PROF. BABBAGE, the next inventor whom we have to mention, being thus introduced to us, we may at once proceed to speak of his performances. It seems to have occurred to him as early as 1812 or 1813, that T. of logarithms might be calculated by machinery. We propose to draw from his own record-Passages from the Life of a Philosopher- such details as may best expound his efforts, and at the same time aid in a general understanding of this interesting subject:

I considered that a machine to execute the more 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 T., it was necessary that the machine which computed T. should also set them up in type, or else supply a mould in which stereotype plates of those T. 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 at the same instant, but reserving a certain mechanical memorandum whenever 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.

He then tells us how his first experiments failed; and the cause of the failure. He proceeds:

The next step was to devise means for printing the T. to be computed by this machine. My first plan was to put together movable type. I proposed to make metal boxes, each containing 3000 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 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. . . . By means of this common notch, precautions were taken to prevent disaster after the type was finally set :

Another plan for printing the T. 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 of the very slight depth of impression from the type would be quite sufficient.

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 co-ordinate machine, because it moved the copper plate and steel punches in the direction of three rectangular co-ordinates. This machine was afterwards found very useful for many other purposes.... In the end Mr. Babbage prepared various sketches, prob. answering each of the ends here described. Between the years 1820 and 1822 he actually constructed a DIFFERENCE ENGINE--the first which had ever been constructed. This, for the sake of distinction, we shall call Difference Engine No. 1.

About 1815 DR. ROGET invented a sliding scale of involution, known by his name. The instrument consists of one fixed and one movable scale, like a sliding rule. On the slide a line is logometrically divided, the divisions of one half being from 1 to 10, and repeated on the second half in the same order. The fixed scale is graduated in such a manner that each of its own divisions is set against its own respective logarithm on the slider; and consequently, all the numbers on the slider will be situated immediately under those numbers in the fixed scale of which they are the logarithms. Thus 3 on the fixed scale will stand under 100, and so on. The instrument is adapted to perform the operations of involution and evolution. [See 1851.]

In 1822 the following pub. appeared :-(1) Note respecting the Application of Machinery to the Calculation of Mathematical Tables. This appeared in the Memoirs of the Astronomical So. (2) A Letter to Sir H. Davy, P.R.S., on the Application of Machinery to the purpose of Calculating and Printing Mathematical Tables. (3) On the Theoretical Principles of the Machinery for Calculating Tables. This appeared in Brewster's Edin. Journ. of Science. Each of the above papers were from the pen of Charles Babbage. (4) Address to the Astronomical So. by Henry Thomas Colebrooke, Esq., F.R.S., President, on Presenting the first Gold Medal of the So. to Charles Babbage, Esq., for the Invention of the Calculating Engine.

In the above letter to Sir Humphry Davy, Mr. Babbage said his machine would "calculate tables governed by laws which have not been hitherto shown to be explicitly determinable, or solve equations for which analytical methods of solution have not yet been contrived." He further said that the machine would take from the several boxes containing type the numbers which it calculated and place them side by side: thus

becoming at the same time a substitute for the compositor and the computor, by which means all error in copying as well as printing is removed.

In his address to the Astronomical So. Mr. Colebrooke said:

The principle which essentially distinguishes Mr. Babbage's invention from all these [the inventions which had preceded him] is, that it proposes to calculate a series of numbers following any law by the aid of differences; and that by setting a few figures at the outset, a long series of numbers is readily produced by a mechanical operation. The method of differences in a very wide sense is the mathematical principle of the contrivance. A machine to add a number of arbitrary figures together is no economy of time or trouble; since each individual figure must be placed in the machine; but it is otherwise when those figures follow some law. The insertion of a few at first determines the magnitude of the next, and those of the succeeding. It is this constant repetition of similar operations which renders the computation of tables a fit subject for the application of machinery. Mr. Babbage's invention puts an engine in the place of the computor; the question is set to the instrument, or the instrument is set to the question; and by simply giving it motion the solution is wrought, and a string of answers is exhibited.

