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The Transactions of the British Association consist of 
three parts ; first, of Reports on the State of Science drawn 
up at the instance of the Association ; secondly, of Miscel- 
laneous Communications to the Meetings ; and thirdly, of 
Recommendations by the Committees, having for their ob- 
jects to mark out certain points for scientific inquiry. 

It is proper to remark, that some of the Reports here 
printed are to be considered in the light of first parts of 
the intended survey of the sciences reviewed in them, the 
continuation being postponed to a future Meeting. Thus, 
the Report on Hydraulics, by Mr. G. Rennie, will be 
completed in a second part, to be presented to the Meeting 
at Edinburgh ; the Report on the mathematical theory of 
the same science, by the Rev. Mr. Challis, which is 
here restricted to problems on the common theory of Fluids, 
will be further extended to the theories which have recently 
been advanced respecting the internal constitution of Fluids 
and the state of their caloric, to account for certain phseno- 
mena of their equilibrium and motion ; and the Report on 
Analytical Science, by the Rev. Mr. Peacock, which in the 
present volume includes Algebra, and the application of 
Algebra to Geometry, is intended to be hereafter concluded 
by a review of the Differential and Integral Calculus and the 
theory of Series. In like manner, to the Report on Botany, 
by Dr. Lindley, which embraces only the physiological 
part of the science, that which Mr. Bentham has under- 
taken on the State and Progress of Systematic Botany will 
be supplemental ; and to the present Report, by Dr. Charles 
Henry, on one branch of Animal Physiology, a more 
general review of the progress of that science will be added 
by the Rev. Dr. Clark. 

With respect to the next part of the Transactions, which 
includes the communications made to the Sections, two 

a 2 


rules have been adopted ; the first is, to print no oral com- 
munications unless furnished or revised by the Author him- 
self. In the former volume this rule was slightly deviated 
from, for the purpose of showing in what manner the 
Meetings were conducted. But however valuable a part of 
the proceedings of the Meetings the verbal communications 
and discussions maybe, it is evidently impossible to publish 
a safe and satisfactory report of them from any minutes 
which can be taken. The second rule is, not to print any 
of the miscellaneous communications at length ; but either 
abstracts of them, or notices* only, the object of the rule 
being to keep the Transactions within the bounds which 
the Association has prescribed to itself, and to prevent any 
interference with the publications of other societies. In 
the present volume, there is one paper printed at lengthf , 
which contains the results of certain experiments instituted 
expressly at the request of the Association. 

The Recommendations of various subjects for scientific 
inquiry agreed upon at Cambridge have been here incor- 
porated with those adopted at former Meetings, and the 
Suggestions which are contained in the Reports on the state 
of science, published in the present and preceding volume, 
have likewise been added; so as to present a general view of 
the desiderata in science to which attention has been invited. 
To this part of the volume are also appended those direc- 
tions for the use of observers which have proceeded from 
Committees appointed to promote particular investigations. 

To the Transactions is prefixed a brief outline of the 
General Proceedings of the Cambridge Meeting, a fuller Rc; 
port of them having been rendered unnecessary by the ac- 
count which has already issued from the University press. 
The observations, however, delivered by the Rev. Mr. Whe- 
WELL on the state of science as it is exhibited in the first 
volume of the Reports of the Association, not having been 
before published, are printed at length. 

• The notices of Communications will be found in the general account of the 
Proceedings of the Sections, p. 353. 

f " Experiments on the Quantity of Rain which falls at different Heights in 
the Atmosphere." 


Proceedings of the Meeting ix 


Report on the State of Knowledge respecting Mineral Veins. By 
John Taylor, F.R.S., Treasurer of the Geological Society and 
of the British Association for the Advancement of Science, &c. 1 

On the Principal Questions at present debated in the Philosophy 
of Botany. By John Lindley, Ph. D., F.R.S., Professor of 
Botany in the University of London 27 

Report on the Physiology of the Nervous System. By William 
Charles Henry, M.D., Physician to the Manchester Royal In- 
firmary .59 

Report on the present State of our Knowledge respecting the 
Strength of Materials. By Peter Barlow, F.R.S., Corr. Memb. 
Inst. France, &c. &c 93 

Report on the State of our Knowledge respecting the Magnetism 
of the Earth. By S. Hunter Christie, M.A., F.R.S., M.C.P.S., 
Corr. Memb. Philom. Soc. Paris, Hon. Memb. Yorkshire Phil. 
Soc. ; of the Royal Military Academy ; and Member of Trinity 
College Cambridge 105 

Report on the present State of the Analytical Theory of Hydro- 
statics and Hydrodynamics. By the Rev. J. Challis, late Fel- 
low of Trinity College Cambridge 131 

Report on the Progress and present State of our Knowledge of 
Hydraulics as a Branch of Engineering. By George Rennie, 
F.R.S., &c. &c 153 

Report on the recent Progress and present State of certain Branches 
of Analysis. By George Peacock, M.A., F.R.S., F.G.S., 
F.Z.S., F.R.A.S., F.C.P.S., FeUow and Tutor of Trinity Col- 
lege Cambridge 185 

I. Mathematics and Physics. 

Professor CErsted on the Compressibility of Water 353 

W. R. Hamilton on some Results of the View of a Characteristic 

Function in Optics 360 

The Rev. H. Lloyd on Conical Refraction 370 



Sir John F. W. Herschel on the Absorption of Light by coloured 
Media, vieM'ed in connexion with the undulatory Theory .... 373 

The Rev. Baden Powell on the Dispersive Powers of the Media 
of the Eye, in connexion with its Achromatism 374 

R. Potter, Jun., on the power of Glass of Antimony to reflect 
Light 377 

on a Phsenomenon in the Interference of Light 

hitherto undescribed 378 

Sir John F. W. Herschel's Explanation of the Principle and Con- 
struction of the Actinometer 379 

M. Melloni's Account of some recent Experiments on Radiant 
Heat 381 

John Prideaux on Thermo-Electricity 384 

W. Snow Harris on some new Phaenomena of Electrical Attrac- 
tion 386 

The Rev. John G. MacVicar on Electricity 390 

The Rev. J. Power's Inquiry into the Cause of Endosmose and 
Exosmose 391 

Michael Faraday on Electro -chemical Decomposition 393 

Dr. Turner's Experiments on Atomic Weights 399 

Prof. Johnston's Notice of a Method of analysing Carbonaceous 
Iron 400 

R. Potter, Jun. A Communication respecting an Arch of the 
Aurora Borealis 401 

John Phillips's Report of Experiments on the Quantities of Rain 
falling at different Elevations above the Surface of the Ground at 
York 401 

II. Philosophical Instruments and Mechanical Arts. 

The Rev. Wm. Scoresby on a peculiar Source of Error in Experi- 
ments with the Dipping Needle 412 

The Rev. W. H. Miller on the Construction of a new Barometer 414 

W. L. Wharton on a Barometer with an enlarged Scale 414 

W. S. Harris on the Construction of a new Wheel Barometer . . 414 

J. Newman on a new Method of constructing a Portable Barometer 417 
The Rev. James Cumming on an Instrument for measuring the 

total heating Effect of the Sun's Rays for a given time 418 

on some Electro-magnetic Instruments 418 

Andrew Ure on the Thermostat, or Heat-governor 419 

Thomas Davison on a Reflecting Telescope 420 

W. L. Wharton on a Steam-engine for pumping Water 421 

E. J. Dent on the Application of a glass Balance-siDring to Chro- 
nometers 421 

E. HoDGKiNsoN on the EflFect of Impact on Beams 421 

on the direct tensile Strength of Cast Iron 423 

J. I. Hawkins's Investigation of the Principle of Mr. Saxton's loco- 
motive differential Pulley, &c 424 

John Taylor's Account of the Depths of Mines 427 

J. Owen on Naval Architecture 430 


III. Natural History — Anatomy — Physiology. 

Professor Agardh on the originary Structure of the Flower, and 
the mutual Dependency of its Parts 433 

Professor Daubeny's Notice of Researches on the Action of Light 
upon Plants 436 

Walter Adam on some symmetrical Relations of the Bones of the 
Megatherium 437 

R Harlan on some new species of Fossil Saurians found in Ame- 
rica 440 

The Rev. L. Jenyns's Remarks on Genera and Subgenera, &c. . . 440 

J. Macartney on some parts of the Natural History of the Com- 
mon Toad 441 

J. Blackwall's Observations relative to the Structure and Func- 
tions of Spiders 444 

W. Yarrell on the Reproduction of the Eel 44G 

C. WiLLCox on the Naturalization in England of the Mytilus cre- 
natus, a native of India, and the Acematichcerus Heros, a native 
of Africa 448 

J. Macartney's Abstract of Observations on the Structure and 
Functions of the Nervous System 449 

H. Carlile's Abstract of Observations on the Motions and Sounds 
of the Heart 454 

H. Earle on the Mechanism and Physiology of the Urethra .... 460 

Burt on the Nomenclature of Clouds 460 

G. H. Fielding on the peculiar Atmospherical Phaenomena as ob- 
served at Hull during April and May 1833, in relation to the 
prevalence of Influenza 461 

IV. History of Science. 

Francis Baily's short Account of some MS. Letters (addressed 
to Mr. Abraham Sharp, relative to the Publication of Mr. Flam- 
steed's Historia Calestis,) laid on the table for the inspection of 
the Members of the Association 462 

Recommendations of the British Association for the Advancement 

of Science 467 

Recommendations of the Committees 469 

Appendix 484 

Prospectus of the Objects and Plan of the Statistical Society of 

London 492 

Objects and Rules of the Association 497 

Index 501 

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The third Meeting of the British Association commenced its 
sittings at Cambridge on Monday, the 24th of June, 1833. It 
was attended by more than nine hundred Members, and was 
honoured with the presence of several foreign philosophers. 
The extent of accommodation provided by the University, and 
by the societies of which it consists, corresponded with the 
magnitude of the Meeting. The public schools, with two 
adjoining halls, were allotted to the use of the Sections and 
Committees, and the Senate-house was appropriated to the 
reception of the general assemblies ; a large proportion of the 
visitors were lodged within the walls of the Colleges, and the 
great halls of the two principal foundations were opened in 
hospitality to a concourse of guests collected from all parts by 
a common interest in scientific pursuits. 


On Monday evening, at eight o'clock, the Members assem- 
bled in the Senate-house : and a public discussion took place 
on the phenomena and theory of the Aurora Borealis. 

On Tuesday, at 1 p. m. a General Meeting was held in 
the Senate-house ; the President of the preceding year, (the 
Rev. Dr. Buckland,) resigned his office. In the course of 
his speech*, he congratulated the Meeting on the proof af- 
forded by the Report recently published, that the Association 
was pursuing a course of peculiar utility to science, whilst at the 

* A fuller account of the speeches delivered at the Meeting -will be found 
annexed to the lithographed signatures, &c., published at Cambridge. 
1833. b 


same time it had fully redeemed its pledge of not interfering with 
the province of other Scientific Societies. 

The President (the Rev. Professor Sedgwick,) stated, in his 
opening speech, that it was the desire of the Vice-Chancellor 
and the Heads of Colleges that everything should be done on 
the present occasion to emulate, as far as circumstances per- 
mitted, the splendid reception which had been given to the 
Association by the sister University of Oxford. He dwelt on 
the advantages which such a Meeting brought with it to the 
places in which it was held, by inducing scientific foreigners to 
visit them, and expressed the delight with which he hailed such 
visits, as an omen that the great barriers which for a length 
of time had served man for man, had now been broken 
down. He described the character of the Reports which 
the Association has published ; and added that he attached so 
much value to these expositions of the state of science, that 
he had requested one of the Secretaries, (the Rev, William 
Whewell,) to present to the Meeting a fuller analysis of their 
contents. The President concluded his speech with the fol- 
lowing gratifying announcement : " There is a philosopher," he 
said, " sitting among us whose hair is blanched by time, but 
possessing an intellect still in its healthiest vigour, — a man whose 
whole life has been devoted to the cause of truth, — my vener- 
able friend Dr. Dalton. Without any powerful apparatus for 
making philosophical' experiments, with an apparatus, indeed, 
which many might think almost contemptible, and with very 
limited external means for employing his great natural powers, 
he has gone straight forward in his distinguished course, and 
obtained for himself in those branches of knowledge which he 
has cultivated, a name not perhaps equalled by that of any- 
other living philosopher in the world. From the hour he came 
from his mother's womb the God of nature laid his hand upon 
him, and ordained him for the ministration of high philosophy. 
But his natural talents, great as they are, and his almost 
intuitive skill in tracing the relations of material phsenomena, 
would have been of comparatively little value to himself and to 
society, had there not been superadded to them a beautiful 
moral simpUcity and singleness of heart, which made him go 
on steadily in the way he saw before him, without turning to 
the right hand or to the left, and taught him to do homage to 
no authority before that of truth. Fixing his eye on the most 
extensive views of science, he has been not only a successful 
experimenter, but a philosopher of the highest order; his 
experiments have never had an insulated character, but have 
been always made as contributions towards some important 


end, as among the steps towards some lofty generalization. 
And with a most happy prescience of the points to which the 
rays of scattered observations were converging, he has more 
than once seen light while to other eyes all was yet in darkness ; 
out of seeming confusion has elicited order ; and has thus 
reached the high distinction of being one of the greatest legis- 
lators of chemical science. 

"It is my delightful privilege this day to announce (on the 
authority of a Minister of the Crown who sits near me,*) that 
His Majesty, King William the Fourth, wishing to manifest 
his attachment to science, and his regard for a character like 
that of Dr. Dalton, has graciously conferi'ed on him, out of 
the funds of the Civil List, a substantial mark of his royal 

The Rev. William Whewell, being called vipon by the 
Pi'esident, delivered the following address : — 

" The British Association for the Advancement of Science 
meets at present under different circumstances from those 
which accompanied its former Meetings. The publication of 
the volume containing the Reports applied for by the Meeting 
at York, in 1831, and read before the Meeting at Oxford last 
year, must affect its proceedings during our sittings on the 
present occasion ; and thus we are now to look for the operation 
of one part of the machinery which its founders have endea- 
voured to put in action. Entertaining the views which sug- 
gested to them the scheme and plan of the Association, they 
must needs hope that such an event as this publication will 
exercise a beneficial influence upon its future career. 

" This hope is derived, they trust, from no visionary or 
presumptuous notions of what institutions and associations can 
effect. Let none suppose that we ascribe to assembled num- 
bers and conjoined labours extravagant powers and privileges 
in the promotion of science ; — that we believe in the omnipo- 
tence of a parliament of the scientific world. We know that 
the progress of discovery can no more be suddenly accelerated 
by a word of command uttered by a multitude, than by a 
single voice. There is, as was long ago said, no royal road to 
knowledge — no possibility of shortening the way, because he 
who wishes to travel along it is the most powerful one ; and 
just as little is there any mode of making it shorter, because 
they who press forward are many. We must all start from 
our actual position, and we cannot accelerate our advance by 

* The Right Honourable T. Spring Rice. 


xii . THIIID REPORT — 1838. 

any method of giving to each man his mile of the march. Yet 
something we may do : we may take care that those who come 
ready and willing for the road, shall start from the proper 
point and in the proper direction ; — shall not scramble over 
broken ground, when there is a causeway parallel to their 
path, nor set off confidently from an advanced point when the 
first steps of the road are still doubtful ; — shall not waste their 
powers in struggling forwards where movement is not progress, 
and shall have pointed out to them all glimmerings of Hght, 
through the dense and deep screen which divides us from the 
next bright region of philosophical truth. We cannot create, 
we cannot even direct, the powers of discovery ; but we may 
perhaps aid them to direct themselves ; we may perhaps 
enable them to feel how many of us are ready to admire their 
success, and willing, so far as it is possible for intellects of a 
common pitch, to minister to their exertions. 

" It was conceived that an exposition of the recent progress, 
the present condition, the most pressing requirements of the 
principal branches of science at the present moment, might 
answer some of the purposes I have attempted to describe. 
Several such expositions have accordingly been presented to 
the Association by persons selected for the task, most of them 
eminent for their own contributions to the department which 
they had to review ; and these are now accessible to Members 
of the Association and to the public. It appears to be suitable 
to the design of this body, and likely to further its aims, that 
some one should endeavour to point out the bearing which the 
statements thus brought before it may and ought to have upon 
its future proceedings, and especially upon the laboiu's of the 
Meeting now begun. I am well persuaded that if the President 
had taken this ofiice upon himself, the striking and important 
views which it may naturally suggest would have been pre- 
sented in a manner worthy of the occasion : he has been 
influenced by various causes to wish to devolve it upon me, and 
I have considered that I should show my respect for the Asso- 
ciation better by attempting the task, however imperfectly, 
than by pleading my inferior fitness for it. 

" The particular questions which require consideration, and 
the researches which most require prosecution, in the sciences 
to which the Reports now before you refer, will be offered to 
the notice of the Sections of the Association which the subjects 
respectively concern, at their separate sittings. It is conceived 
that the most obvious and promising chance of removing 
deficiencies and solving difficulties in each subject, is to be 
found in drawing to them the notice of persons who have paid 


a continued and especial attention to the subject. The con- 
sideration of these points will therefore properly form a part 
of the business of the Sectional Meetings ; and all Members of 
the Association, according to their own peculiar pursuits 
and means, will thus have the opportunity of supplying any 
wanting knowledge, and of throwing light upon any existing 

" But besides this special examination of the suggestions 
which your Reports contain, there are some more general 
reflexions to which they natui-ally give rise, which may perhaps 
be properly brought forward upon this first General Assembly 
of the present Meeting ; and which, if they are well founded, 
may preside over and influence the aims and exertions of many 
of us, both during our present discussions and in our future 
attempts to further the ends of science. 

" There is here neither time nor occasion for any but the 
most rapid survey of the subjects to which your Reports refer, 
in the point of view in which the Reports place them before 
you. Astronomy, which stands first on the list, is not only 
the queen of sciences, but, in a stricter sense of the term, the 
only perfect science ; — the only branch of human knowledge 
in which particulars are completely subjugated to generals, 
eflPects to causes ; — in which the long observation of the past 
has been, by human reason, twined into a chain which binds 
in its links the remotest events of the future ; — in which we 
are able fully and clearly to interpret Nature's oracles, so that 
by that which we have tried we receive a prophecy of that 
which is untried. The rules of all our leading facts have 
been made out by observations of which the science began 
with the earliest dawn of history ; the grand law of causation 
by which they are all bound together has been enunciated for 
150 years; and we have in this case an example of a science 
in that elevated state of flourishing maturity, in which all that 
remains is to determine with the extreme of accuracy the con- 
sequences of its rules by the profoundest combinations of 
mathematics, the magnitude of its data by the minutest scru- 
pulousness of observation ; in which, further, its claims are so 
fully acknowledged, that the public wealth of every nation pre- 
tending to civilization, the most consummate productions of 
labour and skill, and the loftiest and most powerful intellects 
which appear among men, are gladly and emulously assigned 
to the task of adding to its completeness. In this condition of 
the science, it will readily be understood that Professor Airy, 
your Reporter upon it, has had to mark his desiderata, in no 
cases but those where some further developement of calcula- 

Xiv THIRD REPORT — 1833. 

tion, some furthei- delicacy of observation, some further accu- 
mulation of exact facts, are requisite ; though in every branch 
of the subject the labour of calculation, the delicacy of obser- 
vation, and the accumulation of exact facts, have already gone 
so far that the mere statement of what has been done can 
hardly be made credible or conceivable to a person unfamiliar 
with the study. 

" One article, indeed, in his list of recommendations to future 
labourers, read at the last Meeting of the Association, may ap- 
pear capable of being accomplished by more limited labour than 
the rest, — the determination of the mass of Jupiter by obser- 
vations of the elongations of his satellites. And undoubtedly, 
many persons were surprised when they found that on this, so 
obvious a subject of interest, no measures had been obtained 
since those which Pound took at the request of Newton. Yet 
in this case, if an accuracy and certainty worthy of the present 
condition of Astronomy were to be aimed at, the requisite ob- 
servations could not be few nor the calculation easy, when it 
is considered in how complex a manner the satellites disturb 
each other's motions. But the Meeting will learn with pleasure 
that the task which he thus pointed out to others, he has him- 
self in the intervening time executed in the most complete 
manner. He has weighed the mass of Jupiter in the way he 
thus recommended ; and it may show the wonderful perfection 
of such astronomical measures to state, that he has proved with 
certainty, that this mass is more than 322 and less than 323 
times the mass of the terrestrial globe on which we stand. 

" Such is Astronomy : but in proceeding to other sciences, 
our condition and our task are of a far different kind. Instead 
of developing our theories, we have to establish them ; instead 
of determining our data and rules with the last accuracy, we 
have to obtain first approximations to them. This, indeed, 
may be asserted of the next subject on the list, though that 
is, in its principles, a branch of Physical Astronomy ; for that 
alone of all the branches of Physical Astronomy had been al- 
most or altogether neglected by men of science. I speak of 
the science of the Tides. Mr. Lubbock terminated his Report 
on this subject, by lamenting in Laplace's words this unmerited 
neglect. He himself in England, and Laplace in France, were 
indeed the only mathematicians who had applied themselves 
to do some portion of what M^as to be done with respect to this 
subject. Since our Meeting last year, Mr. Dessiou has, under 
Mr. Lubbock's direction, compared the tides of London, Sheer- 
ness, Portsmouth, Plymouth, Brest, and St. Helena ; and the 
comparison has brought to light very remarkable agreements . 


in the law which regulates the time of high water, agreements 
both with each other and with theory ; and has at the same 
time brought into view some anomalies which will give a strong 
impulse to the curiosity with which we shall examine the re- 
cords of future observations at some of these places and at 
many others. I may perhaps here take the Uberty of mention- 
ing my own attempts since our last Meeting, to contribute 
something bearing on this department. It appeared to me that 
our knowledge of one particular branch of this subject, the 
motion of the tide-wave in all parts of the ocean, was in such a 
condition, that by collecting and arranging our existing mate- 
rials, we should probably be enabled to procure abundant and 
valuable additions to them. This, therefore, I attempted to 
do ; and I have embodied the result of this attempt in an 
' Essay towards a First Approximation to a Map of Cotidal 
Lines,' which is now just printed in the Philosophical Transac- 
tions of the Royal Society. If the time of the Meeting allows, 
I would willingly place before you the views at which we have 
now arrived, and the direction of our labours which these 

*' In the case of the science of Tides, we have no doubt about 
the general theory to which the phaenomena are to be referred, 
the law of universal gravitation ; though we still desiderate a 
clear application of the theory to the details. In another sub- 
ject which comes under our review, the science of Light, the 
prominent point of interest is the selection of the general 
theory. Sir David Brewster, the author of our Report on this 
subject, has spoken of ' the two rival theories of light,' which 
are, as you ai'e aware, that which makes light to consist in 
material particles emitted by a luminous body, and that which 
makes it to consist in undulations pi'opagated through a sta- 
tionary ether. The rivalry of these theories, so far as they 
can now be said to be rivals, has been by no means barren of 
interest and instruction during the year which is just elapsed. 
The discussions on the undulatory theory in our scientific 
journals have been animated, and cannot, I think, be considered 
as having left the subject where they found it. The claims of 
the undulatory theory, it will be recollected, do not depend 
only on its explaining the facts which it was originally intended 
to explain ; but on this ; — that the suppositions adopted in 
order to account for one set of facts, fall in most wonderfully 
with the suppositions requisite to explain a class of facts en- 
tirely different ; in the same manner as in the doctrine of gra- 
vitation, the law of force which is derived from the revolutions 
of the planets in their orbits, accounts for the apparently re- 


mote facts of the precession of the equinoxes and the tides. 
To all this there is nothing corresponding in the history of the 
theory of emission ; and no one, I think, well acquainted with 
the subject, would now assert, that if this latter theory had 
been as much cultivated as the other, it might have had a simi- 
lar brilliant fortune in these respects. 

