 1812 or 1813.

About 1819 I was occupied with devising means for accurately dividing astronomical instruments, and had arrived at a plan which I thought was likely to succeed perfectly. I had also at that time been speculating about making machinery to compute arithmetical Tables.

One morning I called upon the late Dr. Wollaston, to consult him about my plan for dividing instruments. On talking over the matter, it turned out that my system was exactly that which had been described by the Duke de Chaulnes, in the Memoirs of the French Academy of Sciences, about fifty or sixty years before. I then mentioned my other idea of computing Tables by machinery, which Dr. Wollaston thought a more promising subject.

I considered that a machine to execute the mere isolated operations of arithmetic, would be comparatively of little value, unless it were very easily set to do its work, and unless it executed not only accurately, but with great rapidity, whatever it was required to do.

On the other hand, the method of differences supplied a general principle by which all Tables might be computed through limited intervals, by one uniform process. Again, the method of differences required the use of mechanism for Addition only. In order, however, to insure accuracy in the printed Tables, it was necessary that the machine which computed Tables should also set them up in type, or else supply a mould in which stereotype plates of those Tables could be cast.

I now began to sketch out arrangements for accomplishing the several partial processes which were required. The arithmetical part must consist of two distinct processes⁠—the power of adding one digit to another, and also of carrying the tens to the next digit, if it should be necessary.

The first idea was, naturally, to add each digit successively. This, however, would occupy much time if the numbers added together consisted of many places of figures.

The next step was to add all the digits of the two numbers each to each at the same instant, but reserving a certain mechanical memorandum, wherever a carriage became due. These carriages were then to be executed successively.

Having made various drawings, I now began to make models of some portions of the machine, to see how they would act. Each number was to be expressed upon wheels placed upon an axis; there being one wheel for each figure in the number operated upon.

Having arrived at a certain point in my progress, it became necessary to have teeth of a peculiar form cut upon these wheels. As my own lathe was not fit for this job, I took the wheels to a wheel-cutter at Lambeth, to whom I carefully conveyed my instructions, leaving with him a drawing as his guide.

These wheels arrived late one night, and the next morning I began putting them in action with my other mechanism, when, to my utter astonishment, I found they were quite unfit for their task. I examined the shape of their teeth, compared them with those in the drawings, and found they agreed perfectly; yet they could not perform their intended work. I had been so certain of the truth of my previous reasoning, that I now began to be somewhat uneasy. I reflected that, if the reasoning about which I had been so certain should prove to have been really fallacious, I could then no longer trust the power of my own reason. I therefore went over with my wheels to the artist who had formed the teeth, in order that I might arrive at some explanation of this extraordinary contradiction.

On conferring with him, it turned out that, when he had understood fully the peculiar form of the teeth of wheels, he discovered that his wheel-cutting engine had not got amongst its divisions that precise number which I had required. He therefore had asked me whether another number, which his machine possessed, would not equally answer my object. I had inadvertently replied in the affirmative. He then made arrangements for the precise number of teeth I required; and the new wheels performed their expected duty perfectly.

The next step was to devise means for printing the tables to be computed by this machine. My first plan was to make it put together moveable type. I proposed to make metal boxes, each containing 3,000 types of one of the ten digits. These types were to be made to pass out one by one from the bottom of their boxes, when required by the computing part of the machine.

But here a new difficulty arose. The attendant who put the types into the boxes might, by mistake, put a wrong type in one or more of them. This cause of error I removed in the following manner:⁠—There are usually certain notches in the side of the type. I caused these notches to be so placed that all the types of any given digit possessed the same characteristic notches, which no other type had. Thus, when the boxes were filled, by passing a small wire down these peculiar notches, it would be impeded in its passage, if there were included in the row a single wrong figure. Also, if any digit were accidentally turned upside down, it would be indicated by the stoppage of the testing wire.

One notch was reserved as common to every species of type. The object of this was that, before the types which the Difference Engine had used for its computation were removed from the iron platform on which they were placed, a steel wire should be passed through this common notch, and remain there. The tables, composed of moveable types, thus interlocked, could never have any of their figures drawn out by adhesion to the inking-roller, and then by possibility be restored in an inverted order. A small block of such figures tied together by a bit of string, remained unbroken for several years, although it was rather roughly used as a plaything by my children. One such box was finished, and delivered its type satisfactorily.