Regarding the printing its own Tables Mr. Colebrooke says:

The usefulness of the instrument is thus more than doubled; for it not only saves time and trouble in transcribing results into a tabular form, and setting types for the printing of the table, but it likewise accomplishes the yet more important object of insuring accuracy, obviating numerous sources of error through the careless hands of transcribers and compositors.

No sooner was the Difference Engine No. I completed, than Mr. Babbage received instructions to construct another and more comprehensive one, for and at the expense of the the English Gov. This [Difference Engine No. 2] he commenced in 1823. It was proposed that this new engine should have six orders of differences, each consisting of about 20 places of figures; and also that it should print the tables it computed. The printing might be by either of the processes already described. While the construction of this second machine is in hand we must proceed with our chronicle. [See 1836.]

In 1824 there appeared in the Memoirs of the Astronomical So. a paper by Charles Babbage, Observations on the Application of Machinery to the Computations of Mathematical Tables.

In 1829 there was pub. a Report by the Committee appointed by the Council of the Royal So. to consider the subject referred in a Communication received by them from the Treasury, respecting Mr. Babbage's Calculating Engine, and to Report thereupon. This Committee, which had been appointed at the instance of the Gov., consisted of Mr. Davies Gilbert, then President; the Secretaries; Sir John Herschel; Mr. Francis Baily; Mr. Brunel, Engineer; Mr. Donkin, Engineer; Mr. G. Rennie, Engineer; Mr. Barton, Comptroller of the Mint; and Mr. Warburton, M. P. The voluminous drawings, the various tools, and the portion of the machinery then executed, underwent a close and elaborate examination by this Committee. We can only give an abstract of the report:

They had declined the consideration of the principle on which the practicability of the machinery depended, and of the public utility of the object which it proposed to attain; because they considered the former fully admitted, and the latter obvious to all who considered the immense advantage of accurate numerical tables in all matters of calculation-especially those which related to astronomy and navigation, and the great variety and extent of those which it is professedly the object of the machinery to calculate and print with perfect accuracy; that absolute accuracy being one of the prominent pretensions of the undertaking, they had directed their attention especially to this point, by careful examination of the drawings, and of the work already executed, and by repeated conferences with Mr. Babbage on the subject; that the result of their inquiry was, that such precautions appear to have been taken in every part of the contrivance, and so fully aware was the inventor of every circumstance which might by possibility produce error, that they had no hesitation in stating their belief that these precautions were effectual, and whatever the machine would do it would do truly. They further stated that the progress which Mr. Babbage had then made, considering the very great difficulties to be overcome in an undertaking of so novel a kind, fully equalled any expectations that could reasonably have been formed; and that although several years had elapsed since the commencement of the undertaking, yet, when the necessity of constructing plans, sections, elevations, and working drawings of every part; of constructing, and in many cases of inventing, tools and machinery of great expense and complexity, necessary to form with the requisite precision parts of the apparatus differing from any which had previously been introduced in ordinary mechanical works; of making many trials to ascertain the value of each proposed contrivance; of altering, improving, and simplifying the drawings;-that considering all these matters, the Committee, instead of feeling surprised at the time which the work had occupied, felt more disposed to wonder at the possibility of accomplishing so much.

The Committee expressed their confident opinion of the adequacy of the machinery to work under all the friction and strain to which it could be exposed; of its durability, strength, solidity, and equilibrium; of the prevention of, or compensation for, wear by friction; of the accuracy of the various adjustments; and of the judgment and discretion displayed by the inventor in his determination to admit into the mechanism nothing but the very best and most finished workmanship; as a contrary course would have been false economy, and might have led to the loss of the whole capital expended on it.

Finally, considering all that had come before them, and relying on the talent and skill displayed by Mr. Babbage, as a mechanist, in the progress of this arduous undertaking, not less for what remained, than on the matured and digested plan and admirable execution of what was completed, the Committee did not hesitate to express their opinion, that in the then state of the engine, they regarded it as likely to fulfil the expectations entertained of it by its inventor.