" But if the undulatory theory be true, there must be solu- 
tions to all the apparent difficulties and contradictions which 
may occur in particular cases ; and moreover the doctrine will 
probably gain general acceptance, in proportion as these solu- 
tions are propounded and understood, and as prophecies of 
untried results are dehvered and fulfilled. In the way of such 
prophecies few things have been more remarkable than the 
prediction, that under particular circumstances a ray of light 
must be refracted into a conical pencil, deduced from the theory 
by Professor Hamilton of Dublin, and afterwards verified ex- 
perimentally by Professor Lloyd. In the way of special diffi- 
culties, Mr. Potter proposed an ingenious experiment which 
appeared to him inconsistent with the theory. Professor Airy, 
from a mathematical examination of this case, asserted that the 
facts, which are indeed difficult to observe, must be somewhat 
different from what they appeai-ed to Mr. Potter ; and having 
myself been present at Professor Airy's experiments, I can 
venture to say, that the appearances agree exactly with the 
results which he has deduced from the theory. Another gen- 
tleman, Mr. Barton, proposed other difficulties founded upon 
the calculation of certain experiments of Biot and Newton ; 
and Professor Powell of Oxford has pointed out that the data 
so referred to cannot safely be made the basis of such calcula- 
tions, for mathematical reasons. There is indeed here, also, 
one question of fact concerning an experiment stated in New- 
ton's Optics : In a part of the image of an aperture where 
Newton's statement places a dark line, in which Mr. Barton has 
followed him, Professors Airy, Powell, and others, have been 
able to see only a bright space, as the theory would require. 
Probably the experiments giving the two different results have 
not been made under precisely the same circumstances ; and 
the admirers of Newton are the persons who will least of all 
consider his immoveable fame as exposed to any shock by these 

" Perhaps, while the undulationist will conceive that his 
opinions have gained no small accession of evidence by this ex- 
emplification of what they will account for, those who think the 
advocates of the theory have advanced its claims too far, will 
be in some degree concihated by having a distinct acknow- 


ledgement, as during these discussions they have had, of what 
it does not pretend to explain. The whole doctrine of the 
absorption of light is at present out of the pale of its calcula- 
tions ; and if the theory is ever extended to these phaenomena, 
it must be by supplementary suppositions concerning the ether 
and its undulations, of which we have at present not the slight- 
est conception. 

" There are various of the Physical subjects to which your 
Reports I'efer, which it is less necessary to notice in a general 
sketch like the present. The recent discoveries in Thermo- 
electricity, of which Professor Gumming has presented you 
with a review, and the investigations concerning Radiant Heat 
which have been arranged and stated by Professor Powell, are 
subjects of great interest and promise ; and they are gradually 
advancing, by the accumulation of facts bound together by 
subordinate rules, into that condition in which we may hope to 
see them subjugated to general and philosophical theories. 
But with regard to this prospect, the subjects I have mentioned 
are only the fragments of sciences, on which we cannot hope 
to theorize successfully except by considering them with refer- 
ence to their whole ; — Thermo-electricity with reference to the 
whole doctrine of electricity ; Radiant Heat with reference to 
the whole doctrine of heat. 

" If the subjects just mentioned be but parts of sciences, 
thei'e is another on which you have a Report before you, which, 
though treated as one science, is in reality a collection of several 
sciences, each of great extent. I speak of Meteorology, which 
is reported on by Professor Forbes. There is perhaps no por- 
tion of human knowledge more capable of being advanced by 
our conjoined exertions than this : some of the requisite ob- 
servations demand practice and skill ; but others are easily 
made, when the observer is once imbued with sound elemen- 
tary notions ; and in all departments of the subject little can 
be done without a great accumulation of facts and a patient in- 
quiry after their rules. Some such contributions we may look 
for at our present Meeting. Professor Forbes has spoken of 
the possibility of constructing maps of the sky by which we 
may trace the daily and hourly condition of the atmosphere 
over large tracts of the earth. If, indeed, we could make a 
stratigraphical analysis of the aerial shell of the earth, as the 
geologist has done of its solid crust, this would be a vast step 
for Meteorology. This, however, must needs be a difficult task : 
in addition to the complexity of these superincumbent masses, 
time enters here as a new element of variety : the strata of the 
geologist continue fixed and permanent : those of the meteoro- 

Xviii THIRD REPORT — 1833. 

legist change from one moment to another. Another difficulty 
is this ; that while we want to determine what takes place in 
the whole depth of the aerial ocean, our observations are neces- 
sarily made almost solely at its bottom. Our access to the 
heights of the atmosphere is more limited, in comparison with 
what we wish to observe, than our access to the depths of the 

" Geology, indeed, is a most signal and animating instance of 
what may be effected by continued labours governed by common 
views. Mr. Conybeare's Report upon this science gives you 
a view of what has been done in it during the last twenty years ; 
and his ' Section of Europe from the North of Scotland to the 
Adriatic,' which is annexed to the Report, conveys the general 
views with regard to the structure of Central Europe, at which 
geologists have now arrived. To point out any more recent 
additions to its progress or its prospects is an undertaking 
more suitable to the geologists by profession, than to the pre- 
sent sketch. And all who take an interest in the subject will 
rejoice that the constitution and practice of the Geological So- 
ciety very happily provide, by the annual addresses of its Pre- 
sidents, against any arrear in the incorporation of fresh acquisi- 
tions with its accumulated treasures. 

" The science of Mineralogy, on which I had the honour of 
offering a Report to the Association, was formerly looked upon 
as a subordinate portion of Geology. It may, however, now be 
most usefully considered as a science co-ordinate and closely 
allied with Chemistry, and the most important questions for 
examination in the one science belong almost equally to the 
other. Mr. Johnston, in his Report on Chemical Science, has, 
as the subject required, dwelt upon the questions of isomor- 
phism and plesiomorphism, which I had noticed as of great im- 
portance to Mineralogy. Dr. Turner and Prof. Miller, who at 
the last Meeting undertook to inquire into this subject, have 
examined a number of cases, and obtained some valuable facts ; 
but the progress of our knowledge here necessarily requires 
time, since the most delicate chemical analysis and the exact 
measurement of 30 or 40 crystals are wanted for the satisfac- 
tory estabhshment of the properties of each species *. In Che- 

• Perhaps I shall not have a more favourable occasion than the present of 
correcting a statement in my Report, which is not perfectly accurate, on a point 
which has been a subject of controversy between Sir David Brewster and Mr. 
Brooke. I have noticed (p. 338.) the sulphato-tricarbonate of lead of Mr. 
Brooke, as a mineral which at first appeared to contradict Sir David Brewster's 
general law of the connexion of crystalline form with optical structure, in as 
much as it appeared to be of the rhombohedral system, and was found to have 


mistry, besides the great subject of isomorphism to which I 
have referred, there are some other yet undecided questions, 
as for instance those concerning the existence and relations 
of the sulpho-salts and chloro-saUs ; and these are not small 
points, for they affect the whole aspect of chemical theory, and 
thus show us how erroneously we should judge, if we were to 
consider this science as otherwise than in its infancy. 

" In every science. Notation and Nomenclature are questions 
subordinate to calculation and theory. The Notation of Cry- 
stallography is such as to answer the purposes of calculation, 
whether we take that of Mohs, Weiss, or Nauman. It appears 
very desirable that the Notation of Chemistry also should be so 
constructed as to answer the same purpose. Dr. Turner in the 
last edition of his Chemistry, and Mr. Johnston in his Report, 
have used a notation which has this advantage, which that 
commonly employed by the continental Chemists does not 

" I have elsewhere stated to the Association how little hope 
there appears at present to be of purifying and systematizing 
our mineralogical nomenclature. The changes of theory in 
Chemistry to which I have already referred, must necessarily 
superinduce a change of its nomenclature, in the same manner 
in which the existing nomenclature was introduced by the pre- 
valent theory ; and the new views have in fact been connected 
with such a change by those who have propounded them. It 
will be for the Chemical Section of the Association to consider 
how far these questions of Nomenclature and Notation can be 
discussed with advantage at the present Meeting. 

" The Reports presented at the last Meeting had a reference, 
for the most part, to physical rather than physiological science. 
The latter department of human knowledge will be more pro- 
minently the subject of some of the Reports which are to come 
before us on the present occasion. There is, however, one of 

two axes of double refraction ; and which was afterwards found to confirm the 
law, the apparently rhombohedral forms being found by Mr. Haidinger to be 
not simple but compound. It seems, however, that the solution of the difficulty 
(for no one now will doubt that it lias a solution,) is somewhat different. There 
appear to have been included under this name two different kinds of crystals 
belonging to different systems of crystallization. Some which Mr. Brooke found 
to be rhombohedral, Sir David Brewster found to have a single optical axis 
with no trace of composition ; others were prismatic with two axes ; and thus 
Mr. Brooke's original determinations were probably correct. The high reputa- 
tion of the parties in this controversy does not need this explanation ; but pro- 
bably those who look with pleasure at the manner in which the apparent excep- 
tions to laws of nature gradually disappear, may not think a moment or two lost 
in placing the matter on its proper footing. 



last year's Reports which refers to one of the widest questions 
of Physiology ; that of Dr. Prichard on the History of the 
Human Species, and its subdivision into races. The other 
lines of research which tend in the same direction will probably 
be brought before the Association in successive years, and thus 
give us a view of the extent of knowledge which is accessible 
to us on this subject. 

" In addition to these particular notices of the aspect under 
which various sciences present themselves to us as resulting 
from the Reports of last years, there is a reflexion which may 
I think be collected from the general consideration of these 
sciences, and which is important to us, since it bears upon the 
manner in which science is to be promoted by combined labour 
such as that which it is a main object of this Association to 
stimulate and organize. The reflexion to which I refer is 
this ; — that a combination of theory with facts, of general views 
with experimental industry, is requisite, even in subordinate 
contributors to science. It has of late been common to assert 
that facts alone are valuable in science ; that theory, so far as 
it is valuable, is contained in the facts ; and, so far as it is not 
contained in the facts, can merely mislead and preoccupy men. 
But this antithesis between theory and facts has probably in 
its turn contributed to delude and perplex ; to make men's ob- 
servations and speculations useless and fruitless. For it is only 
through some view or other of the connexion and relation of 
facts, that we know what circumstances we ought to notice and 
record ; and every labourer in the field of science, however 
humble, must direct his labours by some theoretical views, 
original or adopted. Or if the word theory be unconquerably 
obnoxious, as to some it appears to be, it will probably still be 
conceded, that it is the rviles of facts, as well as facts themselves, 
with which it is our business to acquaint ourselves. That the 
recollection of this may not be useless, we may collect from the 
contrast which Professor Airy in his Report has drawn between 
the astronomers of our own and of other countries. "In En- 
gland," he says, (p. 184,) " an observer conceives that he has 
done everything when he has made an observation," " In 
foreign observatories," he adds, " the exhibition of results and 
the comparison of results with theory, are considered as de- 
serving more of an astronomer's attention, and demanding 
greater exercise of his intellect, than the mere observation of a 
body on the wire of a telescope." We may, indeed, perceive 
in some measure the reason which has led to the neglect of 
theory with us. For a long period astronomical theory was 
greatly a-head of observation, and this deficiency was mainly 


supplied by the perseverance and accuracy of English ob- 
servers. It was natural that the value and reputation which 
our observations thus acquired for the time, should lead us to 
think too disrespectfully, in comparison, of the other depart- 
ments of the science. Nor is the lesson thus taught us con- 
fined to Astronomy ; for, though we may not be able in other 
respects to compare our facts with the results of a vast and yet 
certain theory, we ought never to forget that facts can only 
become portions of knowledge as they become classed and con- 
nected ; that they can only constitute truth when they are in- 
cluded in general propositions. Without some attention to this 
consideration, we may notice daily the changes of the winds 
and skies, and make a journal of the weather, which shall have 
no more value than a journal of our dreams would have ; but 
if we can once obtain fixed measures of what we notice, and 
connect our measures by probable or certain rules, it is no 
longer a vacant employment to gaze at the clouds, or an un- 
profitable stringing together of expletives to remark on the 
weather ; the caprices of the atmosphere become steady dispo- 
sitions, and we are on the road to meteorological science. 

" It may be added — as a further reason why no observer 
should be content without arranging his observations, in what- 
ever part of Physics, and without endeavouring at least to 
classify and connect them — that when this is not done at first, 
it will most likely never be done. The circumstances of the 
observation can hardly ever be properly understood or inter- 
preted by others ; the suggestions which the observations 
themselves supply, for change of plan or details, cannot in any 
other way be properly appreciated and acted on. And even 
the mere multitude of unanalysed observations may drive future 
students of the subject into a despair of rendering them useful. 
Among the other desiderata in Astronomy which Professor 
Airy mentions, he observes, " Bradley's observations of stars," 
made in 1750, " were nearly useless till Bessel undertook to re- 
duce them" in 1818. "In like manner Bradley's and Mas- 
kelyne's observations of the sun are still nearly useless," and 
they and many more must continue so till they are reduced. 
This could not have happened if they had been reduced and 
compared with theory at the time ; and it cannot but grieve us 
to see so much skill, labour and zeal thus wasted. The per- 
petual reference or attempt to refer observations, however nu- 
merous, to the most probable known rules, can alone obviate 
similar evils. 

" It may appear to many, that by thus recommending theory 
we incur the danger of encouraging theoi-eticul speculations 

XXii THIRD REPORT — 1833. 

to the detriment of observation. To do this would be indeed 
to render an ill service to science : but we conceive that our 
purpose cannot so far be misunderstood. Without here at- 
tempting any nice or technical distinctions between theory and 
hypothesis, it may be sufficient to observe that all deductions 
from theory for any other pupose than that of comparison with 
observation are frivolous and useless exercises of ingenuity, so 
far as the interests of physical science are concerned. Specu- 
lators, if of active and inventive minds, will form theories 
whether we wish it or no. These theories may be useful or 
may be otherwise — we have examples of both results. If the 
theories merely stimulate the examination of facts, and are 
modified as and when the facts suggest modification, they may be 
erroneous, but they will still be beneficial ; — they may die, but 
they will not have lived in vain. If, on the other hand, our 
theory be supposed to have a truth of a superior kind to the 
facts ; to be certain independently of its exemplification in par- 
ticular cases ; — if, when exceptions to our propositions occui', 
instead of modifying the theory, we explain away the facts, — 
our theory then becomes our tyrant, and all who work under 
its bidding do the work of slaves, they themselves deriving no 
benefit from the result of their labours. For the sake of ex- 
ample we may point out the Geological Society as a body which, 
labouring in the former spirit, has ennobled and eni-iched itself 
by its exertions : if any body of men should employ themselves 
in the way last described, they must soon expend the small 
stock of a priori plausibility with which they must of course 
begin the world. 

" To exemplify the distinction for a moment longer, let it be 
recollected that we have at the present time two rival theories 
of the history of the earth which prevail in the minds of geo- 
logists ; — one, which asserts that the changes of which we trace 
the evidence in the earth's materials have been produced by 
causes such as are still acting at the surface ; another, which 
considers that the elevation of mountain chains and the transi- 
tion from the organized world of one formation to that of the 
next, have been produced by events which, compared with the 
present course of things, may be called catastrophes and con- 
vulsions. Who does not see that all that those theories have 
hitherto done, has been, to lead geologists to study more ex- 
actly the laws of permanence and of change in the existing 
organic and inorganic world, on the one hand; and on the 
other, the relations of mountain chains to each other, and to 
the phaenomena which their strata present ? And who doubts, 
that, as the amount of the full evidence may finally be, (which 


may, indeed, perhaps require many generations to accumulate,) 
geologists will give their assent to the one or the other of these 
views, or to some intermediate opinion to which both may 
gradually converge? 

" On the other hand — to take an example from a science with 
which I have had a professional concern- — the theory that cry- 
stalline bodies are composed of ultimate molecules which have 
a definite and constant geometrical form, may properly and 
philosophically be adopted, so far as we can, by means of it, 
reduce to rules the actually occurring secondary faces of such 
substances. But if we assume the doctrine of such an atomic 
composition, and then form imaginary arrangements of these 
atoms, and enunciate these as explanations of dimorphism, 
or plesiomorphism, or any other apparent exception to the 
general principle, we proceed, as appears to me, unphilosophi- 
cally. Let us collect and classify the facts of dimorphism and 
plesiomorphism, and see what rules they follow, and we may 
then hope to discern whether our atomic theory of crystalline 
molecules is tenable, and what modifications of it these cases, 
uncontemplated in its original formation, now demand. 

" I will not now attempt to draw forth other lessons which 
the Report of last year may supply for our future guidance ; 
although such offer themselves, and will undoubtedly affect 
the spirit of our proceedings during this Meeting. But there 
is a reflexion belonging to what I may call the morals of science, 
which seems to me to lie on the face of this Report, and which 
I cannot prevail upon myself to pass over. In looking steadily 
at the past history and present state of physical knowledge, we 
cannot, I think, avoid being struck with this thought, — How 
little is done and how much remains to do ; — and again, not- 
withstanding this, how much we owe to the great philosophers 
who have preceded us. It is sometimes advanced as a charge 
against the studies of modern science, that they give men an 
overweening opinion of their own acquirements, of the supe- 
riority of the present generation, and of the intellectual power 
and progress of man ; — that they make men confident and con- 
temptuous, vain and proud. That they never do this, would 
be much to say of these or of any other studies ; but, assuredly, 
those must read the history of science with strange preposses- 
sions who find in it an aliment for such feelings. What is the 
picture which we have had presented to us ? Among all the 
attempts of man to systematize and complete his knowledge, 
there is one science, Astronomy, in which he may be considered 
to have been successful ; he has there attained a general and 
certain theory : for this success, the labour of the most highly- 

Xxiv THIRD REPORT — 1833. 

gifted portion of the species for 5000 years has been requisite. 
There is another science, Optics, in which we are, perhaps, in 
the act of obtaining the same success, with regard to a part of 
the phaenomena. But all the rest of the prospect is compara- 
tively darkness and chaos ; hmited rules, imperfectly known, 
imperfectly verified, connected by no known cause, are all that 
we can discern. Even in those sciences which are considered 
as having been most successful, as Chemistry, every few years 
changes the aspect under which the theory presents the facts 
to our minds, while no theory, as yet, has advanced beyond the 
mere horn-book of calculation. What is there here of which 
man can be proud, or from which he can find reason to be pre- 
sumptuous ? And even if the Discoverers to whom these sciences 
owe such progress as they have made — the great men of the 
present and the past — if they might be elate and confident 
in the exercises of their intellectual powers, who are ive, that 
we should ape their mental attitudes ? — we, who can but with 
pain and eiFort keep a firm hold of the views which they have 
disclosed ? But it has not been so ; they, the really great in 
the world of intellect, have never had their characters marked 
with admiration of themselves and contempt of others. Their 
genuine nobility has ever been superior to those ignoble and 
low-born tempers. Their views of their own powers and achieve- 
ments have been sober and modest, because they have ever felt 
how near their predecessors had advanced to what they had 
done, and what patience and labour their own small progress 
had cost. Knowledge, like wealth, is not likely to make us 
proud or vain, except when it comes suddenly and unlearned ; 
and in such a case, it is little to be hoped that we shall use 
well, or increase, our ill-understood possession. 

" Perhaps some of the appearance of overweening estimation 
of ourselves and our generation which has been charged against 
science, has arisen from the natural exultation which men feel 
at witnessing the successes of art. I need not here dwell upon 
the distinction of science and art ; of knowledge, and the ap- 
plication of knowledge to the uses of life ; of theory and 
practice. In the success of the mechanical arts there is much 
that we look at with an admiration mingled with some feeling 
of triumph ; and this feeling is here natural and blameless. 
For what is all such art but a struggle, — a perpetual conflict 
with the inertness of matter and its unfitness for our purposes? 
And when, in this conflict, we gain some point, it is impossible 
we should not feel some of the exultation of victory. In all 
stages of civilization this temper prevails : from the naked in- 
habitant of the islands of the ocean, who by means of a piece 


of board glides through the furious and apparently deadly line 
of breakers, to the traveller vi'ho starts along a rail-road with 
a rapidity that dazzles the eye, this triumphant joy in suc- 
cessful art is universally felt. But we shall have no difficulty 
in distinguishing this feeling from the calm pleasure which we 
receive from the contemplation of truth. And when we con- 
sider how small an advance of speculative science is implied in 
each successful step of art, we shall be in no danger of im- 
bibing, from the mere high spirits produced by difficulty over- 
come, any extravagant estimate of what man has done or can 
do, any perverse conception of the true scale of his aims and 

" Still, it would little become us here to be unjust to prac- 
tical science. Practice has always been the origin and stimulus 
of theory : Art has ever been the mother of Science ; the 
comely and busy mother of a daughter of a far loftier and 
serener beauty. And so it is likely still to be : there are no 
subjects in which we may look more hopefully to an advance in 
sound theoretical views, than those in which the demands of 
practice make men willing to experiment on an expensive scale, 
with keenness and perseverance ; and reward every addition 
of our knowledge with an addition to our power. And even 
they — for undoubtedly there are many such — who require no 
such bribe as an inducement to their own exertions, may still 
be glad that such a fund should exist, as a means of engaging 
and recompensing subordinate labourers. 

" I will not detain you longer by endeavouring to follow 
more into detail the application of these observations to the 
proceedings of the General and Sectional Meetings during the 
present week. But I may remark that some subjects, circum- 
stanced exactly as I have described, will be brought under 
your notice by the Reports which we have reason to hope for 
on the present occasion. Thus, the state of our knowledge of 
the laws of the motion of fluids is universally important, since 
the motion of boats of all kinds, hydraulic machinery, the tide^, 
the flowing of rivers, all depend upon it. Mr. Stevenson and 
Mr. Rennie have undertaken to give us an account of different 
branches of this subject as connected with practice ; and Mr. 
Challis will report to us on the present state of the analytical 
theory. In like manner the subject of the strength of materials, 
which the multiplied uses of iron, stone and wood, make so inter- 
esting, will be brought before you by Mr. Barlow. These were 
two of the portions of mechanics the earliest speculated upon, 
and in them the latest speculators have as yet advanced little 
beyond the views of the earliest. 


" I mention these as specimens only of the points to which 
we may more particularly direct our attention. I will only 
observe, in addition, that if some studies, as for instance those 
of Natural History and Physiology, appear hitherto to have 
occupied less space in our proceedings than their importance 
And interest might justly demand, this has occurred because 
the Reports on other subjects appeared more easy to obtain in 
the first instance ; and the balance will I trust be restored at 
the present Meeting. I need not add anything further on this 
subject. Among an assembly of persons such as are now met 
in this place, there can be no doubt that the most important 
and profound questions of science in its existing state will be 
those which will most naturally occur in our assemblies and 
discussions. It merely remains for me to congratulate the As- 
sociation upon the circumstances under which it is assembled ; 
and to express my persuasion that all of us, acting under the 
elevating and yet sobering thought of being engaged in the 
great cause of the advancement of true science, and cherishing 
the views and feelings which such a situation inspires, shall 
derive satisfaction and benefit from the occasions of the present 

Mr. Whewell having concluded his Address, the Meeting 
adjourned, after electing by a general vote the candidates who 
had been approved by the Council and by the General Com- 

At eight P.M., the Members having reassembled in the Senate- 
house, Mr. Taylor read a Report on the state of our know- 
ledge respecting Mineral Veins, which was followed by a general 
discussion on the nature and origin of veins. 