Another plan for printing the tables, was to place the ordinary printing type round the edges of wheels. Then, as each successive number was produced by the arithmetical part, the type-wheels would move down upon a plate of soft composition, upon which the tabular number would be impressed. This mould was formed of a mixture of plaster-of-Paris with other materials, so as to become hard in the course of a few hours.

The first difficulty arose from the impression of one tabular number on the mould being distorted by the succeeding one.

I was not then aware that a very slight depth of impression from the type would be quite sufficient. I surmounted the difficulty by previously passing a roller, having longitudinal wedge-shaped projections, over the plastic material. This formed a series of small depressions in the matrix between each line. Thus the expansion arising from the impression of one line partially filled up the small depression or ditch which occurred between each successive line.

The various minute difficulties of this kind were successively overcome; but subsequent experience has proved that the depth necessary for stereotype moulds is very small, and that even thick paper, prepared in a peculiar manner, is quite sufficient for the purpose.

Another series of experiments were, however, made for the purpose of punching the computed numbers upon copper plate. A special machine was contrived and constructed, which might be called a coordinate machine, because it moved the copper plate and steel punches in the direction of three rectangular coordinates. This machine was afterwards found very useful for many other purposes. It was, in fact, a general shaping machine, upon which many parts of the Difference Engine were formed.

Several specimens of surface and copperplate printing, as well as of the copper plates, produced by these means, were exhibited at the Exhibition of 1862.

I have proposed and drawn various machines for the purpose of calculating a series of numbers forming Tables by means of a certain system called “The Method of Differences,” which it is the object of this sketch to explain.

The first Difference Engine with which I am acquainted comprised a few figures, and was made by myself, between 1820 and June 1822. It consisted of from six to eight figures. A much larger and more perfect engine was subsequently commenced in 1823 for the Government.

It was proposed that this latter Difference Engine should have six orders of differences, each consisting of about twenty places of figures, and also that it should print the Tables it computed.

The small portion of it which was placed in the International Exhibition of 1862 was put together nearly thirty years ago. It was accompanied by various parts intended to enable it to print the results it calculated, either as a single copy on paper⁠—or by putting together moveable types⁠—or by stereotype plates taken from moulds punched by the machine⁠—or from copper plates impressed by it. The parts necessary for the execution of each of these processes were made, but these were not at that time attached to the calculating part of the machine.

A considerable number of the parts by which the printing was to be accomplished, as also several specimens of portions of tables punched on copper, and of stereotype moulds, were exhibited in a glass case adjacent to the Engine.

In 1834 Dr. Lardner published, in the Edinburgh Review,12 a very elaborate description of this portion of the machine, in which he explained clearly the method of Differences.

It is very singular that two persons, one resident in London, the other in Sweden, should both have been struck, on reading this review, with the simplicity of the mathematical principle of differences as applied to the calculation of Tables, and should have been so fascinated with it as to have undertaken to construct a machine of the kind.

Mr. Deacon, of Beaufort House, Strand, whose mechanical skill is well known, made, for his own satisfaction, a small model of the calculating part of such a machine, which was shown only to a few friends, and of the existence of which I was not aware until after the Swedish machine was brought to London.

Mr. Scheutz, an eminent printer at Stockholm, had far greater difficulties to encounter. The construction of mechanism, as well as the mathematical part of the question, was entirely new to him. He, however, undertook to make a machine having four differences, and fourteen places of figures, and capable of printing its own Tables.

After many years’ indefatigable labour, and an almost ruinous expense, aided by grants from his Government, by the constant assistance of his son, and by the support of many enlightened members of the Swedish Academy, he completed his Difference Engine. It was brought to London, and some time afterwards exhibited at the great Exhibition at Paris. It was then purchased for the Dudley Observatory at Albany by an enlightened and public-spirited merchant of that city, John F. Rathbone, Esq.

An exact copy of this machine was made by Messrs. Donkin and Co., for the English Government, and is now in use in the Registrar-General’s Department at Somerset House. It is very much to be regretted that this specimen of English workmanship was not exhibited in the International Exhibition.