It would be difficult to imagine a much stronger expression of opinion, and this too by men thoroughly competent to form a practical judgment. The work continued to progress, but in a very slow and interrupted manner, until 1833, when it became entirely relinquished.

In the Edin. Review, for July, 1834, appeared an art. by Dr. Lardner, mainly consisting of a description of the scientific appliances combined in the construction of Mr.

Babbage's "Calculating Engine"; but giving much information on many collateral subjects. From this we have already had occasion to quote. The following passage deserves a place here:

Its

There are nevertheless many persons who, admitting the great ingenuity of the contrivance, have, notwithstanding, been accustomed to regard it more in the light of a philosophical curiosity, than an instrument for purposes practically useful. This mistake-than which it is not possible to imagine a greater-has arisen mainly from the ignorance which prevails of the extensive utility of those numerical tables which it is the purpose of the engine in question to produce. There are also some persons, who, not considering the time requisite to bring any invention of this magnitude to perfection in all its details, incline to consider the delays which have taken place in its progress as presumptions against its practibility. These persons should, however, before they arrive at such a conclusion, reflect upon the time which was necessary to bring to perfection engines infinitely inferior in complexity and mechanical difficulty. Let them remember that-not to mention the invention of that machine-the improvements alone introduced into the steam-engine by the celebrated Watt occupied a period of not less than 20 years of the life of that distinguished person, and involved an expenditure of capital amounting to £50,000. The calculating machine is a contrivance even new in its details. inventor did not take it up already imperfectly formed, after having received the contributions of human ingenuity exercised upon it for a century or more. It has not, like almost all other great mechanical inventions, been gradually advanced to its present state through a series of failures, through difficulties encountered and overcome by a succession of projectors. It is not an object on which the light of various minds has thus been shed. It is, on the contrary, the production of solitary and individual thought-begun, advanced through each successive stage of improvement, and brought to perfection by one mind. Yet this creation of genius, from its first rude conception to its present state, has cost little more than half the time and not one-third of the expense consumed in bringing the steam-engine-previously far advanced in the course of improvement-to that state of comparative perfection in which it was left by Watt. Short as the period of time has been which the inventor has devoted to this enterprise, it has, nevertheless, been demonstrated, to the satisfaction of many scientific men of the first eminence, that the design in all its details, reduced, as it is, to a system of mechanical drawings, is complete; and requires only to be constructed in conformity with those plans to realize all that its inventor has promised.

This was high testimony in favour of the new and more comprehensive machine upon which the inventor was then engaged. But an equally high testimony to the ability of the writer of that art. has to be recorded, in the fact that two separate inventors actually constructed calculating machines from merely reading the descriptions given by Dr. Lardner therein of Mr. Babbage's appliances.

The first of these constructions was by MR. DEACON, of Beaufort House, Strand, a well-known mechanist, who, simply for his own satisfaction, constructed a small model of the calculating part of such a machine, which however was only shown to a few friends, and never made generally known. The other was the Swedish, of which we shall speak in due order.

The reader will have surmised, from the nature of the report by the Royal So., as also from Dr. Lardner's art., that some difficulties had originated regarding the progress of Mr. Babbage's Difference Engine No. 2. This was so. By 1834 the sum of £17,000 or more had been expended, and yet the machine was very far from complete. Mr. Babbage wavered regarding the completion. The Gov. began to waver on the subject of the expense-the limit of which appeared undefined. Several years' correspondence ensued. The reason for all this is now rendered clear. Mr. Babbage was doubtful "whether the discoveries which he was then advancing might not ultimately supersede the work already executed." [Passages, etc., p. 88.] Mr. Babbage had in fact conceived the idea of his Analytical Engine, of which we shall presently have to speak. In a letter to the Chancellor of the Exchequer, under date 20th Jan., 1836, he speaks of his new conception thus:

It is not only capable of accomplishing all those other complicated calculations which I had intended, but it also performs all calculations which were peculiar to the Difference Engine, both in less time, and to a greater extent; in fact it completely supersedes the Difference Engine.