On Wednesday at one p.m., the Chairmen of the Sections hav- 
ing read the minutes of their proceedings to the Meeting, the 
Rev. G. Peacock delivered a brief abstract of his Report on 
the state of the Theory of Algebra. Professor Lindley read a 
Report on the state of Physiological Botany ; and Mr. G. Ren- 
nie on the state of Practical Hydraulics. Auditors were ap- 
pointed to examine the accounts. 

On Thursday, at one p.m., the auditors reported the state 
of the accounts. The Chairmen of the Sections read the mi- 
nutes of their proceedings. Professor Christie read a Report 
on the present state of our knowledge respecting the Magnetism 
of the Earth. A summary of the contents of a Report on the 
state of knowledge as to the Strength of Materials, by Pro- 


fessoi' Bavlow, was given, in the absence of the Author, by the 
-Rev. W. Whewell. 

In the evening, Mr. Whewell delivered a Lecture in the 
Senate-house, on the manner in which observations of the Tide 
may be usefully made to serve as a groundwork for general 
views ; either by observing the time of high water at different 
places on the same day, in order to determine the motion of 
the summit of the tide-wave ; or by continuing the observations 
for a considerable time, and comparing them with the moon's 
transit to obtain the semi-menstrual inequality. He observed, 
that it appears from Mr. Lubbock's recent researches on the 
subject, that the tides of Portsmouth and Brest agree very 
closely in the law of this inequality, and that the tides of Ply- 
mouth and London also agree ; but that there is an anomaly 
which cannot at present be explained in the comparison of Brest 
with Plymouth. Professor Parish explained to the Meeting 
the advantages which he conceived would be derived from ap- 
plying the power of steam to carriages on undulating roads in 
preference to level rail-ways. 

On Friday, at one p.m., the Chairmen of the Sections having 
read the minutes of their proceedings, the Rev. J. Challis made 
a Report on the progress of the Theory of Fluids. The Pre- 
sident stated the appropriation * to certain scientific objects of 
a portion of the funds of the Association to the amount of 
600/. Mr. Babbage, at the President's request, explained his 
views respecting the advantages which wovild accrue to science 
from such a collection of numerical facts as he had formerly 
recommended under the title of " Constants of Nature and 
Art." The President announced, that it had been resolved 
by the General Committee, that the Meeting of 1834 should 
take place at Edinburgh in the early part of the month of Sep- 
tember ; he read the names of the Officers and Members of the 
Council appointed for the ensuing year. 

The thanks of the Meeting were then voted to the Vice- 
Chancellor and the other authorities of the University, to the 
retiring Officers and Members of the Council, to the President, 
the Secretaries for Cambridge, the Local Committee of Manage- 
ment, and the General Secretary. 

The President, in his concluding Address to the Meeting, 
explained an irregularity which had occurred in the formation 
of a new Section. In addition to the five Sections into which 
the Meeting had been divided by the authority of the General 

* For a particular account of these appropriations, see p. xxxvi. 


XXviii THIRD REPORT 1833. 

Committee, he stated that another had come into operation, the 
object of whicli was to promote statistical inquiries. It had 
originated with some distinguished philosophers, but could not 
be regarded as a legitimate branch of the Association till it had 
received the recognition of the governing body ; there could be 
little doubt, however, that the new Section would obtain the 
sanction of the General Committee, with some limitation per- 
haps of the specific objects of inquiry. On this subject he 
made the following observations : — 

" Some remarks may be expected from me in reference to the 
objects of this Section, as several Members may perhaps think 
them ill fitted to a Society formed only for the promotion of 
natural science. To set, as far as I am able, these doubts at 
rest, I will explain what I understand by science, and what I 
think the proper objects of the Association. By science, then, 
I understand the consideration of all subjects, whether of a pure 
or mixed nature, capable of being reduced to measurement and 
calculation. All things comprehended under the categories of 
space, time and number properly belong to our investigations ; 
and all phaenomena capable of being brought under the sem- 
blance of a law are legitimate objects of our inquiries. But there 
ai'e many important subjects of human contemplation which come 
under none of these heads, being separated from them by new- 
elements ; for they bear upon the passions, affections and feel- 
ings of our moral nature. Most important parts of our nature 
such elements indeed are ; and God forbid that I should call 
upon any man to extinguish them ; but they enter not among 
the objects of the Association. The sciences of morals and 
politics are elevated far above the speculations of oiu- philosophy. 
Can, then, statistical inquiries be made compatible with our 
objects, and taken into the bosom of our Society? I think 
they unquestionably may, so far as they have to do with matters 
of fact, with mere abstractions, and with numerical results. 
Considered in that light they give what may be called the raw 
material to political economy and political philosophy ; and by 
their help the lasting foundations of those sciences may be per- 
haps ultimately laid. These inquiries are, however, it is import- 
ant to observe, most intimately connected with moral phaeno- 
mena and economical speculations, — they touch the mainsprings 
of passion and feeling, — they blend themselves with the generali- 
zations of political science ; but when we enter on these higher 
generahzations, that moment they are dissevered from the ob- 
jects of the Association, and must be abandoned by it, if it 
means not to desert the secure ground which it has now taken. 

" Should any one affirm (what, indeed, no one is prepared 


to deny,) that all truth has one common essence, and should 
he then go on to ask why truths of different degrees should be 
thus dissevered from each other, the reply would not be dif- 
ficult. In physical truth, whatever may be our difference of 
opinion, there is an ultimate appeal to experiment and ob- 
servation, against which passion and prejudice have not a 
single plea to urge. But in moral and political reasoning, we 
have ever to do with questions, in which the waywardness of 
man's will and the turbulence of man's passions are among the 
strongest elements. The consequence it is not for me to tell. 
Look around you, and you will then see the whole framework 
of society put in movement by the worst passions of our na- 
ture; you will see love turned into hate, deliberation into dis- 
cord, and men, instead of mitigating the evils which are about 
them, tearing and mangling each other, and deforming the 
moral aspect of the world. And let not the Members of the 
Association indulge a fancy, that they are themselves exempt 
from the common evils of humanity. There is that within us, 
which, if put into a flame, may consume our whole fabric, — 
may produce an explosion, capable at once of destroying all 
the principles by which we are held together, and of dissi- 
pating our body in the air. Our Meetings have been essen- 
tially harmonious, only because we have kept within our proper 
boundaries, confined ourselves to the laws of nature, and 
steered clear of all questions in the decision of which bad 
passions could have any play. But if we transgress our pro- 
per boundaries, go into provinces not belonging to us, and open 
a door of communication with the dreary wild of politics, that 
instant will the foul Daemon of discord find his way into our 
Eden of Philosophy. 

" In every condition of society there is some bright spot on 
which the eye loves to rest. In the turbulent republics of 
ancient Greece, where men seemed in an almost ceaseless war- 
fare of mind and'body, they had their seasons of solemnity, when 
hostile nations made a truce with their bitter feelings, as- 
sembled together, for a time, in harmony, and joined in a great 
festival ; which, however differing from what we now see in 
its magnitude and forms of celebration, was consecrated, like 
our present Meeting, to the honour of national genius. What- 
ever have been the bitter feelings which have so often disgraced 
the civil history of mankind, I dare to hope that they will never 
find their resting-place within the threshold where this Associa- 
tion meets ; that peace and good will, though banished from 
every other corner of the land, will ever find an honoured seat 
amongst us ; and that the congregated philosophers of the 
empire, throwing aside bad passion and party animosity, will, 


year by year, come to their philosophical Olympia, to witness 
a noble ceremonial, to meet in a pacific combat, and share in the 
glorious privilege of pushing on the triumphal car of Truth. 

" The last duty I have to perform this morning would be a 
painful one indeed, were our Assembly to be broken up into 
elements which were not again to be reunited. The Association 
is not, however, dissolved ; its meeting is only adjourned to an- 
other year; and it has been a matter of great joy to me to an- 
nounce to you, that the Committee has elected for your next 
President a distinguished soldier and philosopher ; and that it 
will be your privilege to reassemble in one of the fairest capitals 
of the world, — in a city v/hich has nursed a race of literary and 
philosophic giants, — ^^in a land filled with natural beauties, and 
wedded to the imagination and the memory by a thousand en- 
dearing associations. 

" There is a solemnity in parting words, which may, I think, 
justify me (especially after what has been so well said this morn- 
ing by the Marquis of Nortliampton,) in passing the limits I 
have so far carefully prescribed to myself, and in treading for a 
moment on more hallowed ground. In the first place, I would 
entreat you to remember that you ought above all things to re- 
joice in the moral influence of an Association like the present. 
Facts, which are the first objects of our pursuit, are of compa- 
ratively small value till they are combined together so as to 
lead to some philosophic inference. Physical experiments, con- 
sidered merely by themselves, and apart from the rest of nature, 
are no better than stones lying scattered on the ground, which 
require to be chiselled and cemented before they can be made 
into a building fit for the habitation of man. The true value 
of an experiment is, that it is subordinate to some law, — that it 
is a step toward the knowledge of some general truth. Without, 
at least, a glinnnering of such truth, physical knowledge has no 
true nobility. But there is in the intellect of man an appetency 
for the discovery of general truth, and by this appetency, in 
subordination to the capacities of his mind, has he been led on to 
the discovery of general laws ; and thus has his soul been fitted 
to reflect back upon the world a portion of the counsels of his 
Creator. If I have said that physical phsenomena, unless con- 
nected with the ideas of order and of law, are of little worth, 
I may further say, that an intellectual grasp of material laws of 
the highest oi'der has no moral worth, except it be combined 
with another movement of the mind, raising it to the perception 
of an intelligent First Cause. It is by help of this last movement 
that nature's language is comprehended ; that her laws become 
pregnant with meaning ; that material phaenomena are instinct 
with life ; that all moral and material changes become linked 


together ; and that Truth, under whatever forms she may pre- 
sent herself, seems to have but one essential substance. 

"I have before spoken of the distinctions between moral and 
physical science; and I need not repeat what I have said, unless 
it be once more solemnly to adjure you not to leave the straight 
path by which you are advancing, — not to desert the cause for 
which you have so well combined together. But let no one 
misunderstand my meaning. If I have said that bad passions 
mingle themselves with moral and political sciences, and that 
the conclusions of these sciences are made obscure from the 
want of our comprehending all the elements with which we 
have to deal, I have only spoken the truth ; but still I hold that 
moral and political science is of a higher order than the physical. 
The latter has sometimes, in the estimation of man, been placed 
on a higher level than it deserves, only from the circumstance of 
its being so well defined, and grounded in the evidence of ex- 
periments appealing to the senses. Its progress is marked by 
indices the eye can follow ; and the boundaries of its conquests 
are traced by landmarks which stand high in the horizon of 
man's history. But with all these accompaniments, the moral 
and political sciences entirely swallow up the physical in impor- 
tance. For what are they but an interpretation of the governing 
laws of intellectual nature, having a relation in time pi'esent to 
the social happiness of millions, and bearing in their end on the 
destinies of immortal beings ? 

" Gentlemen, if I look forward with delight to our meeting 
again at Edinburgh, it is a delight chastised by a far different 
feeling, to which, had not these been parting words, I should 
not have ventured to give an utterance. It is not possible 
we should all again meet together. Some of those whose 
voices have been lifted up during this great Meeting, whose 
eyes have brightened at the presence of their friends, and 
whose hearts have beat high during the intellectual commu- 
nion of the week, before another year may not be numbered 
with the living. Nay, by that law of nature to which every 
living man must in his turn yield obedience, it is certain that 
before another festival, the cold hand of death will rest on the 
head of some who are present in this assembly. If a thought 
like this gives a tone of grave solemnity to words of parting, it 
svu-ely ought to teach us, during our common rejoicings at the 
triumphal progress of science, a personal lesson of deep humility. 
By the laws of nature, before we can meet again, many of those 
bright faces which during the past week I have seen around me 
may be laid low, for the hand of death may have been upon 
them ; but wherever we reassemble, God grant that all our 
attainntents in science may tend to our moral improvement; and 


may we all meet at last in the presence of that Almighty Being, 
whose will is the rule of all law, and whose bosom is the centre 
of all power!" 


The Sections assembled daily at eleven a.m., and occasion- 
ally also at half-past eight p.m., at their respective places of 
meeting, in the Schools, the Astronomical Lecture-room, and 
the Hall of Caius College. On Saturday, the Section of Na- 
tural History made an excursion to the Fens. 

Abstracts of most of the Communications which were made 
to the Sections will be found in a subsequent part of the 

In addition to the communications of which abstracts are 
there given, notices of the following transactions appear on the 
minutes : — 

M. Quetelet described the observations which he had made 
on Falling Stars. It was suggested that such observations 
might be available in certain cases for determining differences 
of longitude. 

Mr. Potter communicated some calculations of the height 
of the Aurora BoreaHs, seen on the 21st of March 1833. 

Mr. Hopkins gave an abstract of a paper on the Vibration 
of Air in Cyhndrical Tubes of definite length. 

Dr. Ritchie made some remarks on the Sensibility of the 
Eye, and the errors to which it is subject. 

Mr. Barton gave a view of his opinions on the Propagation 
of Heat in solid bodies. 

A letter was received from Mr. Frend regarding certain 
points in the Theory of the Tides. 

The Rev.W. Scoresby described a Celestial Compass invent- 
ed by Col. Graydon. 

Mr. R. Murphy read some remarks on the utihty of observ- 
ing the Magnetic Dip in Mines. 

M. Quetelet gave an account of some observations made by 
himself and M. Necker de Saussure, which corroborate the 
statements of M. Kuppfer, respecting theinequahty of magne- 
tic intensity at the top and the base of mountains. 

Professor Christie stated his views relative to the cause of 
the Magnetism of the Earth. 

Mr. A. Trevelyan read a paper on certain Vibrations of 
Heated Metals. 

Mr. Brunei exhibited and explained a Model in illustration 
of his method of constructing Bridges without centering. 


A notice of some experiments relative to Isomorphism, by 
Dr. Turner and Professor Miller, was read. 

Dr. Daubeny made a communication on the Gases given off 
from the surfaces of the water in certain thermal springs. 

The Rev. W. V. Harcourt exhibited specimens of Metal taken 
out of the crevices at the bottom of a mould in which a large 
bronze figure had been cast by Mr. Chantrey ; together with 
fragments of the Bronze employed in the casting, from which 
the former specimens differed considerably in colour, frangi- 
bility, &c. 

Mr. Lowe gave an account of various chemical products 
found in the retorts and flues of Gas Works. 

Mr. Pearsall made a communication on the bleaching powers 
of Oxygen. 

Mr. J. Taylor described the character of the Ecton Mine, 
and the occurrence of the copper ore in connected cavities 
which had been explored to a depth of 225 fathoms without 
reaching the termination of them. 

Dr. Buckland described the manner in which fibrous Lime- 
stone occurs in the Isle of Purbeck and other situations. 

Mr. Murchison stated, and illustrated by Maps and Sections, 
the principal results of his inquiries into the sedimentary de- 
posits which occupy the western parts of Shropshire and Here- 
fordshire, and are prolonged in a S.W. direction through the 
counties of Radnor, Brecknock, and Caermarthen, and the in- 
trusive igneous rocks which occur in certain parts of the di- 
strict. He mentioned the occurrence of freshwater Limestone 
in a detached Coal-field of Shropshire. 

Professor Sedgwick described the leading features in the 
Geology of North Wales, the lines of elevation, the relation of 
the trap rocks to the slate system, the cleavage of the slate ; 
pointed out the relations of this tract to that examined by Mr. 
Murchison ; and drew a general parallel between the slate 
formations of Wales and Cumberland. 

Mr. J. Taylor having read to the Section the concluding 
part of his Report on Veins, in the discussion which followed, 
M. Dufrenoy entered into a consideration of some phaenomenaof 
the igneous rocks of Britanny and Central France, viewed with 
reference to the connexion between them and the metalliferous 
veins of those districts, and remarked on the occurrence in 
Central France of mineral veins, only in the narrow zone at the 
junction of the unstratified and stratified rocks. He also made 
some remarks on the association of dolomite and gypsum, with 
the igneous rocks of the Alps and the Pyrenees. 

Professor Sedgwick gave a general account of the Red Sand- 
stones connected with the Coal-measures of Scotland, and the 

XXxiv THIRD REPORT — 1833. 

Isle of Arran, with the view of showing that they are perfectly 
distinct from the similar rocks connected with the Magnesian 

JMr. Hartop exhibited a Map and Sections to illustrate the 
series of Coal Strata in South Yorkshire, and their direction and 
varying dip in the valley of the Dun, and to the north and south 
of that river; described the characters of the strata, and the in- 
fluence of certain great dislocations on the quality of the coal. 

Mr. Greenough exhibited a Map of Western Europe, on 
which the relative levels of land and water were represented by 
means of colours, instead of engraving. Mr. Greenough was 
requested to permit a map on this plan to be published. 

The Rev. J. Hailstone communicated some notices relating 
to Mineral Veins. 

Sections of the Well in the Dock Yard at Portsmouth, and 
of the Well in the Victualling Yard at Weevil, were communi- 
cated by the Rev. Mr. Leggat and Mr. Blackburn, on the pai't 
of the Portsmouth Philosophical Society; and a letter from 
Mr. Goodrich, explanatary of the Sections, was read. 

Mr. Mantell exhibited a perfect Femur of the Iguanodon, 
and explained its distinctive anatomical characters. 

Mr. W. C. Trevelyan exhibited specimens of Coprolites, 
and remains of Fishes, from the Edinburgh Coal-field. 

Mr. Fox exhibited specimens of Fishes from the Magnesian 
Limestone and Marl-slate of Durham. 

Mr. Gray made some remarks on the occurrence of Water 
in the Valves of Bivalve Shells, and exhibited a specimen of 
Spondylus varius, in which water was contained in both the 

Mr. Ogilby gave an account of his views respecting the 
classification of Ruminating Quadrupeds, which he proposed to 
found upon the presence or absence of horns on the female sex ; 
the peculiar form of the upper lip ; and the presence or absence 
of the subocular and submaxillary glands. He showed the ap- 
plication of these views to the division of hollow-horned rumi- 
nating animals without horns in the female sear, which he dis- 
tributed into five new genera. 

The Rev. W. Scoresby communicated some observations on 
the adaptation of the Structure of the Cetacea to their habits 
of life and residence in the Ocean ; and suggested the use 
which might be made of the peculiar forms of the Whalebone 
in their classification. 

Lieutenant Colonel Sykes exhibited a specimen of the Short- 
tailed Manis, and communicated some observations on its mode 
of progression. 

Mr. Brayley communicated a memoir on the laws regulating 


the distribution of the powers of producing Light and Heat 
among Animals. 

Mr. H. Strickland made some remarks on the Vipera Chersea, 
showing its specific difterence from the common Viper. 

The subject of the use of the Pith in Plants, was discussed 
by Professor Burnett, Professor Henslow, Mr. Curtis, and Mr. 

Dr. Roupell exhibited some Drawings representing the 
effects of irritant Poisons upon the hving membrane of the in- 
testinal canal of Men and Animals. 

Mr. Fisher communicated some observations on the physical 
condition of the Brain during sleep. 

Mr. Brooke made some remarks on the physiology of the 
Eye and the Ear. 

Dr. Marshall Hall gave an abstract of his views respecting 
the reflex function of the Medulla oblongat^i and Medulla spi- 


The General Committee met daily at ten a.m., and at other 
hours by adjournment, in the Hall of Trinity Hall. The Com- 
mittees of Sciences met as soon after ten as the business of the 
General Coimnittee permitted, in the rooms of their respective 
Sections. The General Committee made the necessary arrange- 
ments for the conduct of the Meeting ; formed the Sectional 
Committees of Sciences ; determined the place and time of the 
next Meeting ; appointed the new Officers and Council ; and 
passed the following Resolutions : — 

1. That the thanks of the Association be given to the Societies 
and Institutions from which it has received invitations, — in Bris- 
tol, Birmingham, Liverpool, Newcastle and Edinburgh. 

2. That Members of the Association whose subscription shall 
have been due for two years, and who shall not pay it on proper 
notice, shall cease to be Members, powder being left to the Com- 
mittee or Council to reinstate them on reasonable grounds 
within one year, on payment of their arrears. 

3. That the number of Deputies which provincial Institutions 
shall be entitled to send to the Meetings as Members of the 
General Committee, shall be two from each Institution. 

4. That the following instructions be given to each of the Com- 
mittees of Sciences : — 

To select those points of science, which, on a review of the 
former Recommendations of the Committees, or those contained 

XXXvi THIRD REPORT — 18,33. 

in the Reports published by the Association, or from sugges- 
tions made at the present Meeting, they may think most fit to 
be advanced by an application of the funds of the Society, 
either in compensation for labour, or in defraying the expense 
of apparatus, or otherwise. The Committee are requested to 
confine their selections to definite as well as important objects ; 
to state their reasons for the selection, and where they may 
think proper, to designate individuals to undertake the desired 
investigations ; they are to transmit their Recommendations 
through their Secretaries to the General Committee. 

The Committees of Sciences having complied with these in- 
structions, the following Resolutions were passed by the General 
Committee : 

1. That a sum not exceeding 200/. be devoted to the dis- 
cussion of observations of the Tides, and the formation of Tide 
Tables, under the superintendence of Mr. Baily, Mr. Lubbock, 
Rev. G. Peacock, and Rev. W. Whewell. 

2. That a sum not exceeding 50/. be appropriated to the 
construction of a Telescopic Lens, or Lenses, out of Rock Salt, 
under the direction of Sir David Brewster. 

S. That Dr. Dalton and Dr. Prout be requested to institute 
experiments on the specific gravities of Oxygen, Hydrogen, and 
Carbonic Acid ; and that a sum not exceeding 50/. be appropri- 
ated to defray the expense of any apparatus which may be re- 

4. That a series of experiments on the effects of long con- 
tinued Heat be instituted at some iron furnace, or in any other 
suitable situation ; and that a sum not exceeding 50/. be placed 
at the disposal of a Sub-Committee, consisting of Professor 
Daubeny, Rev. W. V. Harcourt, Professor Sedgwick, and Pro- 
fessor Turner, to meet any expense which may be incurred *. 

5. That measurements should be made, and the necessary 
data procured, to determine the question of the permanence or 
change of the relative Level of Sea and Land on the coasts of 
Great Britain and Ireland ; and that for this purpose a sum 
not exceeding 100/. be placed at the disposal of a Sub-Com- 
mittee, consisting of Mr. Greenough, Mr. Lubbock, Mr. G. 
Rennie, Professor Sedgwick, Mr. Stevenson, and Rev. W. 
Whewell ; — the measurements to be so executed, as to furnish 
the means of reference in future times, not only as to the re- 
lative levels of the land and sea, but also as to waste or exten- 
sion of the land. 

• These experiments have been instituted by Mr. Harcourt, in Yorkshire, at 
the Low Moor Iron Works, the property of Messrs. Hird and Co., and at the 
Elsecar Furnace, belonging to Earl Fitzwilliam. 


6. That the effects of Poisons on the Animal Economy should 
be investigated and illustrated by graphic representations ; and 
that a sum not exceeding 251. be appropriated for this object. 
Dr. Roupell, and Dr. Hodgkin were requested to undertake 
this investigation. 

7. That the sensibilities of the Nerves of the Brain should 
be investigated ; and that a sum not exceeding 251. should be 
appropriated to this object. Dr. Marshall Hall and Mr. S. D. 
Broughton were requested to undertake these experiments. 

8. That a sum not exceeding 100/. be appropriated towards 
the execution of the plan proposed by Professor Babbage, for 
collecting and arranging the Constants of Nature and Art*. 

