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which both the differential and integral calculus are employed, and has published them in a separate tract.* The editor dwells so much on the matter and consequences of the manuscripts, that he forgets to satisfy curiosity as to their form, the circumstances of the discovery, &c.: they ought to be republished with proper fac-similes of the hand-writing. Not that we at all doubt them; for, independently of the full credit due to Dr. Gerhardt, we do not believe that human ingenuity could have forged so genuine a mess of spoiled exercises. We cannot attempt a full account of them; but this is of little consequence, since they will of necessity be fully described in more appropriate quarters, so soon as they are better known to exist. These papers are seven in number, dated+ November 11, 21, 22, 1675, June 26, July, November, 1676, and one without date. They are not descriptions of the principles, but study_exercises in the use, of both differential and integral calculus. Except out of the problems themselves we learn nothing of the extent to which the structural operations were in the power of the writer. We find strange mistakes of operation, such as beginners now make: and it is clear that the writer is trying to push his calculus forward into discovery of new results in geometry before he has either sounded its extent or settled its language. In the first of the papers he enters (among other things) upon the examination whether dx.dy is

the same with d (xy) and d (1)

dx

with : at first he inclines to the dy

affirmative, but in the next page decides in the negative. This will not surprise the mathematician of our day, who remembers that these are the private memoranda of a discoverer in the very process of investigation: but nevertheless he will look to find some particular cause of confusion of ideas at the outset. We suspect it to be as follows. Leibnitz frequently supposes dx=1, or dy=1: that is, he establishes two kinds of units, without any symbolic distinction, the unit of finite, and the unit of infinitely small, quantity. In inte

gration, he halts between the use of fy and of fyda, as the ex

pression of an integral. There are also obvious slips of the pen, and operations set down for thought, which lead to nothing.

The first problems treated are in the direct and inverse method of tangents, in which the method of Sluse is referred to by name. The two following extracts, in which the Latin is literally translated, of the date of November 11, 1675, will be as much as we can afford room for. They give two of the earliest problems solved, the first and third.

*Die Entdeckung der Differentialrechnung durch Leibniz. Von Dr. C. J. Gerhardt.' Quarto. No date nor place; preface dated "Salzwedel, im Januar 1848."

+ The editor tells us that some one had been meddling with the date of the first paper, and had turned the 5 of 1675 into a 3. Leibnitz, speaking from recollection in 1714, says that his discovery was made, as near as he could remember, in 1676.

+ Professor Rigaud has published, from the Macclesfield collection, a manuscript draught of Newton, of Nov. 13, 1665. But this is formally written out, proposition, resolution, and demonstration. An earlier essay, of May 20, is not given, which is to be regretted. But from the description we see that Newton used the peculiar notation of fluxions in May, and abandoned it in November. His formal proposition uses distinct letters for fluxions of other letters. In Leibnitz, everything in language is progression: no step gained is ever abandoned.

The problem is to find a curve in which the subnormal (w) is reciprocally proportional to the ordinate. Putting z instead of dx. Leibnitz proceeds thus: "It appears from what I have shown

elsewhere, that Swz

[ocr errors][ocr errors][ocr errors][ocr errors][merged small]

2d'

is the symbol of differentiation of the whole: it frequently happens in the first papers. "But from the quadrature of the triangle this is y." We should write ydy, but Leibnitz tacitly makes dy=1, and he afterwards says he has here thought of making an abscissa

of the ordinate. "Now from the hypothesis w

b

bz

[ocr errors]

whence

=y,

У

У

[ocr errors]

and z=2. But Sz=z. Therefore ft. But =

x.

b

by the quadrature of the parabola; therefore a=;

ყვ

3ba

[ocr errors]

3ba

This a is

not of easy explanation. It is afterwards given to make the subnormal reciprocally proportional to the abscissa.

y2

but fu, whence y=√(2 fw), or v (2

= 2

[merged small][ocr errors][merged small]

cannot be found except by the help of the logarithmic curve. Therefore the figure required is that in which the ordinates are in the subduplicate ratio of the logarithms of the abscissæ."