We must now leave Mr. Babbage some few years to determine upon his future course of action in regard to the prosecution of his rival conceptions. [See 1848.]

In 1840 there was pub., in Paris, General Plan, No. 25, of Mr. Babbage's Great Calculating or Analytical Engine.

In 1842 there appeared in the Bibliothèque Universelle de Genève, General Menabrea's Sketch of the Analytical Engine invented by Charles Babbage. This was afterwards translated by the late Countess Lovelace [Lord Byron's daughter-"Ada, fair daughter of my home and heart"], with extensive notes by the translator.

We have next to speak of the second machine brought into existence by the practical descriptions of the Edinburgh Reviewer. It was constructed by HERR GEORGE SCHEUTZ, at that time the editor of a technological journal at Stockholm, and a practical printer. After reading the article in question, and satisfying himself of the practicability of construction, he laid the matter on one side to wait for an opportunity. Three years later, or in 1837, his son, HERR EDWARD SCHEUTZ, then a student at the Royal Technological Institute at Stockholm, undertook the construction of the machine, his father giving him the use of work-room, lathe and tools, with such other appliances as severe economy enabled him to procure. An application was made to the Gov. for assistance, and refused. By 1840, after many trials and modifications, a model was so far completed as to be able to calculate correctly "series with terms of 5 figures and one difference also of 5 figures." By April, 1842, its power was extended to "calculate similar series, with 2 or 3

orders of differences." In 1843, the machine, with the printing apparatus, was ready for inspection by the Royal Swedish Academy of Sciences, and after various trials a certificate was signed by several of the leading members of that body, under date 18th September, 1843, and containing the following:

The apparatus in question is composed of 3 parts. 1. The Calculating Machine. It cannot compute series of a higher degree than the third, nor does it give complete terms exceeding 5 figures; but in the nature of the mechanism there is nothing to prevent its extension to the working of series of any degree whatever, and to terms of as many figures as the purpose may require. 2. The Printing Machine. Every term given by the calculating apparatus is expressed by printed figures closely arranged in lines, as in a printed table, the lines being impressed on some softer material adapted to receive galvanoplastic or stereotyped copies. All the lines succeed each other very correctly in the same vertical column. 3. The Numbering Machine. With the printing machine another apparatus is combined, which prints the arguments before every term. The machine is put in motion by turning the handle of a winch, by means of which, and without further manipulations, the calculations as well as the printing and arranging of figures and lines are effected.

The inventors, furnished with this certificate, sought orders in various countries, but without success; and the machine remained shut up in its case for the ensuing 7 years.[See 1853 and 1854.]

In 1843 was pub. by Mr. Babbage-Statement of the Circumstances respecting Mr. Babbage's Calculating Engines.

By 1848 MR. BABBAGE had completely mastered the details of his Analytical Engine— that is, had reduced them to diagrams, or working drawings, capable of being understood and executed by skilled workmen. The scope and capabilities of this new machine we propose briefly to notice, and in doing this we shall not depart from the language of its

inventor:

The circular arrangement of the axes of the Difference Engine round large central wheels led to the most extended prospects. The whole of arithmetic now appeared within the grasp of mechanism. The most important part of the Analytical Engine was undoubtedly the mechanical method of carrying the tens. On this I laboured incessantly, each succeeding improvement advancing me a step or two. The difficulty did not consist so much in the more or less complexity of the contrivance as in the reduction of the time required to effect the carriage. Twenty or thirty different plans or modifications had been drawn. At last I came to the conclusion that I had exhausted the principle of successive carriage. I concluded that nothing but teaching the Engine to foresee, and then to act upon this foresight, could ever lead me to the object I desired, namely, to make the whole of any unlimited number of carriages in one unit of time.

Yet he did accomplish even this.