9. That a representation be submitted to Government on the 
part of the British Association, stating that it would tend greatly 
to the advancement of astronomy, and the art of navigation, if the 
observations of the sun, moon and planets, made by Bradley, 
Maskelyne and Pond, were reduced; and that a deputation -j- be 
appointed to wait upon the Lords of the Treasuiy with a re- 
quest, that public provision may be made for the accomplish- 
ment of this great national object. 

Proposals for the formation of a Statistical Section were ap- 
proved. It was resolved, that the inquiries of this Section should 
be restricted to those classes of facts relating to communities of 
men which are capable of being expressed by numbers, and 
which promise, when sufficiently multiplied, to indicate general 

A Committee of Statistical Science was formed %. The Re- 
commendations § of the several Committees of Science were re- 
vised and approved. 


Charles Babbage, F.R.S. Lucasian Professor of Mathe- 
matics, Cambridge. 

R. I. Murchison, F.R.S. V.P.G.S. &c. 
John Taylor, F.R.S. Treas. G.S. &c. 

• For an abstract of Mr. Babbage 's plan, see the Appendix. 

+ The deputation consisted of Professor Airy, Mr. Baily, Mr. D. Gilbert and 
Sir John Herschel. The application was immediately complied with by the Go- 

\ For an account of the proceedings of this Committee, see the Appendix. 

§ These Recommendations will be found marked with an asterisk in the col- 
lection of Recommendations and Suggestions printed in the latter part of the 

XXXviii THIRD REPORT — 1833. 


President. — Rev. Adam Sedgwick, F.R.S. G.S. and Wood- 
wardian Professor of Geology, Cambridge. 

Vice-Presidents. — G. B. Airy, F.G.S. Plumian Professor of 
Astronomy, Cambridge. John Dalton, D.C.L. F.R.S. Instit. 
Reg. Sc. Paris. Corresp. 

President elect.— lAeni. Gen. Sir T. M. Brisbane, K.C.B. 
F.R.S. L. & E. President of the Royal Soc. Edinb. Inst. 
Reg. Sc. Paris. Corresp. 

Vice-Presidents elect. — Sir David Brewster, K.G.H. LL.D. 
F.R.S. L. & E. Rev. J. Robinson, D.D. Astronomer Royal 
at Armagh. 

Treasurer.— John Taylor, F.R.S. Treas. G.S. 

General Secretary. — Rev. W. V. Harcourt, F.R.S. G.S. 

Assistant Secretary. — John Phillips, F.R.S. G.S. Professor 
of Geology in King's College, London. 

Secretaries for Oxford. — Charles Daubeny, M.D. 
L.S. Professor of Botany. Rev. B. Powell, F.R.S. 
Professor of Geometry. 

Secretaries for Cambridge. — Rev. J. S. Henslow, 
G.S. Professor of Botany. Rev. W. Whewell, F.R.S. 

Secretaries for Edinburgh. — John Robison, Sec. 
James D. Forbes, F.R.S. L. & E. F.G.S. Professor of Natural 

Secretary for Dublin. — Rev. Thomas Luby. 


Rev, W. Buckland, D.D, F.R.S. Professor of Geol. and Min. 
Oxford. W. Clift, F.R.S. Rev. T.Chalmers, D.D. Professor of 
Divinity, Edinbvn-gh. S. H. Christie, F.R.S. Professor of Ma- 
thematics at Woolwich. Earl Fitzwilliam, F.R.S. G.S. G. B. 
Greenough, F.R.S. Pres. of the Geol. Society. T. Hodg- 
kin, M.D. London. W. R. Hamilton, Astronomer Royal 
for Ireland. W. J. Hooker, F.R.S. Professor of Botany, 
Glasgow. Robert Jameson, F.R.S. Professor of Natural Hi- 
story, Edinburgh. John Lindley, F.R.S. Professor of Botany 
in the University of London. J. W. Lubbock, Treas. R.S. 
Rev. B. Lloyd, D.D. Treas. Prov. of Trin. Coll. Dublin. 
R. I. Murchison, F.R.S. &c. Patrick Neill, M.D. F.R.S.E. 
Edinburgh. George Rennie, F.R.S. Rev. W. Ritchie, LL.D. 
F.R.S. Professor of Nat. Philosoohy in the University of Lon- 
don. J. S. Traill, M.D. W. Yarrell, F.L.S. &c. Ex officio 
members, — The Trustees and Officers of the Association. 

,S'ecre/fm>.s.— Edward Turner, M.D. F.R.S. Sec. G.S. Rev. 
James Yates, F.L.S. G.S. 








I. Mathematics and General Physics. 

Chairman. — Sir D. Brewster, F.R.S. &c. 

Deputy Chairman. — Rev. G. Peacock, F.R.S. 

Secretary. — Professor Forbes. 

Viscount Aclare, F.R.S. Professor Airy. Professor Bab- 
bage. Francis Baily, V.P.R.S. John Barton, F.R.S. Rev. 
J. Bowstead. Sir T. M. Brisbane, F.R.S. Professor Christie. 
Rev, H. Coddington, F.R.S. E. J. Cooper. Dr. Corrie, 
F.R.S. G. DoUond, F.R.S. Lieut. Drummond. Davies 
Gilbert, D.C.L. F.R.S. Rev. R. Greswell, F.R.S. Pro- 
fessor W. R. Hamilton. Hon. C. Harris, F.G.S. G. Harvey, 
F.R.S. Sir John F. W. Herschel, F.R.S. E. Hodgkinson. 
W. Hopkins. John Hymers. Rev. Professor T. Jarratt. 
Rev. Dr. Lardner, F.R.S. Rev. Dr. Lloyd. Professor Lloyd. 
J. W. Lubbock, Treas. R.S. R. Murphy, F.R.S. Phil- 
pott. R. Pottei-, jun. Professor Powell. Professor Quetelet. 
Professor Rigaud. Rev. Dr. Robinson. Rev. R. Walker, 
F.R.S. W. L. Wharton. C. Wheatstone. Rev. W. Whewell, 
F.R.S. Rev. R. Willis, F.R.S. 

II. Chemistry, Mineralogy, Sj-c. 

Chairman.— 3. Dalton, D.C.L. F.R.S. 

Deputy Chairman. — ^Rev. Professor Cumming. 

Secretary. — Professor Miller. 

Professor Daniell. Professor Daubeny. M. Faraday, 
D.C.L. Rev. W. Vernon Harcourt, F.R.S. W. Snow Harris, 
F.R.S. W. Hatfeild, F.G.S. J. F. W. Johnston, A.M. Rev. 
D. Lardner, LL.D. F.R.S. Rev. B. Lloyd, LL.D. T. J. 
Pearsall. Dr. Prout, F.R.S. Professor W. Ritchie. Rev. W. 
Scoresby, F.R.S. W. Sturgeon. Professor Turner. 

III. Geology and Geography. 

Chairman. — G. B. Greenough, F.R.S. Pres. G.S. 

Deputy Chairmen. — Rev. Dr. Buckland, F.R.S. G.S. R.L 
Murchison, F.R.S. V.P.G.S. 

Secretaries.— W. Lonsdale, F.G.S. John Phillips, F.R.S. 

Dr. Boase. James Bryce, jun. F.G.S. Joseph Carn^j 
F.R.S. G.S. Major Clerke, C.B. F.R.S. M. Dufrenoy. Sir 
Philip Malpas de Grey Egerton, F.R.S. G.S. Dr. Fitton, F.R.S. 
G.S. Rev. J. Hailstone, F.R.S. G.S. Professor Harlan. 
G. Mantell, F.R.S. G.S. Lieut. Murphy, R. E. Marquis of 

xl THIRD REPORT — 1833. 

Northampton, F.R.S. G.S. Rev. Professor Sedgwick. Colonel 
Silvertop, F.G.S, W. Smith. John Taylor, F.R.S. Treas. 
G.S. W. C. Trevelyan, F.G.S. H. T. M. Witham, F.G.S. 
Rev. J. Yates, F.G.S. 

IV. Natural History. 

Chairman.— Kev. W. L. P. Garnons, F.L.S. 

Deputy Chairman. — Rev. L. Jenyns, F.L.S. 

Secretaries. — C. C. Babington, F.L.S. D. Don, F.L.S. 

Professor Agardh. G. Bentham, Sec. Hort. Soc. F.L.S. 
J. Blackwall, F.L.S. W. J. Burchell. Professor Burnett. 
W.Christy, F.L.S. Allan Cunningham, F.L.S. J. Curtis,F.L.S. 
E. Forster, F.R.S. Treas. L.S. G. T. Fox, F.L.S. J. £. 
Gray, F.R.S. Rev. Professor Henslow. Rev. Dr. Jermyn. 
Rev. W. Kirby, F.R.S. L.S. Professor Lindley. W. Ogilby, 
F.L.S. Dr. J. C. Prichard, F.R.S. J. F. Royle, F.L.S. 
J. Sabine, F.R.S. L.S. P. J. Selby, F.L.S. J. F. Stephens, 
F.L.S. H.Strickland. Colonel Sykes, F.R.S. L.S. Richard 
Taylor, F.L.S. G.S. W. G. Werscow. J. O. Westwood, 
F.L.S. W. Yarrell, F.L.S. 

V. Anatomy, Medicine, ^c. 

Chairman. — Dr. Haviland. 

Deputy Chairman. — Dr. Clark. 

Secretaries. — Dr. Bond. Mr. Paget. 

Dr. Alderson. S. D. Broughton, F.R.S. W. Clift, F.R.S. 
G.S. Dr. Dugard. H. Earle, F.R.S. Dr. Marshall Hall, 
F.R.S. Dr. Hewett. Dr. Malcavey. Dr. Macartney. Pro- 
fessor Mayo. Dr. Paris, F.R.S. Dr. Prout, F.R.S. Dr. 
Roget, F.R.S. G.S. Dr. Thackeray. Dr. D. Thorp. 

VI. Statistics. 

Chairman. — Professor Babbage. 

Secretary. — J. E. Drinkwater, M.A. 

H. Elphinstone, F.R.S. W. Empson, M.A. Earl Fitz- 
william, F.R.S. H. Hallam, F.R.S. E. Halswell, F.R.S. 
Rev. Professor Jones. Sir C. Lemon, Bart. F.R.S. J. W. 
Lubbock, Treas. R.S. Professor Malthus. Capt. Pringle. 
M. Quetelet. Rev. E. Stanley, F.L.S. G.S. Colonel Sykes, 
F.R.S. F.L.S. G.S. Richard Taylor, F.L.S. G.S. 

[ 1 ] 


Report on the State of Knowledge respecting Mineral Veins. 
By John Taylor, F.R.S., Treasurer of the Geological So- 
ciety and of the British Associatioii for the Advancement of 
Science, Sfc. 8fc. 

X HAVE found it very difficult to execute the task proposed to 
me in a manner satisfactory to myself, as we have at this time 
no digested account of the views entertained by geologists of 
the present day upon this interesting subject. The most per- 
fect treatise is that of Werner, which deserves much attention 
for the observation of facts which it displays; but as it was 
written to propound a theory, and as that theory depended 
upon views of the structure of the crust of the earth which 
modern geology has at least thrown much doubt upon, so his 
work cannot be taken as an outline of our present state of 

Since his time but little has been attempted respecting vein 
formations; and the subject has been, I think, rather neglected 
by geologists, who have advanced other branches of the science 
with extraordinary skill, industry and success. Detached pa- 
pers have, indeed, appeared by English authors, among which 
that on the veins of Cornwall, by Mr. Joseph Carne, holds a 
distinguished place. 

As some proof that the subject of veins has not been much 
attended to, I would remark, that in the Second Series of the 
Transactions of the Geological Society of London, consisting 
now of the first and second volumes complete, and two Parts of 
the third volume, no paper expressly on veins is to be found. 
In the First Series there are two papers, one by the late Mr. 
W. Phillips, giving an outline of facts more generally obsei'ved 
with respect to veins in Cornwall, from observations made 
principally in the year 1800. Another is by Dr. Berger, on 
the physical structure of Devon and Cornwall, from observa- 
tions made in 1809. The writer adopts the Wernerian theory, 
and mentions cases which he thinks confirmatory of its truth. 

In the four volumes of the Transactions of the Royal Geo- 
logical Society of Cornwall, we shall find this subject more 

1833. B 

2 THIRD REPORT — 1833. 

attended to, and there are several communications relating to 
it : among the authors are Dr. Boase, Mr. Carne, Dr. Davey, 
Mr. K. W. Fox, and Mr. John Hawkins. One of the papers 
by Mr. Carne is that to which I have before alluded. 

One of the most recent works by foreign writers is that of 
the late M. Schmidt of Siegen. He was an experienced prac- 
tical miner, and wrote chiefly with a view to his art, describing 
the various derangements in mineral veins, and tracing the best 
rules to be observed in pursuing researches in difiicult circum- 
stances. He adopts the Wernerian theory of formations, and 
refers to the author of it as the great master of the subject. 

Though no general theory has of late been produced in re- 
gular form, yet with the great attention that has been given to 
geology by so many eminent men, an extended field of observa- 
tion has taken place, leading to a very general change of opi- 
nion on most important points ; many conjectures respecting the 
formation of veins have sprung up, and which, when the facts 
are more investigated, and they shall have been recorded and 
classified, may form the groundwork for a more enlarged and 
rational theory, by which their phaenomena and structure may 
be explained, and the causes of their formation, the manner of 
filling up, and the circumstances of the varied derangements 
and dislocations, may be traced and be better understood. 

The subject is of threefold importance : first, as it relates to 
science, wherein a better knowledge of veins generally must 
very materially contribute to sound investigations as to the 
structure of the rocks that inclose them: secondly, as it is much 
owing to the pursuit of the minerals which are deposited in veins 
that we have acquired and may yet extend our knowledge of 
geology in general : thirdly, in relation to the question some- 
times proposed as to the usefulness of geological science, the 
most ready answer may be given, if it be considered that this 
inquiry will relate to subjects of practical utility, in which man- 
kind are universally and largely interested. 

Before I proceed to any account of the opinions as to the 
formation of veins, I would offer some definition descriptive of 
their character and structure, that in proceeding with our sub- 
ject we may clearly understand what is meant to be treated on. 

Werner lays it down, " That veins are particular mineral re- 
positories, of a flat or tabular shape, which in general traverse 
the strata of mountains, and are filled with mineral matter dif- 
fering more or less from the nature of the rocks in which they 

. " Veins cross the strata, and have a direction different from 
theirs. Other mineral repositories, such as particular strata or 


beds, of whatever thickness they occur, have, on the contrary, 
a similar direction with the strata of the rock, and instead of 
crossing, run parallel with them : this forms the characteristic 

Playfair says : " Veins are of various kinds, and may in ge- 
neral be defined, separations in the continuity of a rock, of a 
determinate width, but extending indefinitely in length and 
depth, and filled with mineral substances different from the 
rock itself. The mineral veins, strictly so called, are those filled 
with sparry or crystallized substances, and containing the me- 
tallic ores." 

Mr. Carne says : " By a true vein I understand the mineral 
contents of a vertical or inclined fissure, nearly straight, and 
of indefinite length and depth. These contents are generally, 
but not always, different from the strata or the rocks which the 
vein intersects. True veins have regular walls, and sometimes 
a thin layer of clay between the wall and the vein ; small 
branches are also frequently found to diverge from them on 
both sides." 

Mr. Carne mentions other veins, which he distinguishes from 
the true ones as being shorter, crooked, and irregular in size ; 
he considers these to have formed in a different manner : but 
this will be discussed hereafter. 

These definitions seem to me to be sufficient for our pur- 
pose ; but it may be advantageous here to introduce some 
further description of circumstances connected with veins, and 
to explain the terms usually employed to describe them. 

Being tabular masses, generally of no great width, any one 
will, whether vertical or inclined, present at its intersection 
with the surface a line nearly straight : this may be from north 
to south, or from east to west, or in any intermediate course. 
This is usually called the direction ; by miners frequently the 
run of the vein, or the course of the vein, and is denoted by the 
points of the compass it may cross. 

The length, as Werner states, is indefinite, it being doubtful 
whether any vein has been pursued to a perfect termination. 

The tabular mass, again, may be either vertical to the plane 
of the earth's surface, or may deviate from this position by in- 
clining to one side or the other of the perpendicular. This 
deviation is called the inclination of the vein ; by the Cornish 
miners the underlie. It is measured by the angle made with 
the perpendicular ; and as the dip will be to one side of the 
direction, the latter being known, the other is easily expressed. 

The depth to which veins descend into the earth is unknown, 
as well as the length, and for the same reason. 


4 THIRD REPORT — 1833. 

The only dimension we can ascertain is that across from one 
side to the other of the tabular mass, and is measured from one 
wall to the other, which is the term used in England for the 
cheeks or sides presented by the inclosing rock. This dimen- 
sion is called the width, or frequently the size of the vein. 

The width varies considerably in the same vein. In Europe 
a vein containing ore is considered to be a wide one if it ex- 
ceeds five or six feet. In Mexico the width of veins is gene- 
rally greater. 

In metalliferous veins the deposits of ore are extremely irre- 
gular, forming masses of very diversified form and extent, and 
are separated from each other by intervening masses of vein- 
stone or matrix, either entirely devoid of ore, or more or less 
mixed with it. It is rare to find a vein entirely filled with ore 
in any part. 

In this respect they differ from most beds, where, as in those 
of coal, the whole is a uniform mass. 

The layer of clay, which, as Mr. Carne says, is frequent in 
such veins, will deserve particular notice when we consider 
their general structure and the theories of their formation : 
this is called Saal-bande by the Germans, and flookan by the 
Cornish miners. 

The clearest idea of a vein will be obtained by imagining a 
crack or fissiu'e in the rocks, running in nearly a straight line, 
extending to great and unknown length and depth, and filled 
with various substances. 

I do not intend by this description to convey any theoretic 
opinion as to the manner in which such fissures may have been 
formed, or as to the mode of their being furnished with their 
present contents. These are subjects on which the greatest 
diversity of opinion has existed in former times, and this diver- 
sity is continued to the present period. It is the main business 
of this Report to state these opinions, and to describe our pre- 
sent state of knowledge of this difficult subject. I feel great 
distrust of my power to do it justice; but I am encouraged by 
the idea that a feeble sketch may induce abler hands to pursue 
the design, and throw more and more light upon this interesting 
branch of geology. 

It would be of little use to go into details of the conjectures 
of ancient authors, or into the mysteries with which this sub- 
ject was enveloped in the age of alchemy. 

The earliest writer who is worthy to be consulted is Agricola 
(whose proper name was Bauer) : he resided in the Saxon Erz- 
gebirge, and died in the middle of the sixteenth century. He 
has been called the father of mineralogy, and of the science of 


mining. He had the rare merit of emerging from the mists and 
clouds of an absurd school of philosophy, which had till then 
obscured the objects which it pretended to illustrate ; and he 
first subjected them to inquiries prompted by sound reason and 
just views of nature. 

His writings were numerous, and in such pure Latin that 
they are said to be entitled to a place among the classics. He 
treats of veins in a work called Bermannus, but more particularly 
in the third book of his great work De Re MetaUica. 

Agricola being held to be the first who has written anything 
certain on the formation of veins, and his theory of the manner 
of their being filled up having, with some modifications, been 
for a long period generally received, and in part even adopted 
by Werner, I shall commence from his time the notice of the 
opinions promulgated by various writers antecedent to Werner 
and Hutton. 

Some have maintained. That veins and their branchings are 
to be considered as the branches and twigs of an immense trunk 
which exists in the interior of the globe : 

That from the bowels of the earth metallic particles issued 
forth in the form of vapours and exhalations through the rents, 
in the same manner as sap rises and circulates in vegetables. 

This speculation was proposed by Von Oppel, captain-ge- 
neral of the Saxon mines, who wrote in 1749. He was a skilful 
miner and an accurate observer; and it is singular that this opi- 
nion is not consistent with most that he has elsewhere said on 
the subject, which generally rather agreed with the views which 
were adopted by Werner and others. 

Henkel, who wrote in the early part of the seventeenth cen- 
tury, and who has been held to be the father of mineralogical 
chemistry, first attributed the formation of the contents of veins 
to peculiar exhalations : he supposed the basis of each metal 
and mineral to have existed in the substance of the rock, and 
to have been developed by a peculiar process of nature. 

Becher about the same time supported very similar views. 
Stahl, who commented upon the writings of Becher, had ad- 
vanced a somewhat similar opinion; but he afterwards rejected 
this theory, and considered veins, as well as the substances of 
which they are composed, as having been formed at the same 
time with the earth itself. 

Zimmerman, chief commissioner of mines in Saxony, who 
died in 1747, had an idea that the variety of minerals contained 
in veins had been produced by a transformation of the sub- 
stance of the rock. 

Charpentier, in 1778, supported nearly similar opinions, and 

6 THIRD REPORT — 1833. 

combated strenuously against the theory which considers veins 
to have been rents that were afterwards filled up by different 
mineral substances. 

This is the theory, however, which, from the time of Agricola 
to the present day, has been most generally received, namely, 
that veins were Jissiires which have been since filled up by de- 
grees tvith mineral matters. 

The causes of such fissures, and the mode of their contents 
being deposited, have been variously stated, and have given rise 
to much conjecture; and allowing for these differences, the main 
proposition has been supported by many writers. Among these 
I would name Agricola; Balthazar Rosier, an eminent miner of 
Freyberg, who died in 1673; Hoffman, a commissioner of mines 
at the same place, in 1746; Von Oppel, before mentioned, who, 
though he had indulged in other speculations, distinctly lays 
down in his Introduction to Subterranean Geometry , (Dres- 
den, 1749,) that veins were formerly fissures, open in their su- 
perior part, and that they traverse and intersect the strata. 

Bergman entertained opinions very similar, which were also 
supported by Delius, an author on mining, of considerable ce- 
lebrity, who wrote about 1770. 

Gerhard, in his Essay on the History of the Mineral Ki?igr 
dom, (Berlin, 1781,) gives a collection of interesting facts con- 
cerning veins, and considers them to have originally been rents, 
which were afterwards filled up with mineral substances. 

To this list may be added Lasius, in his Observations on the 
Mountains oftheHartz,in 1787; and Linnaeus is stated "to have 
wondered at the nature of that force which split the rocks into 
those cracks ; and adds, that probably the cause is very familiar, 
— that they were formed moist, and cracked in drying*." 

In England we have testimony to the same opinion from 
Dr. Pryce, who wrote his Mineralogia Cormibiensis in 1778. 
He says, "When solid bodies were separated from fluid, certain 
cracks, chinks and fissures in various directions were formed, 
and as the matter of each stratum became more compact and 
dense by the desertion of moisture, each stratum within itself 
had its fissures likewise, which, for the most part, being in- 
fluenced by peculiar distinct laws, were either perpendicular, 
oblique," &c. 

He afterwards adds, that those very fissures are the wombs 
or receptacles of all metals, and most minerals. He assigns the 
derangements of veins to the effect of fracture by violence, and 
quotes subsidence as one of the probable causes of such dislo- 
cations. He says there can be no doubt that many alterations 

* Hill. 


have happened to various parts of the earth before, at, and 
after the Flood, from inundations, earthquakes, and the dis- 
solvent powers of subterranean fire and ivater, which variety of 
causes and circumstances must infalUbly have produced many 
irregularities in the disposition and situation of circumjacent 
strata and lodes *. 

He describes twelve kinds of lodes or veins in Cornwall, 
naming them from their chief contents. But the most remark- 
able observation of Dr. Pryce is respecting the relative age of 
veins, of which he seems to have given the first intimation. 
Werner, long after, states this as a discovery of his own, and 
as an essential part of his theory. His translator, however, 
(Dr. Anderson,) does Pryce justice, and remarks that his ob- 
servations must have been unknown to Werner, who showed 
much anxiety in all cases to confer on every writer the merit 
which was due to him. 