Explanation of the Difference Engine

Those who are only familiar with ordinary arithmetic may, by following out with the pen some of the examples which will be given, easily make themselves acquainted with the simple principles on which the Difference Engine acts.

It is necessary to state distinctly at the outset, that the Difference Engine is not intended to answer special questions. Its object is to calculate and print a series of results formed according to given laws. These are called Tables⁠—many such are in use in various trades. For example⁠—there are collections of Tables of the amount of any number of pounds from 1 to 100lbs. of butchers’ meat at various prices per lb. Let us examine one of these Tables: viz.⁠—the price of meat 5d. per lb., we find

Number. Lbs.	Table. Price.
	s.	d.
1	0	5
2	0	10
3	1	3
4	1	8
5	2	1

There are two ways of computing this Table:⁠—

1. 1st. We might have multiplied the number of lbs. in each line by 5, the price per lb., and have put down the result in £ s. d., as in the 2nd column: or,

2. 2nd. We might have put down the price of 1lb., which is 5d., and have added five pence for each succeeding lb.

Let us now examine the relative advantages of each plan. We shall find that if we had multiplied each number of lbs. in the Table by 5, and put down the resulting amount, then every number in the Table would have been computed independently. If, therefore, an error had been committed, it would not have affected any but the single tabular number at which it had been made. On the other hand, if a single error had occurred in the system of computing by adding five at each step, any such error would have rendered the whole of the rest of the Table untrue.

Thus the system of calculating by differences, which is the easiest, is much more liable to error. It has, on the other hand, this great advantage: viz., that when the Table has been so computed, if we calculate its last term directly, and if it agree with the last term found by the continual addition of 5, we shall then be quite certain that every term throughout is correct. In the system of computing each term directly, we possess no such check upon our accuracy.

Now the Table we have been considering is, in fact, merely a Table whose first difference is constant and equal to five. If we express it in pence it becomes⁠—

	Table.	1st Difference.
1	5	5
2	10	5
3	15	5
4	20	5
5	25

Any machine, therefore, which could add one number to another, and at the same time retain the original number called the first difference for the next operation, would be able to compute all such Tables.

Let us now consider another form of Table which might readily occur to a boy playing with his marbles, or to a young lady with the balls of her solitaire board.

The boy may place a row of his marbles on the sand, at equal distances from each other, thus⁠—

He might then, beginning with the second, place two other marbles under each, thus⁠—

He might then, beginning with the third, place three other marbles under each group, and so on; commencing always one group later, and making the addition one marble more each time. The several groups would stand thus arranged⁠—

He will not fail to observe that he has thus formed a series of triangular groups, every group having an equal number of marbles in each of its three sides. Also that the side of each successive group contains one more marble than that of its preceding group.

Now an inquisitive boy would naturally count the numbers in each group and he would find them thus⁠—

1. 1,

2. 3,

3. 6,

4. 10,

5. 15,

6. 21

He might also want to know how many marbles the thirtieth or any other distant group might contain. Perhaps he might go to papa to obtain this information; but I much fear papa would snub him, and would tell him that it was nonsense⁠—that it was useless⁠—that nobody knew the number, and so forth. If the boy is told by papa, that he is not able to answer the question, then I recommend him to pay careful attention to whatever that father may at any time say, for he has overcome two of the greatest obstacles to the acquisition of knowledge⁠—inasmuch as he possesses the consciousness that he does not know⁠—and he has the moral courage to avow it.13

If papa fail to inform him, let him go to mamma, who will not fail to find means to satisfy her darling’s curiosity. In the meantime the author of this sketch will endeavour to lead his young friend to make use of his own common sense for the purpose of becoming better acquainted with the triangular figures he has formed with his marbles.

In the case of the Table of the price of butchers’ meat, it was obvious that it could be formed by adding the same constant difference continually to the first term. Now suppose we place the numbers of our groups of marbles in a column, as we did our prices of various weights of meat. Instead of adding a certain difference, as we did in the former case, let us subtract the figures representing each group of marbles from the figures of the succeeding group in the Table. The process will stand thus:⁠—

Number of the Group.	Table.	1st Difference.	2nd Difference.
	Number of Marbles in each Group.	Difference between the number of Marbles in each Group and that in the next.