This new and rapid system of carrying the tens when two numbers are added together reduces the actual time of the addition of any number of digits, however large, to nine units of time for the add., and one unit for the carriage. Thus, in ten units of time, any two numbers, however large, might be added together. A few more units of time, perhaps 5 or 6, were required for making the requisite previous arrangements. Having thus advanced as nearly as seemed possible to the minimum of time requisite for arithmetical operations, I felt renewed power and increased energy to pursue the far higher object I had in view.

To describe the successive improvements of the Analytical Engine would require many vols. To those who are acquainted with the principles of the Jacquard loom, and who are also familiar with analytical formulæ, a general idea of the means by which the Engine executes its operations may be obtained without much difficulty. . . . It is known as a fact that the Jacquard loom is capable of weaving any design which the imagination of man may conceive. It is also the constant practice for skilled artists to be employed by manufacturers in designing patterns. These patterns are then sent to a peculiar artist, who, by means of a certain machine, punches holes in a set of pasteboard cards in such a manner that when those cards are placed in a Jacquard loom, it will then weave upon its produce the pattern designed by the artist.

The analogy of the Analytical Engine with this well-known process is nearly perfect. The A. E. consists of two parts: 1. The store in which all the variables to be operated upon, as well as all those quantities which have arisen from the result of other operations, are placed. 2. The mill into which the quantities about to be operated upon are always brought. Every formula which the Analytical Engine can be required to compute consists of certain algebraical operations to be performed upon given letters, and of certain other modifications depending on the numerical value assigned to those letters. There are therefore two sets of cards: the first, to direct the nature of the operations to be performed these are called operation cards; the other, to direct the particular variables on which those cards are required to operate these latter are called variable cards. Now the symbol of each variable or constant is placed at the top of a column capable of containing any required number of digits.

Under this arrangement, when any formula is required to be computed, a set of operation cards must be strung together, which contain the series of operations in the order in which they occur. Another set of cards must then be strung together, to call the variables into the mill, in the order in which they are required to be acted upon. Each operation card will require 3 other cards, two to represent the variables and constants and their numerical values upon which the previous operation card is to act, and one to indicate the variable on which the arithmetical result of this operation is to be placed. But each variable has below it, on the same axis, a certain number of figure-wheels marked on their edges with the ten digits; upon these any number the machine is capable of holding can be placed. Whenever variables are ordered into the mill, these figures will be brought in, and the operation indicated by the preceding card will be performed upon them. The result of this operation will then be replaced in the store.

The Analytical Engine is therefore a machine of the most general nature. Whatever formula it is required to develope, the law of its development must be communicated to it by two sets of cards. When these have been placed, the Engine is special for that particular formula. The numerical value of constants must then be put on the columns of wheels below them, and on setting the Engine in motion, it will calculate and print the numerical results of that formula.

Every set of cards made for any formula will at any future time recalculate that formula with whatever constants may be required. Thus the A. E. will possess a library of its own. Every set of cards once made will at any future time reproduce the calculations for which it was first arranged. The numerical value of its constants may then be inserted.

Besides the sets of cards which direct the nature of the operations to be performed, and the variables

or constants which are to be operated upon, there is another class of cards, called number cards. These are much less general in their uses than the others, although they are necessarily of much larger size.

The A. E. will contain: 1. Apparatus for printing on paper, one, or if required, two copies of its results. 2. Means for producing a stereotype mould of the T. or results it computes. 3. Mechanism for punching on blank pasteboard cards or metal plates the numerical results of any of its computations. Of course the Engine will compute all the T. which it may itself be required to use. . . .

So much for the mechanism of this almost Human Engine. We have yet to glance at its powers of operation; and here again we shall follow the learned Professor. We are now introduced to a conversation between the inventor and his friend Prof. MacCullagh, late of Dublin, on this very subject:

After a long conversation on the subject, he inquired what the machine could do, if, in the midst of algebraic operations, it was required to perform logarithmic or trigonometrical operations. My answer was, that whenever the A. E. should exist, all the developments of formulae would be directed by this condition-that the machine should be able to compute their numerical value in the shortest possible time. I then added that if this answer were not satisfactory, I had provided means by which, with equal accuracy, it might compute by logarithmic or other T. I explained that the T. to be used must, of course, be computed and punched on cards by the machine, in which case they would undoubtedly be correct. I then added that when the machine wanted a tabular number, say the logarithm of a given number, that it would ring a bell and then stop itself. On this the attendant would look at a certain part of the machine and find that it wanted the logarithm of a given number, say of 2303. The attendant would then go to the drawer containing the pasteboard cards representing its T. of logarithms. From amongst these he would take the required logarithmic card, and place it in the machine. Upon this the Engine would first ascertain whether the assistant had or had not given him the correct logarithm of number; if so, it would use it and continue its work. But if the Engine found the attendant had given him a wrong logarithm, it would then ring a louder bell and stop itself. On the attendant again examining the Engine, he would observe the words "wrong tabular number," and then discover that he really had given the wrong logarithm, and he would have to replace it by a right one.

Tables are used for saving the time of continually computing individual numbers. But the computations to be made by the Engine are so rapid that it seems most prob. that it will make shorter work by computing directly from proper formulæ than by having recourse even to its own T.

Next we have some insight into the scope of its operations:

The A. E. I propose will have the power of expressing every number it uses to fifty places of figures. It will multiply any two such numbers together, and then, if required, will divide the product of one hundred figures by number of fifty places of figures. Supposing the velocity of the moving parts of the Engine to be not greater than 40 feet p. minute, I have no doubt that 60 additions or subtractions may be completed and printed in 1 minute. One multiplication of two numbers, each of 50 figures in 1 minute. One division of a number having 100 places of figures by another of 50 in 1 minute.

Again we are told that "two great principles were embodied to an unlimited extent:" 1. The entire control over arithmetical operations, however large, and whatever might be the number of their digits. 2. The entire control over the combinations of algebraic symbols, however lengthened those processes may be required. The inventor fairly states: "The possibility of fulfilling these two conditions might reasonably be doubted by the most accomplished mathematician as well as by the most ingenious mechanician." He then proceeds:

The difficulties which naturally occur to those capable of examining the questions, as far as they relate to arithmetic, are these: (a). The number of digits in each constant inserted in the Engine must be without limit. (6). The number of constants to be inserted in the Engine must also be without limit. (c). The number of operations necessary for arithmetic is only 4; but these 4 may be repeated an unlimited number of times. (d). These operations may occur in any order, or follow an unlimited number of laws.

Next we learn that the following conditions relate to the algebraic portion of the Analytical Engine:

(e). The number of litteral constants must be unlimited. (f). The number of variables must be without limit. (g). The combinations of the algebraical signs must be unlimited. (h). The number of functions to be employed must be without limit.

This enumeration included 8 conditions, each of which is absolutely unlimited as to the number of its combinations. Now it is obvious that no finite machine can include infinity. It is also certain that no question necessarily involving infinity can ever be converted into any other in which the idea of infinity under some shape or other does not enter. It is impossible to construct machinery occupying unlimited space; but it is possible to construct finite machinery, and to use it through unlimited time. It is this substitution of the infinity of time, for the infinity of space, which I have made use of, to limit the size of the Engine, and yet to retain its unlimited power.

The inventor then proceeds briefly to point out the means by which he had effected this change:

Since every calculating machine must be constructed for the calculation of a definite number of figures, the first datum must be to fix upon that number. In order to be somewhat in advance of the greatest number that may ever be required, I chose 50 places of figures as the standard for the Analytical Engine. The intention being that in such a machine 2 numbers, each of 50 places of figures, might be multiplied together, and the resultant product of 100 places might then be divided by another number of 50 places. It seems to me probable that a long period must elapse before the demands of science will exceed this limit.

He then enters upon a number of scientific details, for the purpose of elucidating the 8 conditions above indicated. We cannot follow these; but we may with advantage cite an occasional passage:

The same reasoning will show that if the numbers of digits of each factor are between 100 and 150, then the time required for the operation will be nearly 9 times that of a pair of factors having only 50 digits....