Dr. Anderson quotes the passage as one of much importance. 

" Because the cross gossans or cross flookans run through 
all veins of opposite directions, without the least interruption 
from them, but, on the contrary, do apparently disjoint and 
dislocate all of them, it seems reasonable to conclude, that the 
east and west veins were antecedent to cross veins, and that 
some great event, long after the Creation, occasioned those 
transverse clefts and openings. But how or when this should 
come to pass, we cannot presume to form any adequate ideaf ." 

Kirwan supports the doctrine that some veins were originally 
open, as appears from the rounded stones and petrifactions 
found in them. Thus, in the granitic mountain of Pangel in 
Silesia there is a vein filled with globular basalt. So also in 
veins of wacken, in Joachimstahl in Bohemia, trees and their 
branches have been found. 

But he deems it improbable that all veins were originally 
open to day, and filled from above. He inclines to the theory 
of veins being filled by the percolation of solutions of the me- 
tals and earths. 

Having now taken a cursory view of the opinions held before 
Werner published his Theory of Veins, and seen something of 
the state of knowledge relating to this subject, we may bear 
in mind the materials which he had to work with, and take 
into account his well-known views as to the origin of rocks from 
aqueous deposition, and we shall comprehend the system which 
he developed, with respect to veins, in the only work, I believe, 
which proceeded from his own hand, and which was published 

* ' Lode ' is the term used in Cornwall for a metalliferous vein. 

+ Miiwialoffia Cornubicnsis, ji. 101. '' 

8 THIRD REPORT — 1838. 

at Freyberg in 1791. Werner adopts, in the first place, the 
proposition that the spaces now occupied by veins were origi- 
nally rents formed in the substance of rocks, and states that 
this is not a new opinion. 

He claims the merit of having ascertained in a more positive 
manner the causes which have produced these rents, and of 
having brought forward better proofs of it than had formerly 
been done. 

He admits that rents may be produced by many different 
causes, but he assigns the greater part to subsidence. He lays 
it down, that when the mass of materials of which the rocks 
were formed by precipitation in the humid way, and which was 
at first soft and moveable, began to sink and dry, fissures must 
of necessity have been formed, chiefly in those places where 
mountain chains and high land existed. He adds, that rents 
and fissures are still forming from time to time in mountains 
which have a close resemblance to those spaces now occupied 
by veins, and that this happens in rainy seasons and from 

He adduces as a proof of his assertions, that veins, in respect 
of their form, situation and position, bear a strong resemblance 
to rents and fissures which are formed in rocks and in the 
earth ; that is to say, both have the same tabular figure, and 
the deviations which they make from their general direction 
are few in number and very inconsiderable ; and he remarks, 
that all the veins of a mining district, more particularly when 
they are of the same formation, have a similar direction, which 
shows them to have been produced by the same general cause. 

But what Werner claimed as altogether new, and what he 
challenges as his own particular discovery is, 

1. To have determined and described in a more particular 
manner the internal structure of veins, as well as the formation 
of the different substances of which they are composed, and to 
have settled the relative age of each. 

2. To have given the most accurate observations and most 
perfect knowledge of the meetings and intersections of veins, 
and to have made these observations subservient to the deter- 
mining their relative ages. 

3. To have determined the different vein formations, parti- 
cularly metalliferous veins, as well as their age. 

4. To have been the first who entertained the idea that the 
spaces which veins occupy were filled by precipitations from 
the solutions, which at the same time formed by other precipi- 
tations the beds of mountains, and to have furnished proofs of 
this : and, 


5. To have determined the essential differences that are 
found between the structure of veins and that of beds. 

Werner illustrates his propositions by many observations, 
which his intimate acquaintance with the extensive mining di- 
stricts in which he was engaged gave him the power of observing 
and recording ; and it must be conceded, at least, that his state- 
ment of facts, and his arrangement of them, give him a manifest 
superiority over most writers upon this subject. Every one 
who has had opportunity to see much of these storehouses of 
nature will be struck with the accuracy of most of his descrip- 
tions, whether they admit the theory by which they are ex- 
plained, or not. 

He allows that the enrichment of veins, or their being filled 
with ores or metals, may have taken place by, 

1. a. A particular filling up from above. 

b. By particular internal canals. 

c. By infiltration across the mass of the vein. 

2. A metallic vein may be increased by the junction of a new 
metalliferous vein, 

3. Though rarely, the richness of a vein may be the effect of 
an elective attraction or affinity of the neighbouring rock. 

The mode assigned by Werner for the formation of the 
spaces now occupied by veins is still further demonstrated, in 
his opinion, by the relation which veins have to one another ; as, 
Their intersecting one another. 
Their shifting one another. 
Their splitting one another into branches. 
Their joining and accompanying one another. 
Their cutting oft" one another. 
All these peculiarities, he remarks, are produced by the ef- 
fects of a new fissure upon one that is older. 

Subsidence having been the cause of fissures he thinks is 
proved by the difference in the level in the parts of the same 
stratum or bed in which a vein is inclosed ; and this throwing 
up or down, as the miners term it, bears a proportion to the 
size of the vein. 

The interior structure of many veins is quoted to show that 
the fissures had been originally open, and which had been af- 
terwards filled by degrees. 

Such veins are composed of beds, arranged in a direction pa- 
rallel to their sides ; their crystallizations are supposed to show 
these beds to have been deposited successively on each other, 
and that those next the walls have been first formed. A cir- 
cumstance much relied on, also, is the existence of rolled masses 
or water-borne stones, fragments of the adjacent rock, some- 

10 THIRD REPORT — 1833. 

times forming a breccia, remains or impressions of organic bo- 
dies, coal and rock salt substances of recent formation, and other 
matters, which should appear to have come in from above. 

This theory obtained considerable attention, and was very 
generally adopted from the time of its being made known ; and 
it has, I believe, many adherents at this day, particularly among 
miners or those who have much opportunity of actual observa- 

Hutton's Theory of the Earth was published afterwards, in 
1795; and as his views regarding the operations employed in 
the formation of the structure of the rocks differed entirely 
from those who assigned to them an aqueous origin, so it will 
readily be supposed that he would promulgate a new explana- 
tion of the formation of veins. 

According to Playfair, this theory embraced the following 
propositions : — 

It allowed that veins are of a formation subsequent to the 
hardening and consolidation of the strata which they traverse, 
and that the crystallized and sparry structure of the substances 
contained in them shows that these substances must have con- 
creted from a fluid state. 

It assumes that this fluidity was simple like that of fusion by 
heat, and not compound like that of solution in a menstruum. 

It is inferred that this is so from the acknowledged insolu- 
bility of the substances that fill the veins in any one menstruum, 
and from the total disappearance of the solvent, if there was 
any, it being argued that nothing but heat could have escaped 
from the cavities. 

It is further maintained, that as the metals generally appear 
in veins in the form of sulphurets, the combination to which 
their composition is owing could only have taken place by the 
action of heat. And, furthei", that metals being also found na- 
tive, to suppose that they could have been precipitated pure 
and uncombined from any menstruum, is to trespass against all 
analogy, and to maintain a physical impossibility. 

It is therefore inferred, that the materials which fill the mi- 
neral veins were melted by heat, and forcibly injected in that 
state into the clefts and fissures of the strata. 

The fissures must have ai'isen, not merely from the shrinking 
of the strata while they acquired hardness and solidity, but 
from the violence done to them when they were heaved up and 
elevated in the manner which the theory has laid down. 

Slips or heaves of veins, and of the strata inclosing them, 
are to be explained from the same violence which has been 


It is admitted as interesting to remark, that in the midst of 
the signs of disturbance which prevail in the bowels of the 
earth, there reigns a certain symmetry and order, which indi- 
cates a force of incredible magnitude, but slow and gradual in 
its eftects. 

Further, that as a long period was required for the elevation 
of the strata, the rents made in them are not all of the same 
date, nor the veins all of the same formation. A vein that forces 
the other out of its place, and preserves its own direction, is 
evidently the more recent of the two. 

The parallel coats lining the walls or sides of the vein, which 
are attributed by Werner and others to aqueous deposition, are 
ascribed to successive injections of melted matter. 

Veins have been considered as traversing only the stratified 
parts of the globe. They do, however, occasionally intersect 
the unstratified parts, particularly the granite ; the same vein 
often continuing its course across rocks of both kinds without 
suffering material change. 

It is asserted that all the countries most remarkable for their 
mines are primary, and that Derbyshire is the most considera- 
ble exception to this rule that is known. 

This preference which the metals appear to give to the pri- 
mary strata, is considered as consistent with Dr. Hutton's 
theory ; and particularly as these strata, being the lowest, have 
also the most direct communication with those regions from 
which the mineral veins derive all their riches. 

In arguing further upon this theory, it is assumed that no- 
thing of the substances which fill the veins is to be found any- 
where at the surface ; and that, contrary to the allegation of 
some that mineral veins are less rich as they go further down, 
it is stated that this is not generally so, and that the mines in 
Derbyshire and Cornwall are richest in depth, as they would 
be if filled with melted matter from below. 

Again, it is said that if veins were filled from above, and by 
water, the materials ought to be disposed in horizontal layers 
across the vein; and that this opinion is sufficiently refuted by 
the fact that rarely any metallic ore is found out of the vein, or 
in the rock on either side of it, and least of all where the vein 
is richest. 

The foregoing seem to be the most important allegations in 
support of the Huttonian theory ; and I have taken them nearly 
in the order in which they are given in Professor Playfair's il- 
lustrations of this celebrated system. 

There is yet another doctrine regarding the formation of 
veins, which, though it is not of modern date, and has had but 


few supporters among writers upon the subject, has yet claims to 
be considered, and particularly as it has of late been urged upon 
our notice, and by some whose observations have been made in 
districts where veins of vai'ious order are abundant. 

This theory is, in short. That veins were formed at the same 
time with the rocks themselves ; that the whole was a contem- 
poraneous creation ; and that there have been neither fissures 
subsequent to the consolidation of the mass, nor filling up from 
above or below, or disturbances to produce the heaves or shifts 
which we see. 

When this hypothesis was first proposed I do not know, but 
that it was long since we may infer, as Agricola regards the 
opinion which supposes veins such as we now see them to have 
been formed at the same time with our globe, to be at variance 
with fact, and he calls it the opinion of the vulgar. The same 
hypothesis was indeed supported by Stahl ; but he seems to 
have adopted it rather on account of the difficulties attendant 
on any other explanation that had been proposed, than for any 
good reason that he had to give. 

Such are, however, but assertions, to be received with doubt 
by any one who inquires freely and without prejudice. Partial 
evidence may appear for some such formations ; but it is another 
affair to attribute all veins to such an origin, and thus to sweep 
away at once the difficulty of explaining many complicated ap- 

The doctrine of a contemporaneous formation of veins has 
lately found an advocate in Dr. Boase, in his paper on the 
geology of Cornwall. After commenting on the division into 
different orders, which Mr. Carne had indicated as to veins, 
according to certain appearances in their direction and the 
character of the substances with which they are filled, he says 
he cannot detect any characters which are not common to all 
the Cornish veins ; and since some of them are generally ac- 
knowledged to be contemporaneous with the rock, he concludes 
that they have all the same origin. 

Dr. Boase, however, candidly sets out by stating that he had 
purposely refrained from making inquiries at the mines con- 
cerning the phasnomena of veins, and that bis experience is 
therefore principally confined to jhose which occur in cliffs, 
quarries, and natural sections that are exposed to open view. 

Lest this admission should create surprise, he remarks that 
such sources of information are invaluable as the only ones 
easily available to exercise the senses on the nature of veins ; 
for, unless to those much accustomed to descend into mines, 
they may as well be visited blindfold. 


He remarks, however, as to the veins of Cornwall, that their 
great irregularity in size and in form, their fi'equent ramifica- 
tions, their similarity of composition and intimate connexion 
with the rocks which they traverse, and, above all, the large 
masses of slate which they envelop, are all circumstances to 
disprove their origin from fissures, and to support their con- 
temporaneous origin. 

Dr. Boase suggests that veins follow the arrangement of the 
joints of the rocks, and that it may thus be explained why the 
different series of veins cross each other, and why the veins of 
each series are respectively parallel. 

And he thinks that thus we may suppose how veins which 
are crossed may seem to abut or terminate against those that 
are opposed thereto ; having, when in the same line, that pecu- 
liar appearance that has been attributed to intersection, and 
the appearance of being heaved when on the opposite sides of 
the cross vein, they are not on the same line, but occur in the 
parallel joints of distant layers. 

The latter occurrence, he remarks, although very common, 
is not however universal ; for, in some instances, the part of the 
vein supposed to have been intersected has never been found. 

As Mr. Carne had observed, that when contemporaneous 
veins meet each other in a ci'oss direction, they do not exhibit 
the heaves and interruptions of true veins, bvit usually unite. 

Dr. Boase says that this statement is opposed to his obser- 
vations, and that the phaenomenon of intersection is common to 
all kinds of veins. Further, he expresses a doubt whether 
heaves in veins are not after all rather apparent than real, but 
explains that he does not mean to assert that they do not ex- 
hibit these phaenomena, but that this arrangement, as in the 
case of small veins, only gives the appearance of being moved 
from the original positions. 

I have now stated the opinions which, as far as I know, have 
been generally received on the subject of the formation of veins, 
from which it wiU appear that there are three leading hypo- 

1st. That which supposes them to have been open fissures, 
caused by disruption, and occasioned principally by subsidence 
of parts of the rocks, which fissures were afterwards filled up 
with various matters by deposits from aqueous solution, chiefly 
from above. 

Modifications of this theory are. That such rents in the earth 
may have been caused in other ways, such as earthquakes, or 
certain great convulsions, as well as by subsidence : 

That they may have been filled by the infiltration of sola- 

i'4 THIRD REPORT 1833. 

tions, which deposited the substances with which they were 
charged in the veins, or by the process of subhmation from 

The second theory allows that veins were formed subse- 
quently to the consolidation of the rocks ; but the cause prin- 
cipally assigned for such fissures is the violence done to the 
strata by the elevation or upheaving of other rocks from 

And it is an essential part of this theory that the materials 
which fill the veins were forcibly injected upwards in a state of 
complete fusion by heat. 

The third theory is that denying any subsequent processes 
which might either cause rents and fissures, or might fill them 
with matter which differs from the rocks which inclose them : 
the whole formation was contemporaneous with the rocks them- 
selves, the mineral substances which we find in veins having 
separated and arranged themselves into the forms in which we 
now see them to exist. 

The advocates of these theories have each zealously asserted 
the truth of his own system, and refused to admit of causes or 
explanations which appeared to militate against it ; and thus a 
boundary has been set, as it appears to me, to that freedom of 
inquiry which is so desirable in such cases, and a limit drawn 
round the reasoning faculties of man upon evidence which may 
come before him. 

It will appear, from what has already been said, that veins 
have very different characters and appearances ; and this might 
be made more clear, if it were here the proper place to enlarge 
upon the subject and point out the distinctions. For our pur- 
pose, however, it may be sufficient to remark upon two or three 
principal varieties. First, then, are those which have beyond 
all comparison been most explored and examined, on account 
of the stores which they contain, — the metalliferous veins. As 
these have been penetrated in all directions to the greatest ex- 
tent that human power and ingenuity have been able to effect, 
so their structure is better known and more accurately ob- 

Similar to these, and occurring with them, and therefore well 
known, are others, which, though baiTen of metals, are yet 
often called true veins ; and these, as well as the first, come 
pretty fully under the view of the miner. 

Next there are veins, regular in their structure to a great 
extent, filled with matter which has the character of being de- 
rived from igneous origin, such as are usually called dykes of 
trap, whinstone, &c., &c. ; to which would be added by most 


geologists of the present day, the veins of granite, porphyry, 
quartz, &c. 

Some of these have been examined below the svirface, where 
they pass through coal-fields, or other deposits of useful mine- 
rals, but containing in themselves nothing to reward the toU of 
exploring them : little has been seen of their contents and con- 
figuration, and our knowledge of them is more limited. 

Lastly, there are tortuous and irregular veins or ramifications 
in most rocks, extending to limited distances, as far as our ob- 
servations permit us to judge, seldom offering a valuable return 
for any effort to explore them, and of which, therefore, our 
knowledge is but superficial. 

Such veins, according to Mr. Carne, have been usually di- 
stinguished from true veins by their shortness, crookedness, 
and irregularity of size, as well as by the similarity of the con- 
stituent parts of the substances which they contain to those of 
the adjoining rocks, with which they are generally so closely 
connected as to appear a part of the same mass. Two other 
distinctive marks may be added ; one is, that when they cross 
they do not exhibit the heaves of true veins, but usually unite ; 
the other is, that when there is an apparent heave it is easy to 
perceive that what appear to be separate parts of the same vein 
are different veins terminating at the cross vein. 

Such may be, probably, of contemporaneous formation ; and 
there may be deposits of ore also which it would be difficult to 
refer the structure of to any other hypothesis, particulai-ly such 
as contain ores so intimately mixed with the rocks as to form a 
constituent part of them. 

I would suggest, that if from any one of these classes we were 
to form a judgement as to the whole, error would probably be 
the consequence, or, at any rate, the view would be a narrow 
and contracted one, and our decisions would be defective in 
many important respects. 

To have conducted the inquiry in this manner seems to me 
to have been the error in many who have preceded us in for- 
warding the state of knowledge on vein formations. Nor do I 
mean to detract from the great merit of many of them on this 
accovmt ; the field of observation is too vast to become fully 
acquainted with it ; it extends over the most rugged parts of 
the earth's surface, and its boundaries are not reached in the 
deep recesses of its bowels. It is no wonder that in the earlier 
stages of such inquiries men should be strongly impressed with 
what lay immediately before them, and should view with dis- 
trust what they might only learn from description. 

Such impressions may be traced in looking at the authors of 

16 THIRD REPORT — 1833. 

the systems which we have reviewed. Werner expressly tells 
us, that we are indebted to miners for the theories which he 
deemed most worthy of acceptation, and he names as such 
Agricola, Rosier, Henkel, HoiFman, Von Oppel, Charpentier, 
and Trebra. We may add his own name and that of Dr. Pryce, 
in our own country, as intimately acquainted with mining. Now 
all such men would be more acquainted with the metalliferous 
veins and such as accompany them; and from these they would 
derive much evidence in favour of the opinions which they ad- 
vocated ; at least, partaking, as I probably do, in the same pre- 
judices, so it would appear to me, if by the labour of other 
inquirers I did not know that there were other facts requiring 
a different explanation. 

Again, Dr. Hutton and his commentators had largely ob- 
served veins which may fairly be attributed to injection; they 
had found dykes of trap passing through coal-beds, and con- 
verting them into cinder. Such evidence of the effects of heat 
and of a filling up by matter in fusion is not to be resisted ; but 
when we look at what is said of the metalliferous veins by some 
of the writers on this side of the question, we observe great 
want of practical knowledge and many errors, arising out of 
the attexnpt to make all bend to a single method of solving the 

For the third hypothesis of contemporaneous formation there 
is this to be said, — that some veins exist which seem to admit of 
no other explanation ; and that this being allowed to such as 
will have but one theory, this is at once the easiest, because it 
gets rid of many difficulties without further trovible ; but we 
can hardly be satisfied to adopt it as universal upon experience 
that has been principally confined to sections in quarries and 
in cliffs, or to such as are exposed to open view. 

Our present state of knowledge as to the formation of veins 
should therefore, in my opinion, be allowed to admit that most 
of the causes which have been stated have operated at various 
periods and through a long succession of time, some prevailing 
at one epoch, and some at another, modified by circumstances 
which we can but imperfectly comprehend or explain. 

In this view we may allow of a classification of veins accord- 
ing to their probable mode of origin ; and such a classification 
has been thought of by some of our ablest geologists of the 
present day, and was indeed propounded in one of our sections 
at Oxford last year by our present learned President, who ex- 
pressed his opinion that there were three different sets of veins : 
— 1. Those which have been plainly mere fissures or cracks, 
and which have been subsequently filled ; 2. Those of injec- 


tion ; 3. The contemporaneous veins, which might more aptly 
be termed veins of segregation. 

Here I might close this Report, which is already much too 
tedious, were it not that I may be expected to notice briefly 
some of the facts adduced by the advocates of the respective 
theories, and, by comparing them, show how far they are enti- 
tled to be considered as objections on one side, or as proofs 
on the other, with the confidence which has been assigned to 

Werner and Hutton agree in allowing that rents took place 
subsequently to the consolidation of the rocks, or at the time of 
their consolidation. They differ as to the cause of the rents : 
Werner ascribes it to subsidence, or to sinking and shrinking 
of the solid materials of our globe ; Hutton, to violent upheaving 
of matter from below, breaking up the superinjacent strata. 

Either of these causes seems adequate to the effect, and in 
either case corresponding strata might be found having different 
levels of position on opposite sides of the fissure, as is constantly 
the case. This by miners in the North of England is called the 
throw of the vein ; and it is clear that one side may as well be 
thrown up as the other thrown down. Mr. Fox and Dr. Boase 
urge the great irregularity of the width of veins, the difficulty 
of supposing the sides to be supported, and some other objec- 
tions to the hypothesis of open fissures. Irregularity of width 
is but a comparative term ; and taking into consideration the 
immense extent of their dimensions in length and depth, it 
amounts in my opinion to but little. 

The other objections are in a great degree anticipated and 
answered by Werner ; and, after all, difficulties can hardly be 
urged against the positive testimony of some veins having been 
open, which is afforded by the substances found in them, such 
as rolled pebbles, petrifactions, &c. 

The parallelism of veins of one formation is insisted upon by 
Werner as a proof of his view of the subject ; and I confess that 
there appears to me to be considerable difficulty in explaining 
this, on the supposition that fissures were caused by a mass 
protruded upwards through strata already foi'med. From such 
a cause one should expect not to have a number of cracks pa- 
rallel to each other, but rather to see them radiating from the 
centre of the greatest disturbance. In the metalhferous veins 
we may certainly observe this parallelism to a great extent. 
Mr. Carne has beautifully illustrated this in Cornwall, and has 
shown how the productive veins generally have an east and 
west course ; how, as they differ in their contents, they diff'er 
also in their direction, each class being, however, parallel in 

18^3. c 

18 THIRD REPORT — 1833. 

itself; ami how tliese facts illustrate relative ages of forma- 

This tendency to an east and west direction of the metallife- 
rous veins may be observed not only in Cornwall but in the 
stratified parts of England, in the mining districts of Europe, 
and in the range of the great veins of Mexico. 

Mr. Robert Fox, having discovered galvanic action to ensue 
by the connexion of an apparatus, constructed to detect it, with 
portions of metalHferous veins, suggests whether some analogies 
may not be traced between electro-magnetic ciu'rents and the 
directions of veins : nothing upon which any hypothesis can be 
built seems, however, as yet to have been proposed ; and it may 
be doubted whether, when this test is apphed to masses of ore, 
the experiment is not liable to many objections. A principal 
one seems to be, that by the very act by which we gain access 
to the vein, we lay it open to atmospheric action, and conse- 
quently to decomposition. Chemical agency commences, and 
with it, very naturally, galvanic influences are excited. 

Veins containing ores little subject to decomposition have, I 
apprehend, been found to give httle or no indications of this 

It may, however, be that this general direction of metallife- 
rous veins may not obtain as to veins of injection ; and in that 
case we shall have additional reason to admit more causes than 
one to have been in operation. This is a matter deserving ex- 
tensive observation. 

Other veins have been stated to cross the metalliferous veins : 
they are generally filled in a different manner. If they contain 
any ores, they are frequently of difterent metals from those in 
the former. They pass through or traverse the other veins, 
cutting them through, and sviffering a disturbance to take place 
in their linear direction, or what the miners significantly term 
a heave. 