If the Committee of the Royal Society had had these papers before them, they would have justly contended that the calculus of Leibnitz, of which the principles and algorithm were settled, received a great accession of working power when Newton communicated the binomial theorem in the epistola prior to Oldenburg; which epistola prior, by the rule of contraries already instanced, has been much less insisted on than the epistola posterior with its anagrams.

On August 27, 1676, Leibnitz acknowledged the receipt of this communication; and his paper of November, 1676, shows that Newton's algebra had borne its fruit. Previously to this date, we cannot find any fractional power differentiated except the square root. In pure algebraical discovery, Leibnitz does not rank with Newton: and he always acknowledged that in the method of series (the phrase by which the algebraical improvements of the day were designated) Newton was before him and beyond him. We have every right to presume, from his conduct, and from the manner in which all subsequent disclosures establish his veracity, that had he lived to publish his own Commercium Epistolicum, he would have pointed out the difference between the invention of the differential calculus and the improvement of the algebra which gives it language and guides its mechanism, and would have illustrated from his own papers the power which Newton's improvements in algebra enabled him to add to his existing differential calculus. We believe (with John Bernoulli) that Newton might have made a similar acknowledgment to Leibnitz as to the idea of a fixed and uniform

method of denoting operations in the fluxions of which he had

already possession.

We have not alluded to the faults on the other side of the controversy, partly because they were much less gross in character, partly because they have been amply insisted on in this country. Nor have we, indeed, in this paper, given anything like a history of the unfair proceedings in this country, but have, for the most part, confined ourselves to points which are particularly affected by recent information. Whether there be anything still to be drawn out must be matter of conjecture, and will be matter of suspicion, until we can be well assured that all the private depositories of information have been exhausted.

University College, London,
October 2, 1851.

A. DE MORGAN.

II. GREAT EXHIBITION OF 1851: FACTS AND

FIGURES.

THE greatest event which the world has witnessed in 1851-the holding of the Industrial Exhibition in Hyde Park—has had many historians, and will doubtless have many more. The origin of the Exhibition, the construction of the Building, the description of the goods exhibited, and the probable results to manufacturing skill, will not be treated in this work; but it may be fitting to keep a permanent record of some among what may be termed the statistics, the facts and figures, of the enterprise-a few of which are marvels, others are curiosities, while all are interesting.

First we will give a brief summary of

DATES AND OCCURRENCES.

1848, May. The Society of Arts ask for Government aid towards the holding of triennial Exhibitions of Manufactures; but with no decisive result.

1849. Mar. The Society publicly announce the probability of an Exhibition in 1851: the Council having sketched the outline of a plan.

April. The Council petition Parliament for aid.

June. Interviews between Prince Albert and several members of the Council, respecting the Exhibition.

July. Prince Albert applies to the Government for the appointment of a Royal Commission, to manage an International Exhibition.

Aug. Messrs. Munday offer a guarantee, concerning an Exhibition Building.

Sept. Prince Albert requests the Lord Mayor to hold a public meeting to make the scheme known; meeting held in October. Oct. Contract signed by the Society of Arts with Messrs. Munday. 1850. Jan. Royal Commission appointed; at their first meeting, the Commissioners resolve to abandon the contract, but to compensate Messrs. Munday.

1850. Jan. 24. Commissioners appoint a Building Committee. 25. Great Meeting at the Mansion House, which commenced the subscriptions.

Feb. 21. Commissioners publicly announce the general plan, and communicate the Queen's consent to the holding of an Exhibition in Hyde Park.

Mar. 13. The Building Committee invite designs for an Exhibition Building.

23. Commissioners invite designs for Medals.

May 9. The Building Committee report on 233 plans, rejecting all, and proposing one of their own.

June 11. Mr. Paxton first mentions his plan of the Crystal Palace; on the 19th proposes it to Mr. Robert Stephenson; and on the 22nd to Mr. Fox.