This fact is relied upon as proving that veins are of different 
ages, as first asserted by Pryce, much insisted upon by Werner, 
and allowed by Hutton and Playfjiir. 

Those who dispute this inference, therefore, are the advo- 
cates for the sole operation of contemporaneous causes : they 
object that rules which have been proposed for ascertaining 
the exact tendency of such disturbances having been found to 
be subject to exceptions, tlie proof of dislocation is wanting, or 
that dislocation has taken place without motion. The latter 
proposition, at any rate, appears to me to be very difficult to 
understand ; and I think if any part of this intricate subject is 
clear and intelligible, it is that the relative age of veins is made 


out by these fiicts, even although we may not yet be able to 
apply rules for every case, — a subject which has been con- 
sidei'ed as highly important in its practical application to the 
art of mining. 

The greatest controversy, however, relates to the mode in 
which veins have been filled. Here, again, we must remark, 
how the opinions of observers have been influenced by the facts 
cominff under their immediate observation. 

Werner, and the mining authors on whom he relies, drew 
their inferences from metalliferous veins. Hutton and his fol- 
lowers regarded chiefly those of another class ; and this great 
author and his commentator Professor Playfair were evidently 
ill informed as to metalliferous veins. 

That certain veins have been filled by injection from below, 
and with matter in igneous fusion, seems to be rendered certain 
by evidence, which is clearer than most we possess on such sub- 
jects, and must be admitted at once. Thus, when we see a 
trap dyke traversing a bed of coal and charring the combusti- 
ble matter, and afl^ecting the rock itself with visible efifects of 
great heat, we must assent to the cause assigned ; and when we 
see matter of igneous origin not only filling the veins, but over- 
flowing on the surface, or insinuating itself between adjacent 
beds, the case is plainer than most that occur in geological re- 

But though one class of theorists have proposed this as the 
universal cause of the filling up of veins, ought we to admit this 
to be true, when we find so many in which no similar appear- 
ances are to be traced? 

Why, for instance, if the ores were forced from below, did 
the power which injected them just limit itself to raising them 
within a short distance of the surface,— for where shall we find 
an instance of their being protruded above it ? 

If the metallic contents of veins were injected from below, we 
ought to be able to trace something like the direction of the 
currents in which the matter flowed ; we ought to see some 
continuity in the operation, and some connexion between the 
masses of ore which occur in veins ; whereas the contrai'y of 
each is notoriously evident to every observer. 

It would seem also to be very probable, if the enrichment was 
from below, and the matter was forced in from those regions 
whence their treasures are supposed to be derived, that by a 
nearer approach to the depths of the earth we should find the 
riches more abundant. 

Professor Playfair admits this inference, and disposes of the 
difficulty by arguing that it is so ; and says, that though mines 


so THIRD REPORT 1833. 

in Mexico and Peru are said to be less rich as they descend 
further, those of Derbyshire and Cornwall exhibit the very 

He is unfortunate in this allegation, and the facts will not 
bear him out, as every one of common experience must know ; 
and thus, as I have before observed, we have hypotheses sup- 
ported by a limited knowledge of the facts. 

The theory of the filling up of veins by precipitation from 
aqueous solutions, is defective in not being able to show what 
menstruum could render such substances soluble in water ; 
and this difficulty must remain an important one, unless en- 
larged knowledge should hereafter afford the means of ex- 
plaining it. 

But when we are told that the supposition is absurd, that 
water cannot arrange its deposits in planes highly inclined, that 
no appearance of stalactites is to be found in veins, nor can 
we see in them any substance like those on the earth's surface, 
which aqueous action has removed, — it must be recollected that 
we know silex is soluble in water at high temperatures ; that 
crystals do arrange themselves on the sides of vessels in planes 
highly inclined; that stalactites of chalcedony, of quartz, and of 
iron pyrites, have been found deep in the veins in Cornwall, and 
that much of the substance of the surrounding rocks, and such 
as we see on the surface, and adjoining and inclosing the veins 
themselves, is found in them, occupying much of their space, 
previously having been worn down into fragments, into loose 
sand, and into clay or mud, the latter of which is so common 
that, as I have before observed, it is relied on by the miner as 
a distinguishing character of regular veins*. 

The action of water may, I think, be as fairly assumed as 
that of fire ; and we may consider what their joint powers might 
be, when compelled, as it were, to act together, under circum- 
stances that immense pressure might produce. 

But in examining the contents of veins, we are, I think, likely 
to be struck, not only by the appearance of a complication of 
causes, but by evidence of their succession, admitting the pro- 
bability not only of different agents having been employed, but 
of their having done their work separately as well as conjointly, 

• Mr. Weavei' describes the contents of the great vein of Bolanos in Mexico 
thus : " The chief mass of this vein may be said to consist of the detritus of the 
adjacent rocks, more or less consolidated, and generally hard ; nay, in places, 
it is actually composed of a conglomerate. Proper vein-stones, such as fluov 
or calc spar, are, comparatively speaking, casualties. In this basis the finer 
delicate silver ores and native silver are dispersed, in common with the harder 
and coarser ores of blende, iron, and copper, besides lead ores." 


—of having operated at different periods, and of one having 
produced effects for which another was inadequate. 

As we cannot easily conceive how the metahic ores can have 
been deposited from solution in water, and appearances are 
much against their having been injected in a state of fusion, 
there is another supposition which, though not free from diffi- 
culties, has yet probability enough in its favour to have gained 
it many supporters, — which is, that these and some other sub- 
stances have been raised from below by sublimation. This is 
not a new opinion, for though the older writers expressed it in 
an indistinct manner, and spoke of metallic vapours and exha- 
lations, — and thus we shall find it proposed by Becher, Stahl, 
Henkel, and others, — yet their meaning evidently was, that sub- 
stances had been volatilized by heat, and assumed their places 
in veins by condensation, or by combining with other materials. 

We know for certain that some of the metallic sulphurets 
may be so volatilized, and will reassume their form and be 
produced in a crystallized state ; and so far nothing is assumed 
beyond our knowledge : but as we find these sulphurets, which 
compose by far the greater part of the metdllic contents of veins, 
in insulated masses, surrounded on all sides by other substances, 
which we can hardly conjecture to have been sublimed, we en- 
counter much difficulty in explaining how the process can have 
taken place ; and it becomes even more difficult when we see 
how very much these different classes of substances are incor- 
porated, and how they completely, in most instances, envelop 
and inclose each other. 

The hypothesis of filling up by sublimation would also seem 
to require that the deepest portions of veins should be richer, 
especially considering the very small extent to which after all 
they have been perforated ; but yet, shallow as our workings 
into the earth really have been, there is much appearance of 
their having in many instances gone below the richest deposits 
of the metals. 

This seems to have been the case in some of the deepest 
mines in Mexico, and in several in our own country. It is im- 
possible, indeed, to say that greater deposits may not exist still 
lower down ; and though veins have not been traced to their 
termination, they have in many instances been pursued until 
the indications of metallic produce have become faint and hope- 
less. And these unfavourable appearances have increased very 
commonly with increasing depth, which is as much, perhaps, as 
we are likely to know about it, as the operations of the miner 
are thus arrested, and the inducement to further experiment is 
taken away. 

22 THIRD REPORT — 1833. 

The agency of sublimation has lately been advocated by Pro- 
fessor Necker of Geneva, in a paper read before the Geological 
Society of London * ; and he has extended an ingenious hypo- 
thesis of Dr. Bovi^, who would bring under a general law the 
relation of metalliferous veins and deposits to those crystalline 
rocks which, by the majority of modern geologists, are consi- 
dered to have been produced by fire ; and thus to lead to the 
inference that the metals were deposited in the former by sub- 
limation from the latter. 

M. Necker inquires, 1. Whether there is near each of the 
known metalliferous deposits any unstratified rock 'i 

2. If none is to be found in the immediate vicinity, is there 
no evidence which would lead to the belief that an unstratified 
rock may extend under the metalliferous district ? 

3. Do there exist metalliferous deposits entirely disconnected 
from unstratified rocks ? 

Professor Necker answers these questions by showing that in 
various countries there are such relations as he supposes, and 
admits, in reply to the last, that there are cases where the depo- 
sits seem to be unconnected with any trace of unstratified rock. 

If metalliferous deposits are commonly in crystalline rocks 
which are attributed to igneous origin, it must be allowed also 
that there are others abundantly rich where no apparent con- 
nexion is to be traced. M. Necker mentions the mountain 
limestone as such ; but he does not seem aware of the extent 
of those deposits, which, with the beds of grit and shale which 
alternate with it, present numberless regular veins abounding 
with certain ores. 

As this fact is indisputable, it seems necessary to show not 
only that unstratified rocks may be under them — which there is 
little doubt about, — but that there should be some connexion 
between the veins which contain the metals and similar chan- 
nels or passages in the rocks below. No such evidence, I be- 
lieve, at present exists ; and I am not aware of any veins having 
yet been found to penetrate from the stratified rocks into those 
upon which they rest. 

This supposition must therefore, like many others, be taken 
as a mere probability to account for some appearances in certain 
places, but not to explain all the phaenomena. 

There is one point which, before I conclude, I would endea- 
vour to press on the attention and consideration of future ob- 
servers, because, in the first place, it does not appear to have 
been much regarded by writers on the subject ; and next, be- 

« March 28th, 1832. 


cause, though it seems to offer objections to some received 
theories, it may, when better understood, assist in developing 
the truth. 

This is the relation that the contents of a vein bear to the 
nature of the rock in which the fissure is situated. 

Thus in the older rocks, we see the same vein intersecting 
clay-slate and granite: it is itself continuous, and there is no 
doubt of its identity ; and yet the contents of the part inclosed 
by the one rock shall differ very much from what is found in 
the other. In Cornwall, a vein that has been productive of 
copper ore in the clay-slate, passing into the granite becomes 
richer, or, what is more remarkable, furnishes ores of the same 
metal differently mineralized. If we pursue it further into the 
granite, the produce of metal frequently is found to diminish. 

Veins in some cases cut through the elvan courses, as well as 
the clay-slate inclosing these porphyries : the ores are rich and 
abundant in the latter ; in other instances they fail altogether. 

Less striking differences in the structure of the rock seem to 
aff'ect the contents of the veins ; and appearances as to the tex- 
ture and formation of the strata are often regarded by miners 
with mox'e anxiety ihan the indications presented by the vein 
itself; and a change of ground is relied upon with an assu- 
rance, derived from experience, as a more certain basis to au- 
gur upon, for better or for worse, than almost any other which 
the difficult art of mining has to offer. 

Numberless facts might be collected and adduced to show 
that this is not mere speculation ; but it will nowhere appear 
more clearly than if we examine the various beds of limestone 
grit, &c., in the great lead mines in the North of England. 

Here we shall find a series of stratified rocks, and that por- 
tion of the series which has been most productive of lead ores, 
occupying a thickness of nearly 280 yards. It is divided into 
55 distinct beds, which are accurately described in Mr. West- 
garth Forster's section, each having its name known to the 
miners of the country. Nine of these beds are of limestone, 
about 18 are of gritstone or siliceous sandstone, and the re- 
mainder are plate or black shale, with thin beds of imperfect 

Now the lead veins pass through all these beds, and have 
been worked more or less into all of them; and it has thus been 
proved, that though the fissure is common to all, yet lead ore is 
only found abundantly in particular beds, and those very much 
the same, if we examine the immense number of mines which 
are working in this district. 

Where the veins pass through the shale, little or no ore is to 

24 THIRD REPORT — 1833. 

be found in them ; where they are inclosed by the gritstones, 
there they become more productive ; but it is in one of the beds 
of limestone, and one only, that the great deposit of lead ore is 
to be found. 

In the great mining field of Alstone Moor, this bed is called 
the great limestone, and yet its thickness is only about 23 
yards out of the 280 which the series of lead measures occupy; 
and notwithstanding this, four-fifths of all the lead ore found 
in the district is derived from such parts of the veins as are 
inclosed by this particular stratum. 

The veins equally passing through the other beds, and traced 
by innumerable workings through them, are yet only rich in 
metallic treasure where they repose in this favoured stratum *. 

Though perhaps few cases are so striking as this, yet it is 
evident that the same thing takes place to a certain extent 
with all the metals, in all rocks and in all countries. 

If it is a fact and correctly stated, it must be considered in 
reference to the theories propounded to us, and it seems directly 
opposed to the doctrine of forcible injection; but it may admit 
of probable explanation by calling in certain affinities, either by 
the advocates of precipitation from water, or by those who may 
contend that sublimed vapours might be attracted to particular 

* To illustrate the comparative bearing of the different beds in the manor of 
Alstone Moor, Mr. Thomas Dickinson, the Moor Master for Greenwich Hospi- 
tal, extracted for me an exact account of the ore produced from each bed in all 
the mines of the manor in the year 1822, which gave the following results: — 

Limestone Beds. — Great limestone 20,827 bings. 

Little limestone 287 

Four-fathom limestone 91 

Scar limeston e 90 

Tyne bottom limestone 393 

■ 21,688 

Gritstone Beds. — High slate sill 107 

Lower slate sill 289 

Firestone 262 

Pattinson's sill 259 

High coal sill 327 

Low coal sill 154 

Tuft 306 

Quarry hazel 44 

Nattrass Gill hazel 21 

, Six-fathom hazel 576 

Slaty hazel 18 

Hazel imder scar limestone 2 


Whole produce of the mines of the manor 24,053 bings. 


That metallic ores are found to repose in rocks which seem 
congenial to them, and that their combinations are modified by 
changes in the rocks, will not I think be disputed by practised 
miners, or by those who have most narrowly searched into the 
hidden recesses of the earth. 

Facts must be observed and compared, effects must be traced 
to probable causes, and difficulties must be explained or can- 
didly admitted, if we would enlarge and generalize our know- 
ledge of vein formations. There are obstacles to the progress 
of this knowledge ; for, as Dr. Boase has remarked, it is not easy 
for a person unaccustomed to it to use his eyes with much ad- 
vantage, in the places where the study can best be pursued. 

It is the miners' business, however, not only to see clearly, but 
to consider all the inti-icate appearances that veins exhibit ; and 
I would exhort them not to be satisfied merely with the obser- 
vations their art may seem to require, but to extend them to a 
larger view of the subject, and to contribute, as many of their 
eminent predecessors have done, to the common stock of general 

If the imperfect view which I have thus endeavoured to give 
of prevalent opinions should assist in such endeavours, or 
should stimulate any persons in undertaking a further pursuit 
of the subject, it would be to me a source of great gratification; 
as the desire of promoting such inquiries must be my apology 
for attempting the task which I have undertaken. 

[ ~^7 ] 

On the Principal Questions at present debated in the Pliiloso- 
j)hy of Botany. By John Lindley, Ph.D., F.R.S., 8fc., 
Professor of Botany in the University of London. 

If we compare the state of Botany at the end of the last cen- 
tury with its present condition, we shall find that it has become 
so changed as scarcely to be recognised for the same science. 
Improvements in the construction of the microscope, the disco- 
veries in vegetable chemistry, the exchange of artificial methods 
of arrangement for an extended and universal contemplation of 
natural affinities, the reduction of all classes of phagnomena to 
general principles, and, above all things, the adoption of the 
philosophical views of Gothe, together with the recognition of 
an universal unity of design throughout the vegetable world, 
are undoubtedly the principal causes to which this change is 
to be ascribed. 

As the general nature of recent discoveries, and a sufficient- 
ly definite notion of the present state of botanical science, may 
be collected from the introductory works which have appeared 
in this country within the last three years, it is presumed that 
the object of the British Association will be attained if the 
present Report is confined to the most interesting only of those 
subjects upon which botanists have been recently occupied, 
and to an indication of the points to which it is more particularly 
desirable that inquiries should now be directed. I have also 
excluded everything that relates to mere systematic botany, in 
the hope that some one will take that subject as the basis of a 
separate Report. 

Elementary Organs. — This country has, till lately, been re- 
markably barren of discoveries in vegetable anatomy, since the 
time of Grew, who was one of the fathers of that branch of 
science. Whatever progress has been made in the determina- 
tion of the exact nature of those minute organs, by the united 
powers of which the functions of vegetation are sustained, it 
has been chiefly in foreign countries that it has taken place : 
the names of Mirbel, Moldenhauer, Kieser, Link and Amici, 
stand alone during the period when their works were published ; 
and it has only been within a very few years that those of 
Brown, Valentine, Griffith and Slack have entered into com- 
petition with the anatomists of Germany and France. 

By the researches of these and other patient inquirers, mc 

28 THIRD REPORT — 1833. 

have already reduced our knowledge of the exact internal struc- 
ture of plants to a state of very considerable precision; although 
it must be confessed that vegetable anatomy is still the field 
where the greatest discoveries may be expected. 

It is now generally agreed that the old opinions, that the tis- 
sue of plants is either a membrane doubled together in endless 
folds, or a congeries of cavities formed in solidifiable mucus 
by the extrication of gaseous matter, are equally erroneous, 
and that it really consists of distinct sacs or cells, pressed to- 
gether and adhering to each other by the sides where they are 
in contact. 

It is considered that this is proved by the following circum- 
stances. 1. By the action of some powerful solvent, such as 
nitric acid, the cells may be artificially separated from each 
other. 2. In parts which become succulent, the cells separate 
spontaneously, as in the receptacle of the strawberry, the berry 
of the privet, &c. 3. When the parts are young, their tissue 
may be easily separated by pressure in water. 4. It is con- 
formable to what has been observed in the growth of plants. 
Amici found that the new tubes of Chara appear like young buds 
from the points or axillae of pre-existing tubes ; an observation 
that has been confirmed by Mr. Henry Slack *. It has been 
distinctly proved by M. Mirbelf , that the same thing occurs in 
the case of Marchantia polymorplia. That learned botanist, in 
the course of his inquiries into the structure of this remarkable 
plant, may be said to have been present at the birth of its cel- 
lular tissue ; and he found that in all cases one tube or utricle 
generated another, so that sometimes the young masses of tis- 
sue had the appearance of knotted or branched cords. He satis- 
fied himself, by a beautifully connected series of observations, 
that new parts are not formed by the adhesion of vesicles origi- 
nally distinct, as many have asserted, but by the generative 
power of one first utricle, which engenders others endowed with 
the same property. 

It appears that when first formed the sacs are completely 
closed up, so that there is no communication between the one 
and the other, excepting through the highly permeable mem- 
brane of which they are composed. This, indeed, is not con- 
formable to the observations of those who have described and 
represented pores or passages of considerable magnitude pierced 
in the sides of the sacs; but it has been satisfactorily shown by 
Dutrochet, that the spaces supposed by such observers to be 

* Transactions of the Society of Arts, vol. xlix. 

f " Recherches Anatomiques et Physiologiques sur le Marchantia po/t/mor- 
pha," in Nouv. Ann, du Museum, vol. i. p. 93. 


pores are nothing more than grains of amylaceous matter stick- 
ing to the sides of the sacs ; for he found that by immersing the 
latter in hot nitric acid, the supposed pores became opake, and 
by afterwards moistening them with a weak solution of caustic 
potash, they recovered their transparency : we also find that 
the supposed pores are readily detached from the sides of the 
sacs to which they adhere ; and I think it may be added, that 
our microscopes are now alone sufficient to show what they are. 

The question as to the perceptible porosity of vegetable tis- 
sue may therefore be considered, I think, disposed of as a 
general fact; for the objection that Dr. Mohl has taken to this 
explanation *, — namely, that in a transverse section we ought to 
find such grains projecting from the sides of the cells Hke little 
eminences, — cannot surely be entitled to much weight, if we op- 
pose to this negative observation the positive evidence already- 
mentioned, and especially if we consider that it is next to im- 
possible for the keenest knife to make a section of svich delicate 
parts without carrying away such particles upon its edge. There 
are, nevertheless, cases in which the point is still open to in- 

Thus Mirbel, in his second memoir on the Marchantia f , 
positively declares that the curious cells which line the anther 
of the common gourd, are continuous membranes till just be- 
fore the expansion of the flower, when they very suddenly en- 
large, and their sides divide into the narrow ribands or threads 
which give their name to what we call fibrous cells. In this, 
and the multitudes of similar cases with which Purkinje has 
made us acquainted, there can be no doubt that the sides of 
the cells consist ultimately of nothing but openwork; but still 
it seems certain that during the principal part of their existence 
they were completely closed up. 

It is also probable that in other cases the sides of the cells 
or vessels ultimately give way and slit ; but this rending seems 
to be a phaenomenon attendant upon the cessation of the ordi- 
nary functions of tissue, and independent of their original con- 

In coniferous plants the wood is in a great measure com- 
posed of closed tubes, tapering to each end, the sides of which 
are marked with circles, containing a smaller circle in their cen- 
tre. These circles have long been considered undoubted pores^ 
and it does not appear possible to prove them otherwise by any 
of the tests already mentioned. 

* Ueher die P or en des Pflanzen- Zell'jewehes, p. 11. Tubingen, 1828. 
t Archives de Botaniqiie, vol. i. 

30 TIITRD REPORT — 1833. 

I have endeavoured to show * that they are glands of a pecu- 
liar figure, which stick to the sides of the tubes ; and I have 
ascertained that the large round holes that are certainly found 
in coniferous tissue are caused by the dropping or rubbing off 
of such supposed glands. But a very different opinion is en- 
tertained by Dr. Mohlf, whose observations have been con- 
firmed by Dr. Unger;}:. In the opinion of the former of these 
botanists the supposed glands of coniferous tissue are circular 
spaces where the membrane of the tube becomes abruptly ex- 
tremely thin ; and it is said that transverse slices of coniferous 
wood, made at an angle of forty-five degrees, demonstrate the 
fact. Dr. Mohl is also of opinion, as has been already said, 
that the porous appearances above mentioned, and ascribed to 
the adhesion of amylaceous matter to the sides, are of a similar 

It has been shown by Mr. Griffiths, that in the kind of tissue 
called the dotted duct, the suspicion of Du Petit Thouars that 
this form of tissue is composed of short cylindrical cells placed 
end to end, and opening into each other, is correct ; their com- 
munication, however, is not by means of an organic perfo- 
ration, but is produced by the absorption and rupture of the 
ends which come in contact. Mr. Slack has also stated, 
in a very good paper upon Vegetable Tissue §, that in other 
cases the vessels of plants open into each other where they 
come in contact ; as, for example, at the conical extremities, 
where ducts join each other ; but he represents this to be owing 
to the obliteration of their membrane at that point ; the internal 
fibre, of which they are in part composed, remaining like a 
grating stretched across the opening where the enveloping 
membrane has disappeared. 

In a short paper, published in the Journal of the Royal In- 
stitution in December 1831, I have endeavoured to show that 
membrane and fibre are to be considered the organic elements 
of vegetable tissue, contrary to the more usual opinion that 
membrane only is its basis : this was attempted to be proved, 
not only by the fact that the simple cells of the testa of Mau~ 
randia, &c., are apparently formed by a fibre twisted spirally 
in the inside of their membrane, but also by the elastic spires I 
had discovered on the outside of the seed of Collomia, in which 
it is plain that no membrane whatever is generated. 

* Introduction to Botany, p. 16. t. 2. f. 7. 

t Ueher die Poren des Pjiemzen-Zellgeivehes. 

X Botanische Zeittmg, October 7, 1832. 

§ Transactions of the Societij of Arts, vol. xlix. 


It would, however, appear from the researches of Mirhel*, 
that the presence of a twisted fibre within a cell is not always 
the cause of the spiral or fibrous character so common in tissvie. 
He finds, as has been already stated, that the cells that line 
the anther of a gourd are at first membranous and closed, and 
that they continue in this state till just before the bursting of the 
anther, when they suddenly divide in such a way as to assume 
the appearance of delicate threads, curved in almost elliptical 
rings, which adhere to the shell of the anther by one end; these 
rings ai-e placed parallel with each other in each cell, to which 
they give an appearance like that of a little gallery with two 
rows of pilasters, the connecting arches of which remain after 
the destruction of the roof and walls. He also watched the 
development of the curious bodies called elaiers in the Mar- 
chantia, which he describes to the following efi^ect. At first 
they are long slender tubes, pointed at each end ; at a subse- 
quent stage their walls thicken, and become less transparent, 
and are marked all round through their entire length with two 
parallel, very close, spiral streaks ; later still the tubes enlai'ge, 
and their streaks become slits, which divide the walls all round, 
from one end to the other, into two filaments ; and, finally, the 
circumvolutions of the filaments separate, assume the appear- 
ance of a corkscrew, acquire a rust colour, and the elater is 
complete. These elaters he considers organically identical with 
the spiral vessel, and hence he concludes that every description 
of vessel is a cell, differing from ordinary cells in being larger. 

Upon the general accuracy' of these observations I am dis- 
posed to place great confidence ; and I would even add, that the 
theory of pierced or open cellular tissue being produced by 
the spontaneous rending of its membrane, is apparently con- 
nected with an observation of my ownf, that in some plants 
simple vegetable membrane will tear more readily in one direc- 
tion than another. It is nevertheless to be observed, that the 
theory of fibre being one of the organic elements of tissue does 
not seem to have occurred to the experienced physiologist to 
whose observations I am referring, and that some of the ap- 
pearances he mentions at a stage pi-eceding transformation are 
very like those of the development of an internal fibre. 

The opinion of the organic identity of all the forms of tissue 
has also been maintained by Mr. Slack, in the paper already 
referred to, and by Dr. Mohl, in his memoir on the comparative 
anatomy of the stem of Cycadece^ Coniferce, and Tree Ferns. 

» Archives de Botantque, vol, i. 
t Introduction to Botany, p. 2. 

S3 THIRD REPORT — 1833. 

The latter considers that the dotted tubes of Cycadece un- 
doubtedly pass directly into the vessels called by the Germans 
vasa scalarlformia \ but my own observations do not confirm 
this statement. ' 

Circulation. — Whether or not plants have a circulation ana- 
logous to that of animals, is a topic that was more open to con- 
jecture at a time when the real structure of the former was un- 
known, than it can be at the present day. Knowing, as we 
now do, that a tree is more analogous to a Polype than to a 
simple animal ; that it is a congeries of vital systems, acting 
indeed in concert, but to a great degree independent of each 
other, and that it has myriads of seats of life, we cannot expect 
that in such productions anything absolutely similar to the mo- 
tion of the blood of animals from and to one common point 
should be found. The idea of circulation existing in plants 
must therefore be abandoned ; but that a motion of some kind 
is constantly going on in their fluids was sufficiently proved by 
the well-known facts of the flow of the sap, the bleeding of the 
vine, the immense loss plants sustain by evaporation, and by 
similar phaenomena. The motion was for the first time beheld 
by Amici, the Professor at Modena, who discovered it in the 
Char a. He found that in this plant the cylindrical cells of the 
stem are filled with fluid, in which are suspended grains of 
green matter of irregular form and size. These grains were 
distinctly seen to ascend one side of each tube, and descend the 
other, after the manner of a jack-chain, and to be continually 
in action, in the same manner, as long as the cell retained its 
life ; the motion of the grains was evidently due to the ascend- 
ing and descending current in the fluid contained within the 
tube-like cell. It could not be ascertained that any kind of 
communication existed between the cells, but each was seen to 
have a motion of its own. 

The observations of Amici have been verified in this country 
chiefly upon species of Nitella ; and from the investigations of 
Mr. Solly, Mr. Varley, and Mr. Slack*, the nature of the phee- 
nomenon has been determined with considerable precision. 

Among other things, it has been ascertained that in Nitella 
the currents have always a certain relation to the axis of growth, 
the ascending current vmiformly passing along the side of the 
cell most remote fi-om the axis, and the descending current 
along the side next the axis. 

Similar motions have been seen in several other plants. In 
the cells of Hydrocharis Morsus-Ranae the fluid has been ob- 

» Transactions of the Society of Arts, vol. xlix. 


served to move round and round their sides in a rotatory man- 
ner, which, however, has not been seen to follow any particular 
law. In the joints of the hairs of Tradescmitia virginica 
several currents of a similar nature exist ; and in the hair of 
the corolla of a species of Pentstemon, Mr. Slack has observed 
several currents taking various directions, some continuing to 
the summit of the hair, whilst others turn and descend in va- 
rious places, two currents frequently uniting in one channel. 

It may hence, possibly, be assumed that in the cells of plants, 
when filled with fluid, there is a very general rotatory move- 
ment, which is confined to each particular cell. This, it is ob- 
vious, can form no part of the general circulation of the system, 
which must often occur with great rapidity, and which must 
take place from the roots to the extremities. The rotatory 
motion may perhaps be considered a sort of motion of di- 
gestion, and connected with the chemical changes which matter 
undergoes in the cells from the united action of light, heat, 
and air. 

What has been supposed to be a discovery of the universal 
motion of sap has been made by Professor Schultz of Berlin, 
who remarked two torrents, one of which was progressive, and 
the other retrogressive, in what he calls the vital vessels (ap- 
parently the woody fibre) in the veins of Cheliclonium majus, 
and in the stipulae of Ficus elastica. 

His observations have been repeated by a Commission of the 
Institute, composed of MM. Mirbel and Cassini, who have 
reported* that they have also seen the motion described by 
Professor Schultz ; and I have myself witnessed it as is repre- 
sented by those observers. But it appears probable, from se- 
veral circumstances, that the motion that has been seen has 
either been owing merely to the vessels in which it was re- 
marked having been cut through, and emptying themselves of 
their contents, as Mr. Slack has suggested, or else was nothing 
])ut the common rotatorymotion imperfectly observed. 

Sfri.'cfure of the Axis. — From the period when INl. Desfon- 
taines first demonstrated the existence of two totally distinct 
modes of increase in the diameter of the stems of plants, it has 
been received as a certain fact that monocotyledonous plants 
increase by addition to the centre of their stem, and dicotyle- 
donous by addition to the circumference. Nothing has yet 
arisen to throw any doubt upon the exactness of this notion in 
regard to dicotyledonous plants ; but Dr. Hugo Mohl has 
endeavoured to showf that monocotyledonous stems are not 

* Aniinlea ih.i Sr.'ienres, vol. xxii. p. 80. 

i Molil, "])i> I'aliiiavum Stnictura," in Marlhis's Genrra ct Species Palmarum. 

IS;}.']. D 

34f THIRD REPORT — 1833. 

formed in the manner that has been supposed. According to 
him, the new matter from which the wood results is not a mere 
addition of new matter to the centre, but consists of bundles 
of wood, which, originating at the base of the leaves, take first 
a direction towards the centre, and then a course outwards 
towards the circumference, forming a curve ; so that the stem 
of a Palm is, in fact, a mass of woody arcs intersecting each 
other, and having their extremities next the circumference of 
the trimk. I regret that I have not been able to consult Dr. 
von Martius's splendid work on Palms since this Report was 
commenced, and that I am therefore unable to state upon what 
evidence Dr. Mohl has rested his theory. 

The same writer has stated* that Cycadece — that singular 
tribe, which is placed, as it were, on the boundary line between 
cellular and vascular plants, — are not in a great measure desti- 
tute of vessels as is commonly supposed, but, on the contrary, 
are composed exclusively of spiral vessels and their modifica- 
tions, without any mixture of woody fibre. I have already ad- 
verted to this hypothesis in speaking of the same author's state- 
ment, that the dotted tubes of Cycadece are a slight modi- 
fication of vasa scalariformia. Dr. Mohl is also of opinion 
that Cycadeae are not exogenous in their mode of growth, as 
seems to be indicated by their appearance when cut, and by 
their dicotyledonous embryo, but that they are more like 
Palms in their manner of forming their wood, which is essen- 
tially endogenous. He asserts that the stem of Cycadece, in 
regard to its anatomical condition, must be considered inter- 
mediate between that of Tree Ferns and Coniferce, just as their 
leaves and fructification undoubtedly are. He states that in 
Cycadece a body of wood is gradually formed of the fibres con- 
nected with the central and terminal bud ; that so long as this 
original wood is soft, and capable of giving way to the fibres 
that are continually passing downwards, no second cylinder of 
wood is formed ; but in time the original wood becomes hard- 
ened, and then the new fibres find their way outward and down- 
ward, collecting into a second cylinder on the outside of the 
original wood. It is obvious that this explanation is not so sa- 
tisfactory as could be desired ; for, in the first place, such a 
distinction between Cycadece and Exogence as that which Dr. 
Mohl states to exist, is verbal rather than real, since he admits 
that the second cylinder of wood is formed externally to the 
first ; and secondly, it is obvious that if that structure which is 
represented in the 21st plate of the third volume of the Hortus 

* Uebcr den Bau des Cycadeen Stammex und sein Verhaltniss xu den Slamni 
der Coniferen und Banmfarrn. 4to. Munich, 1832. 


Malabaricus be correct, where the stem of Cycas c'lrcinal'is is 
shown to have several concentric zones, precisely as in other 
exogenous trees, it must follow that Di\ Mohl's explanation 
would be still more inadmissible ; accordingly, this author dis- 
credits the fact of the stem of Cycas circinalis having numer- 
ous concentric zones. It is, however, certain, from the speci- 
mens brought to England by Dr. WalUch, that the structure 
of this Cycas is really such as is shown in the Hortus Malaba- 
ricus. It is nevertheless extremely well worth further inquiry 
whether there is not some important but as yet vmdiscovered 
peculiarity in the mode of forming their stem by Cycadecs; for 
it must be confessed that growth by a single terminal bud, 
after the manner of Palms, is not what we should expect to 
meet with in exogenous trees. 

Pi-ofessor Schultz of Berlin has indicated* the existence of a 
group of plants, the structure of whose stems he considers at 
variance with all the forms at present recognised ; and to this 
group he refers Cycadece : but the assemblage of orders which 
he collects under what he calls the same plan of growth is so 
extremely incongruous as to lead to no other conclusion than 
that subordinate modifications of internal structure are of no 
general importance, but are merely indicative of individual pe- 

Dr. Mohl further states, that Cryptogamic plants of the 
highest degree of organization, such as Ferns, Lycopodiacece, 
Alarsileacece, and Mosses, in all which a distinct axis is found, 
have a mode of growth neither exogenous nor endogenous, but 
altogether of a peculiar nature. In these plants, when once 
the lower part of the stem is formed it becomes incapable of 
any further alteration, but hardens, and the stem continues to 
grow only by its point, which lengthens merely by the progres- 
sive development of the parts already formed, without sending 
downwards any fibrous or woody bundles, as both in exoge- 
nous and endogenous plants. 

M. Lestiboudois, the Professor of Botany at Lille, distin- 
guishes Monocotyledons from Dicotyledons, upon principles 
different from those generally adniitted. According to this 
writer, dicotyledonous trees have two systems, one, the central, 
consisting of the medullary sheath and the wood ; the other, 
the cortical, composing the bark. These two systems increase 
separately, so that in Dicotyledons there are two surfaces of 
increase, that of the central system, which adds to its outside, 
and that of the cortical system, which adds to its inside : but 

* NuiurHchpn Sijstem des Pflanxenreichs nach seiner inncrcn Organization. 
8vo. Berlin, 1^32." 

36 THIRD REPORT— 18i]o. 

in the stem of Monocotyledons there is only one surfoce of in- 
crease, namely, that on the inside ; and hence he concludes 
that such plants have only a cortical system, and consist of 
bark alone. It must be obvious that there are too many ana- 
tomical objections to this theory to render it deserving of any 
other than this incidental notice*. 

The cause of the formation of wood has always been a sub- 
ject upon which physiologists have been unable to agree ; and 
if the opinions held by the writers of the last century have been 
disproved, it cannot be added that those of the present day are 
by any means settled. It is now, indeed, admitted on all hands 
that wood is a deposit in some way connected with the action 
of leaves ; for it has been proved beyond all question that the 
quantity of wood that is formed is in direct proportion to the 
number of leaves that are evolved, and to their healthy action, 
and that where no leaves are formed, neither is wood deposited. 
But it is a subject of dispute whether wood is actually or- 
ganized matter generated by the leaves, and sent downwards 
by them, or whether it is a mere secretion, which is deposited 
in the course of its descent from the leaves to the roots. The 
former opinion has been maintained in different forms by De 
la Hire, Darwin, Du Petit Thouars, Poiteau, and myself, and 
would perhaps have been more generally adopted if it had not 
been too much mixed up with hypothetical statements, to the 
reception of which there are in the opinion of many persons 
strong objections. For example, it has been assei'ted that the 
wood of trees is an aggregation of the roots of myriads of buds 
in a state of action, and that consequently a tree is an asso- 
ciation of individuals having a peculiar organic adhesion and 
a common system of growth, but each its own individual life. 
To this view it is no doubt very easy to raise objections, some 
of which it may be difficult, in the present state of our know- 
ledge, to answer ; and therefore it is better for the moment to 
leave this part of the proposition out of consideration, and to 
confine it to the simple statement that wood is organized matter, 
generated by the leaves, and sent downvt^ards by them. In 
support of this it is argued : 1 st, That an anatomical examination 
of a plant shows that the woody systems of the leaf and stem 
are continuous : Sndly, That this is not only the fact in exogenous 
plants, but in all endogenous and cellular plants that have 
been examined ; so that it may be considered a universal law : 
ordly. That in the early spring, and for some time after plants be- 
gin to grow, the woody matter is actually to be seen and traced 

• Achille Richai'd, Nouveaux EUmens de la Botaniqtte, 5me edit. p. 119. 


descending in parallel tubes from the origin of the leaves, and 
from no other place : 4thly, That in all cases where obstacles 
are presented to the descent of such tubes, they turn aside, and 
afterwards resume their parallelism when the obstacle has been 
passed by : 5thly, That in endogenous plants, such as Palms, and 
in some exogenous trees, such as Lignum Vita, they cross and 
interlace each other in a manner which can only be accounted 
for by their passing downwards, the one over the other, as the 
leaves are developed : and, finally, That the perfect organization 
of the wood is incompatible with a mere deposit of secreted 
matter. To all which the following evidence has been added 
by M. Achille Richard. He states * that he saw in the pos- 
session of Du Petit Thouars a branch of Robinia Pseudacacia, 
on which Robinia hispida had been grafted. The stock had 
died, but the scion had continued to grow, and had emitted 
from its base a sort of plaster, formed of very distinct fibres, 
which surrounded the extremity of the branch to some distance, 
and formed a sort of sheath ; thus demonstrating incontestibly 
that fibres do descend from the base of the scion, to overlay 
the stock. 

To this several objections have been taken, the most im- 
portant of which are the following. If wood were really or- 
ganized matter, emanating from the leaves, it must necessarily 
happen that in grafted plants the stock ought in time to acquire 
the nature of the scion, because its wood would be formed en- 
tirely by the addition of new matter, said to be furnished by 
the leaves of the scion ; so far, however, is this from being the 
fact, that it is well known that in the oldest grafted trees there 
is no action whatever exercised by the scion upon the stock, 
but that, on the contrary, a distinct line of organic demarcation 
separates the wood of the one from the other, and the shoots 
emitted from the stock by wood said to have been generated 
by the leaves of the scion, are in all respects of the nature of 
the stock. Again, — if a ring of bark from a red-wooded tree 
is made to grow in the room of a similar ring of bark of a white- 
wooded tree^ as it easily may be made, the trunk will increase 
in diameter, but all the wood beneath the ring of red bark will 
be red, although it must have originated in the leaves of the 
tree which produces white wood. It is further urged, that in 
grafted plants the scion often overgrows the stock, increasing 
much the more rapidly in diameter, or that the reverse takes 
place, as when the Pavia littea is grafted upon the common 
Horse-chestnut, — and that these circumstances are inconsistent 

» yuufcaiu- E/emcns de la BoiaiiiqKC, 5mc edit. p. 105. 

38 THIRD REPORT — 1833. 

with the supposition that the wood is organic matter engendered 
by leaves. To these statements there is nothing to object as 
mere facts, for they are true; but they certainly do not warrant 
the conclusions tliat have been drawn from them. One most 
important point is overlooked by those who employ these argu- 
ments, namely, that in all plants there are two distinct simul- 
taneous systems of growth, the ceUular and the fibro-vascular, 
of which the former is horizontal, and the latter vertical. The 
cellular gives origin to the pith, the medullary rays, and the 
principal part of the cortical integument ; the fibro-vascular, to 
the wood and a portion of the bark ; so that the axis of a plant 
may be not inaptly compared to a piece of linen, the cellular 
system being the woof, the fibro-vascular the warp. It has also 
been proved by Mr. Knight* and M. De Candollef that buds 
are exclusively generated by the cellular system, while roots are 
evolved from the fibro-vascular system. Now if these facts are 
rightly considered, they will be found to ofFer an obvious expla- 
nation of the pha?nomena produced by those botanists who think 
that wood cannot be matter generated in an organic state by the 
leaves. The character of wood is chiefly owing to the colour^ 
quantity, size, and distortions of the medullary rays, which be- 
long to the horizontal system; it is for this reason that there is 
so distinct a line drawn between the wood of the graft and 
stock, for the horizontal systems of each are constantly pressing 
together with nearly equal force, and uniting as the trunk in- 
creases in diameter. As buds from which new branches elon- 
gate are generated by cellular tissue, they also belong to the 
horizontal system ; and hence it is that the stock will ahvays 
produce branches like itself, notwithstanding the long super- 
position of new wood which has been taking place in it froni 
the scion. 

The case of a ring of red bark always forming red wood be- 
neath it, is precisely of the same nature. After the new bark 
has adhered to the mouths of the medullary rays of the stock, 
and so identified itself with the horizontal system, it is gradually 
pushed outwards by the descent of woody matter from above 
through it : but in giving way it is constantly generating red 
matter from its hoi-izontal system, through which the wood de- 
scends, which thus acquires a colour that does not properly 
belong to it. With regard to the instances of grafts over- 
growing their stock, or vice versa, it is obvious that these are 
susceptible of explanation upon the same principle. If the hori- 
zontal system of both stock and scion has an equal power of 

• Philosophical Transaciiuns, 1S05, p. 257. 
•{■ Physiohgic Vcnitcde, p. 158. 


lateral extension, the diameter of each will remain the same ; 
but if one grows more rapidly than jthe other, the diameters 
will necessarily be different : where the scion has a horizontal 
system that develops more rapidly than that of the stock, the 
latter will be the smaller, and vice versa. It is* however, to be 
observed, that in these cases plants are altogether in a morbid 
state, and will not live for any considerable time. 

Those who object to the theory of wood being generated by 
the action of leaves, either suppose — 1st, that liber is developed 
by alburnum, and wood by liber ; or, 2ndly, that " the woody and 
cortical layers originate laterally from the cambium furnished 
by preexisting layers, and nourished by the descending sap *." 
The first of these opinions appears to be that of M. Turpinf , 
as far as can be collected from a long memoir upon the grafting 
of plants and animals ; but I must fairly confess that I am not 
sure I have rightly understood his meaning, so much are his 
facts mixed up with gratuitous hypothesis and obscure specu- 
lations upon the action of what he calls globuline. The second 
is the opinion commonly entertained in France, and adopted 
by M. De Candolle in his latest published work. 

The objections to the views of M. Turpin need hardly be 
stated in a Report like this, where conciseness is so much an 
object. Those which especially bear upon the view taken by 
M. De Candolle are, that his theory is not applicable to all 
parts of the vegetable kingdom, but to exogenous plants only; 
that it is inconceivable how the highly organized parallel tubes 
of the wood, which can be traced anatomically from the leaves, 
and which are formed with great rapidity, can be a lateral de- 
posit from the liber and alburnum ; that they are manifestly 
formed long before it can be supposed that the leaves have 
commenced their office of elaborating the descending sap ; and, 
finally, that endogenous and cryptogamic plants, in which there 
is no secretion of cambium, nevertheless have wood. 

Such is the state of this subject at the time I am writing. To 
use the words of M. De Candolle, " The whole question may 
be reduced to this, — Either there descend from the top of a tree 
the rudiments of fibres, which are nourished and developed by 
the juices springing laterally from the body of wood and bark, 
or new layers are developed by preexisting layers, which are 
nourished by the descending juices formed in the leaves |." 

As this is one of the most curious points remaining to be 
settled among botanists, and as it is still as much open to dis- 

* De Candolle, Phyaiologie Vegetale, p. 165. 

+ Sue Annales des Sciences, vols. xxiv. and xxv., particularly vol. x.w. p. •13. 

J De Candollu, Physiologie f'rgelale, p. 157. 

40 THIRD REPORT 1833. 

cussion as ever, I have dwelt upon it at an unusual length, in 
the hope that some Member of the British Association may 
have leisvire to prosecute the inquiry. Perhaps there is no 
mode of proceeding to elucidate it which would be more likely 
to lead to positive results, than a very careful anatomical exami- 
nation of the progressive development of the Mangel Wiu'zel 
root, beginning with the dormant embryo, and concluding with 
the perfectly formed plant. 

Arrangemeni of Leaves. — It has for a long time been thought 
that the various modes in which leaves, and the organs which 
are the result of them, are arranged upon a stem might be re- 
duced to the spiral, and that all deviations from this law of 
arrangement are to be considered as caused by the breaking of 
spires into verticilli. In the Pine Apple, for instance, the Pine 
Cone, the Screw Pine, and many other plants, the spiral arrange- 
ment of the leaves is so obvious that it cannot be overlooked ; 
in trees with alternate leaves this same order of arrangement 
may be discovered if a line is drawn from the base of one leaf 
to that of another, always following the same direction; even 
in verticiilate plants we not unfrequently see that the whorls 
are dislocated by the praBternatural elongation of their axis, and 
then become converted into a spire ; and the same pha3nome- 
non is of common occurrence among the verticilli of leaves in 
the form of calyx, corolla, stamens, and carpella, which com- 
pose the flower. This will be the more distinctly apparent if we 
consider that, as M. Adolphe Brongniart has shown*, what we 
calLvvhorls in a flower often are not so, strictly speaking, but 
only a series of parts placed in close approximation, and at dif- 
ferent heights, upon the short branch that forms their axis. 

Dr. Alexander Braun has endeavoured f to prove mathema- 
tically that the spiral arrangement of the parts of plants is not 
only universal, but subject to laws of a very precise nature. 
His memoir is of considerable length, and would be wholly un- 
intelligible without the plates that illustrate it. It is therefore 
only possible on this occasion to mention the results. Setting 
out witli a contemplation of the manner in which the scales of 
a Pine Cone are placed, to which a long and ingenious method 
of analysis v/as applied, he found that several different series of 
spires are discoverable, between which there invariably exist 
peculiar arithmetical relations, v/hich are the expression of the 
various combinations of a certain number of elements disposed 
in a regular manner. All these spires depend upon the posi- 

* Annales des Sciences, vol. xxiii. p. 226. 

+ Verglekhendr Vritersuchunj iibcr die Ordininfj der Schuppen an den Tan- 
nen~apfcn. Ito. IPjd. 


tion of a fundamental series, from wliicli the otliers are devia- 
tions. The nature of the fundamental series is expressed by a 
fraction, of which the numerator indicates the whole number 
of turns required to complete one spire, and the denominator 
the number of scales or parts which constitute it: thus ^-^ in- 
dicates that eight turns are made round the axis before any 
scale or part is exactly vertical to that which was first formed, 
and the number of scales or parts that intervene before this 
coincidence takes place is twenty-one. 