26. Commissioners invite tenders for the Committee's design. July 10. Twenty tenders for building; Fox and Henderson also send in a tender for Paxton's design.

16. Paxton's design, and Fox and Henderson's tender, verbally accepted; and contract signed on the 26th.

30. Contractors obtain possession of site.

Aug. Charter granted to Royal Commissioners.
Sept. 26. First column of the Crystal Palace fixed.

Oct. 25. Dinner at York to the Prince and the Commissioners. 31. Last day of Application for Exhibition space.

Nov. 18. Goldsmiths' Company announce prizes.
Dec. 4. First transept rib raised at the Crystal Palace.
21. Her Majesty's first visit to the Building.

31. Lecture to the Society of Arts in the Building. 1851. Feb. 3. Building taken possession of by Commissioners, and closed against visitors till May 1.

May 1. Opening Ceremony.

2. First One Pound day.
5. First Five Shilling day.
26. First Shilling day.
30. First Half-Crown day.

Aug. 2. Last Five Shilling day.

Oct. 7. Greatest number of Visitors.

9. Last Shilling day.

11. Last Public day.

15. Jury Awards, and Closing Ceremonial.

We next direct attention to the

PERSONS EMPLOYED.

The number of persons who, in various positions, figure in the management of the Great Exhibition, is very great. In the first place there were twenty-five Royal Commissioners, with Prince Albert at their head. The secretaries to the Commissioners were two; and there were also two Special Commissioners, who were the channel of communication between the Board and the Executive. There were six members of the Executive Committee, of whom one was chairman and one secretary. The Finance Committee was

composed of eight, of whom six were Royal Commissioners. The Building Committee also numbered eight, of whom five were Commissioners. The Medal Committee were seven, and the Inscription Committee four; of these eleven only one was a Commissioner. The Committee appointed to communicate with the local committees consisted of fourteen.

Then, having relation to the goods sent in for Exhibition, there were Committees of Sections appointed to determine on regulations for the classification and admission of specimens. Section one,

divided into four sub-sections, had eighteen members; section two, in two sub-sections, twenty-one; section three, twenty-three; and section four, eleven. These seventy-three Committeemen comprised nearly all the Royal Commissioners, and a fine array of the most talented men in the country.

In the provinces, district committees were appointed in all the principal towns, to conduct the arrangements relating to the application and allotment of space in the building, and to the transmission of the goods. Many of these committees had both chairmen and secretaries, and included a considerable number of persons. Sometimes the local committee which managed these matters for a particular town, was the same as that which had collected subscriptions at an earlier period; but in other cases the two bodies were different. It would be scarcely possible to give the numbers of those who served upon these very numerous local committees.

When the Royal Commissioners had made a formal communication to the various Foreign Governments, the latter in most cases appointed Commissioners to conduct all the relations of those countries in respect to this great undertaking. In the lists given in the Official Catalogue, these Commissioners are mostly mentioned by name, but in some instances they are the members generally of some learned society or corporation. The total number of Commissioners thus actually named was about 250. About 40 persons arrived in this country to serve as Acting Commissioners for the various foreign countries, besides foreign attendants and workmen,

Notwithstanding the number of persons engaged in managing this wonderful collection, it was deemed advisable to increase the number after the first month or two, by the appointment of District Superintendents. The whole building was divided into twelve districts; a small enclosed space was allotted as an office for each district; and each district had a superintendent and assistants appointed to it. The duties of these superintendents were-to make a daily inspection of every part of their district; to see that the goods of exhibitors were kept properly cleaned and arranged; to see that the catalogue numbers and other tickets were affixed to the articles to which they belonged; to see that no articles were admitted without proper sanction; to see that no accumulations of dust or rubbish were permitted; to keep a register of all accidents or damage to the building or the goods; to report every leakage which occurred in the roof; and to keep a book for entering complaints or suggestions.

When these various additions were made, the number of persons daily employed within the building in duties having a defined and specified

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