It does not appear that this inquiry has as yet led to any 
practical application, although one might have expected that 
as the natural affinities of plants are determined, in a great de- 
gree, by the accordance that is observable in the relative posi- 
tion of their parts, the spires of which those parts are composed 
might have had something in common M'hich would be suscep- 
tible of being expressed by numbers. If any practical applica- 
tion can be made of Dr. Braun's fractions, it seems likely to be 
confined to the distinction of species. His observations seem, 
however, to have established the truth of the doctrine that, be- 
ginning with the cotyledons, the whole of the appendages of 
the axis of plants, — leaves, calyx, corolla, stamens, and car- 
pella, — form an uninterrupted spire, governed by laws which 
are almost constant. 

Structtire of Leaves. — The leaves of plants have been found 
by M. Adolphe Brongniart to be not merely expansions of 
the cellular integument of stems, traversed by veins originating * 
in the woody system, but to be organs in which the inteqnal 
parenchyma is arranged with beautiful uniformity, in the man- 
ner most conducive to the end of exposui'e to light and air, and 
of elaboration, for which the leaves are chiefly destined. In 
their usual structure leaves have been found by this observer 
either to consist of two principal layers, — of which the upper, 
into which the ascending sap is first introduced, is formed of 
compact cells, more or less perpendicular to the plane of the cu- 
ticle, and the under, into which the returning sap is propelled, 
is formed of very lax cavernous tissue, more or less parallel with 
the cuticle of the lower surface, — or else of two layers perpen- 
dicular to the cuticle, with a central parallel stratum. 

The observations of Drs. Mohl and Meyen generally confirm 
this ; but at the same time the latter instances several cases in 
which the texture of the leaf has been found to be nearly the 
same throughout. 

Dutrochet* states, in addition, that the interior of the leaf 

* Anitales des Sciences, vol. xxv. p. 215. 

42 THIRD REPORT 1833. 

is divided completely by a number of partitions caused by the 
ribs and jirincipal veins, so that the air cavities have not actually 
a free communication in every direction through the parenchy- 
ma, but are to a certain extent cut off from each other. This 
is conformable to what M. Mirbel has described in Marchantia, 
who finds the leafy expansions of that plant separated by par- 
titions into chambers, between which he is of opinion that 
there is no other communication than what results from the per- 
meability of the tissue *. 

The statement of M. Adolphe Brongniart, that all leaves in- 
tended to exist in the air are furnished with a distinct cuticle 
on their two faces, while those which are developed under 
w^ater have no cuticle at ail, has not been disproved, unless in 
the case of Marchantia f , whose under surface can scarcely be 
said to have a distinct cuticle; but this jilant, which can only 
exist in humid shady places, is perhaps rather a proof of the 
accuracy of the theory of M. Brongniart than an exception 
to it. 

That the stomata in all cases open into internal cavities in 
the leaf, where the tissue is extremely lax and cavernous, ap- 
pears also exti'emely probable. It was especially found to be 
the case by M. Mirbel in his so often quoted remarks upon 

With regard to the stomata themselves, no one appears yet 
to have confirmed the observation of Dr. Brown;}:, that their 
apparent orifice is closed up by a membrane. On the contrary, 
the observations of M. Mirbel on Marchantia, if they are to 
be taken as illustrative of the usual structure of those singular 
organs, go to establish the accuracy of the common opinion 
that the stomata are apertures in the cuticle. That most skil- 
ful physiologist, while watching the development of Marchantia, 
remarked the very birth of the stomata, which he describes as 
taking place thus : — The appearance of a little pit in the middle 
of four or five cells placed in a ring is a certain indication of 
the beginning of a stoma. The pit evidently increases by the 
enlargement and separation of the svirrounding cells. If the 
nascent stoma consists of five cells, of which one is surrounded 
by four others, then the central one is destroyed; but if it con- 
sists of three or four cells adjusted so as to form a disk, then 
the stoma is caused by the separation of their sides in the cen- 
tre, by which means a sort of star is created. It is true that 

* " Reclierclies Anatomiques et Pliysiologiques sur le Marchantia poli/ruor- 
pha," in Nouvcaux Annalcs du Museum, vol. i. p. 7. 
+ Ibid. p. 93. 
J Suppl. prlmttm Pradromi Floros Novce IloUaiidicc, p. 3. 


the stomata of Marchantia are in some respects different from 
what are found upon flowering plants ; yet I think we can hardly 
doubt that the plan upon which they are all formed is essen- 
tially the same. 

Dutrochet also confirms * the statement of Amici, that the 
stomata are perforations; for he finds that when leaves are de- 
prived of their air by the aii'-pump, it is chiefly on the under 
side, where the greatest number of stomata is found, that little 
air bubbles make their appearance ; and that it is through the 
stomata that water rushes into the cavernous parenchyma to 
supply the loss occasioned by the abstraction of air. 

Anther, Sj-c. — Some curious remarks upon the nature of the 
tissue that lines the cells of the anther have been published by 
Dr. John E. Purkinje, Professor of Medicine at Breslau. His 
researches are chiefly directed to the determination of the na- 
ture of the tissue that is in immediate contact with the pollen ; 
and he has demonstrated in an elaborate Essay f, that the opi- 
nion emitted by Mirbel in 1808 J, that the cause of the dehis- 
cence of the anther is its lining, consisting of cellular tissue cut 
into slits and eminently hygrometrical, is substantially true. 
He shows tjiat this lining is composed of cellular tissue chiefly 
of the fibrous kind, which forms an infinite multitude of little 
springs, that when dry contract and pull back the valves of the 
anthers by a powerful accumulation of forces which are indivi- 
dually scarcely appreciable : so that the opening of the anther 
is not a mere act of chance, but the admirably contrived result 
of the maturity of the pollen, — an epoch at which the surround- 
ing tissue is necessarily exhausted of its fluid by the force of 
endosmosis exercised by each particular grain of pollen. 

That this exhaustion of the circumambient tissue by the en- 
dosmosis of the pollen is not a mere hypothesis, has been 
shown by Mirbel in a continuation of the beautiful memoir I 
have already so often referred to§. He finds that, on the one 
hand, a great abundance of fluid is directed into the utricles, 
in which the pollen is developed a little before the maturity of 
the latter, and that by a dislocation of those utricles the pollen 
loses all organic connexion with the lining of the anther ; and 
that, on the other hand, these utricles are dried up, lacerated, 
and disorganized, at the time when the pollen has acquired its 
full development. 

* Annales des Sciences, vol. xxv. p. 24?, 

t De Cellulis Anther arum fihrous. 4to. Wratislaviae, 1830. 

X " Observations sur iin Systcnie d'Anatomie Comparee de Vegetaiix, fondes 
sur rOrganization de la Fleur," in Mcmoire.s de I'lnstitut, 1808, "p. W.'A. 

§ "Complement des ObsciTations sur le Marchantia jwlymorphu," in Ar- 
chives de Botanique, vol. i. 

44 THIRD REPORT — 1833. 

The Origin of the Pollen, connected as it intimately is with 
the singular pha;nomena of vegetable sexuality, has naturally 
been of late an object of some inquiry. To the important dis- 
coveries of the younger Brongniart and of Dr. Robert Brown, 
M. Mirbellias added some observations*, detailed with that 
admirable clearness and precision which give so great a value 
to all his writings, and wiiich are the more interesting as they 
serve to explain what was before obscure, and to correct what 
appears to have been either inaccurately or imperfectly de- 
cribed. This he has been enabled to do by beginning his in- 
quiry at the very earliest period when the organization of the 
anther can be discovered : his subject was the common Gourd. 
At a very early time the whole tissue of the anther is of the 
most perfect uniformity, consisting of cellules, the transverse 
section of which represents nearly regular hexagons and penta- 
gons. In every cell, without even excepting those which com- 
pose the superficies of the anther, are found little loose bodies, 
so exceedingly minute that a magnifying power of 500 or 600 
diameters is scarcely sufficient to examine them : they may be 
compared to transparent, nearly colourless vesicles, more or 
less round, and of unequal size. At a stage but little more ad- 
vanced, you may observe on each side of the medial line of a 
transverse section of a lobe of an anther, a collection of cellules 
rather larger than the remainder: it will afterwards be seen 
that it is here that the pollen is engendered; such cells are 
therefore called pollen-cells. In a bud, a line and a half or two 
lines in diameter, some remarkable alterations were found to 
have taken place ; the pollen-cells had enlarged and their gra- 
nules had so much increased in number, that they nearly filled 
the cells in opake masses. These granules and pollen-cells 
formed together a greyish mass, connected with the rest of the 
tissue by the intervention of a cellular membrane, which, not- 
withstanding its organic continuity with the surrounding parts, 
is at once distinguishable ; for while the cells of the surrounding 
parts elongate parallel to the plane of the surface, and to the 
plane of the base of the anther, those of the cellular membrane 
elongate from the centre to the circumference. In more ad- 
vanced anthers, the sides of the pollen-cells, from being thin 
and dry, had changed to a perceptible thickness, and their sub- 
stance, gorged with fluid, resembled a colourless jelly. When 
the buds were three or four lines long, an unexpected phaeno- 
menon presented itself. At first the thick and succulent walls 
of each pollen-cell dilated so as to leave an empty space between 
the inner face and the granules, not one of which sepiu'ated 
* " Complement des Observations," i^c, as above quoted. 


from the general mass, which showed that some power kept 
them united. Shortly after, four appendages, like knife-blades, 
developed at equal distances upon the inner face of the cell, 
and gradually projected their edges towards the centre, till at 
last they divided the granular mass into four little triangvdar 
bodies; when the appendages had completely united at their 
edges, they divided the cavity of the pollen-cells into four di- 
stinct boxes, which then began to rounden, and finally became 
little spherical masses. Each of these was the rudiment of a 
grain of pollen, subsequently acquired a membranous integu- 
ment, hardened, became yellow, and thus arrived at maturity. 

Wliat is perhaps most important in these observations is the 
demonstration of the original organic continuity of all the parts 
of the anther, against the statement of M. Adolphe Brongniart, 
and also against what appears to be the opinion of Dr. Brown, 
as far as can be collected from the manner in which he speaks 
of the evolution of the pollen in Tradescantia rirginica *. 

Although it is not directly shown by these observations whe- 
ther the perfect grain of pollen has one or two integuments, — a 
question that may still be said to be unsettled, — it nevertheless 
appears from other instances that M. Mii'bel admits the exist- 
ence of an outer not distensible coat, and of an inner highly 
extensible lining. A curious paper upon this point f has been 
published by a Saxon botanist named Fritzsche. By means of 
a mixture of two parts by weight of concentrated sulphuric 
acid, and five parts of water, he found that the grains of pollen 
can be rendered so transparent as to reveal their internal struc- 
ture, and that the whole process of the emission of the pollen- 
tubes can be distinctly traced. He describes the universal pre- 
sence of two coatings to the grains of pollen ; and he also finds 
that the pollen contains a quantity of oily particles in addition 
to the moving corpuscles, — a fact which has also been noticed 
by Dr. Brown. 

Although the generalizations in this work are less satisfactory 
than could be desired, it must nevertheless be considered a 
most valuable collection of facts, and as containing the best 
arrangement that has as yet appeared of the various forms un- 
der which the pollen is seen. 

Fert/lhation. — The road which some years since was so 
happily opened by Amici to the discovery of the exact manner 
in which vegetable fertilization takes effect, is every day be- 
coming more and more direct. The doubts of those vvho could 
not discern the tubes that are projected into the style by the 

* Observations upon Orchidon? n?id Asclrpiacfea?, p. 21. 
f Beitrage ?:ur Kenntnks dcs Pollen. 4to. Berlin, 1S33. 

46 THIRD REPORT — 1833. 

pollen, have been removed; the important demonstration by 
Dr. Brown of the universal presence of a passage through the 
integuments of the ovulum at the point of the nucleus has been 
extended and confirmed by M. Mirbel in a paper of the high- 
est interest * ; the fact that it is at the point of the nucleus (where 
this passage exists,) that the nascent embryo makes it appear- 
ance, is now undisputed; the passage of the contents of the 
pollen down the pollen-tubes, and the curious discovery of a 
power of motion in the granules that are thus emitted, are also 
recognised : it now only remains to be proved that the pollen- 
tubes come in contact with the nucleus, and the whole secret of 
fertilization is revealed. A few remarkable contributions to 
this part of the subject have lately been made. 

Some plants have the passage or foramen in their ovulum so 
remote from any part through which the pollen-tubes can be 
supposed to convey their influence, as to have thrown consider- 
able difficulty in the way of the supposition that actual contact 
between the point of the nucleus and the fertilizing tissue is 

The manner in which, notwithstanding the apparent difficulty 
of such contact taking place, this happens in Statice Armeria, 
was long since made out by Dr. Brown, in whose possession I 
several years ago saw drawings illustrating this phsenomenon ; 
it has since been explained by M. Mirbel. Another case, pre- 
senting similar apparent difficulties, occurs in Helianthemutn. 
In plants of that genus the foramen is at that end of the ovulum 
which is most remote from the hilum ; and although the ovula 
themselves are elevated upon cords much longer than are usually 
met with, yet there are no obvious means of their coming in con- 
tact with any part through which the matter projected into the 
pollen-tubes can be supposed to descend. It has, however, 
been ascertained by M. Adolphe Brongniartf, that at the time 
when the stigma is covered with pollen, and fertilization has 
taken effect, there is a bundle of threads, originating from the 
base of the style, which hang down in the cavity of the ovarium, 
and, floating there, are abundantly sufficient to convey the in- 
fluence of the pollen to the points of the nuclei. So again in 
AsclepiadecE. In this tribe, from the peculiar conformation of 
the parts, and from the grains of pollen being all shut up in a 
sort of bag, out of which there seemed to be no escape, it was 
supposed that this tribe must at least form an exception to the 
general rule. But before the month of November 1828:}:, the 

* Nouvelles Piecherehes sur la Sfnicture dc VOoule Vegetal et sur ses Deve- 
loppements. Also Additions aiix 'Nouvelles Rec/irrclies,' ^c. 

•f- Annales des Sciences, vol. xxiv. p. 123. X Linntpa, vol. iv. p. 94. 


celebrated Prussian traveller and botanist Ehrenberg had dis- 
covered that the grains of pollen of Asclepiadece acquire a sort 
of tails which are all directed to a suture of their sac on the side 
next the stigma, and which at the period of fertilization are 
lengthened and emitted ; but he did not discover that these 
tails are only formed subsequently to the commencement of a 
new vital action connected with fertilization, and he thought 
that they were of a different nature from the pollen-tubes of 
other plants ; he particularly observed in Asclepias syriaca 
that the tails become exceedingly long and hang down. 

In 1831 the subject was resumed by Dr. Brown* in this 
country, and by M. Adolphe Brongniartf in France, at times so 
nearly identical, that it really seems to me impossible to say 
with which the discovery about to be mentioned originated : it 
will therefore be only justice if the Essays referred to are spoken 
of collectively instead of separately. These two distinguished 
botanists ascertained that the production of tails by the grains 
of pollen was a phaenomenon connected with the action of ferti- 
lization ; they confirmed the existence of the suture described 
by Ehrenberg ; they found that the true stigma of Asclepia- 
dece is at the lower part of the discoid head of the style, and 
so placed as to be within reach of the suture through which the 
pollen-tubes or tails are emitted ; they remarked that the latter 
insinuate themselves below the head of the style, and follow its 
surface vmtil they reached the stigma, into the tissue of which 
they buried themselves so perceptibly that they were enabled to 
trace them, occasionally, almost into the cavity of the ovarium ; 
and thus they established the highly important fact, that this 
family, which was thought to be one of those in which it was 
impossible to suppose that fertilization takes place by actual 
contact between the pollen and the stigma, offers the most 
beautiful of all examples of the exactness of the theory, that it 
is at least owing to the projection of pollen-tubes into the sub- 
stance of the stigma. In the more essential parts these two 
observers are agreed : they, however, differ in some of the de- 
tails ; as, for instance, in the texture of the part of the style 
which I have here called stigma, and into which the pollen- 
tubes are introduced. M. Brongniart both describes and figures 
it as much more lax than the contiguous tissue, while on the 
other hand Dr. Brown declares that he has in no case been able 
to observe " the slightest appearance of secretion, or any dif- 

* Observations on the Organs and Mode of Fecundation of Orchideae and 
Asclepiadece. London, October 1831. 

f Annft/fis des Sciences for October and November 1831 ; from observations 
made in July, August and September of that yoar. 


ferences whatever in texture between tliat part and the general 
surface of the stigma" (meaning what I have described as the 
discoid head of the style) : but this is not the place for entering 
into the discussion of these subordinate points. 

Orchidece are another tribe in which similar difficulties have 
been found in reconciling structure with the necessity of con- 
tact between the pollen and stigma in order to effect impregna- 
tion. Indeed it seems in these plants as if every possible pre- 
caution had been taken by nature to prevent such contact. 
Nevertheless it is represented by M. Adolphe Brongniart, in a 
paper read before the Academy of Sciences of Paris in July 
1831 *, that contact is as necessary in these plants as in others, 
and that in the emission of pollen-tubes they do not differ from 
other plants. These statements have been followed up by Dr. 
Brown f, in itn elaborate Essay upon the subject, in which the 
results that are arrived at by our learned countryman are es- 
sentially to the same effect. To these tliere is at present no- 
thing equally positive to oppose ; but as the indirect observa- 
tions of Mr. Bauer %, and the general structure of the order, are 
very much at variance with the probability of actual contact 
being necessary, and especially as Dr. Brown is obliged to have 
recourse to the supposition that tlie pollen of many of these 
plants must be actually carried by insects from the boxes in 
which it is naturally locked up, — it must be considered, I think, 
that the mode of fertilization in Orchidece is still far from being 
determined. I must particularly remark that the very proble- 
matical agency of insects, to m hich Dr. Brown has recourse in 
order to make out his case, seems to be singvilarly at variance 
with his supposition § that the insect forms, which in Oplirys 
are so striking, and which he finds resemble the msects of the 
countries in which the plants are found, are intended rather to 
repel than i^^ attract. It may be true, as Dr. Brown observes, 
that there is less necessity for the agency of insects in such 
flowers as the European OpJinjdecs ; but what other means 
than the assistance of insects can be supposed to extricate the 
pollen from the cells in the insect flowers of such plants as 
ReiioufJiera Arac/iniies, the whole genus Oncidiian, Teiramicra 
rigldn, several species of Epidendrinn, Cipnbidlum tenuijolium, 
Vanda peduncidans, and a host of others ? 

* Annales des Sciences, vol. xxiv. p. 113. 

t Observations upon the Organs and Mode of Fecundation of Orcliidese and 

X Illustrations of the Genera and Species of Orchideovs Plants. Part II. 
" Fructification," tabb. 5. 12. 13. ] k 

§ " Proceedings of tlie Linnean Society," June o, 1832, as given in the Lon- 
don and Edinburgh Philosopliical iMagcuiitc and Journal. 


Origin of Organs. — There is no part of vegetable physiology 
SO obscure as that which relates to the origin of organs. We 
find a degree of simplicity that is perfectly astonishing in the 
fundamental structure of the whole vegetable kingdom ; we 
are able to prove by rigorous demonstration that every one of 
the appendages of the axis is a modification of a leaf, to which 
there is a constant tendency to revert; we see that in some cases 
a part which usually performs one function assumes another, 
as in the Alstromerias, whose leaves by a twist of their petiole 
turn their under surface upwards : but we are entirely ignorant 
of the causes to which these changes are owing. An impor- 
tant step in elucidating the subject has been lately taken by 
M. Mirbel, in his memoir upon the structure of Marchantia po~ 
lymorplia. The young bulbs by which this plant is multiplied 
are originally so homogeneous in structure, that there is no 
apparent character in their organization to show which of their 
faces is destined to become the upper surface, and which the 
under. For the purpose of ascertaining whether there existed 
any natural but invisible predisposition in the two faces to un- 
dergo the changes which subsequently become so apparent, 
and by means of which their respective functions are performed, 
or whether the tendency is given by some cause posterior to 
their first creation, the following experiments were instituted. 
Five bulbs were sown upon powdered sandstone, and it was 
found that the face which touched the sandstone produced 
roots, and the opposite face formed stomata. It Avas, however, 
possible that the five bulbs might have all accidentally fallen 
upon the face which was predisposed to emit roots ; other 
experiments of the same kind were therefore tried, first with 
eighty and afterwards with hundreds of little bulbs, — and the 
result was the same as with the five. This proved that either 
face was originally adapted for producing either roots or sto- 
mata, and that the tendency was determined merely by the po- 
sition in which the surfaces were placed. The next point to 
ascertain was, whether the tendency once given could be after- 
wards altered ; some little bulbs, that had been growing for 
twenty-four hours only, had emitted roots ; they were turned, 
so that the upper surface touched the soil, and the under was 
exposed to light. In twenty-four hours more the two faces 
had both produced roots ; that which had originall)'^ been the 
under surface went on pushing out new roots ; that which had 
originally been the upper surface had also produced roots : 
but in a few days the sides of the young plants began to rise 
from the soil, became erect, turned over, and finally recovered 

1833. E 

50 THIRD REPORT — 1833. 

in this way their original position, and the face which had ori- 
ginally been the uppermost, immediately became covered with 
stomata. It, therefore, appears that the impulse once given, 
the predisposition to assume particular appearances or func- 
tions is absolutely fixed, and will not change in the ordinary 
course of nature. This is a fact of very high interest for 
those who are occupied in researches into the causes of what is 
called vegetable metamorphosis, an expression which has been 
justly criticised as giving a false idea of the subject to which 
it relates. 

Morphology. — When those who first seized upon the im- 
portant but neglected facts out of which the modern theory of 
morphology has been constructed, asserted that all the appen- 
dages of the axis of a plant are metamorphosed leaves, more 
was certainly stated than the evidence would justify ; for we 
cannot say that an organ is a metamorphosed leaf, which in 
point of fact has never been a leaf. What was meant, and that 
which is supported by the most conclusive evidence, is, that 
every appendage of the axis, whether leaf, bractea, sepal, petal, 
stamen, or pistillum, is originally constructed of the same ele- 
ments, arranged upon a common plan, and varying in their 
manner of development, not on account of any original differ- 
ence in structure, but on account of special and local predis- 
posing causes : of this the leaf is taken as the type, because it 
is the organ which is most usually the result of the develop- 
ment of those elements,^ — is that to which the other organs 
generally revert, when from any accidental disturbing cause 
they do not assume the appearance to which they were originally 
predisposed, — and, moreover, is that in which we have the most 
complete state of organization. 

This is not a place for the discussion of the details upon 
which the theory of morphology is founded ; it is sufficient to 
state that it has become the basis of all philosophical views of 
structure, and an inseparable part of the science of botany. Its 
practical importance will be elucidated by the following circum- 
stance. Fourteen or fifteen years ago I was led to take a 
view of the structure of Reseda very different from that usually 
assigned to the genus ; and when a few years afterwards that 
view was published, it attracted a good deal of attention, and 
gained some converts among the botanists of Germany and 
France. It was afterwards objected to by Dr. Brown upon 
several grovmds ; but I am not aware that they were considered 
sufficiently valid to produce any change in the opinions of those 
who had adopted my hypothesis. Lately, however, Professor 


Henslow has satisfactorily proved*, in part by the aid of a 
monstrosity in the common Mignonette, and in part by a severe 
appHcation of morphological rules, that my hypothesis must 
necessarily be false ; and I am glad to have this opportunity of 
expressing my full concurrence in his opinion. 

It has long been known that the ligulate and tubular corollas 
of CompositcB are anatomically almost identical, and that their 
difterence consists only in the five petals of the tubular corolla 
all separating regularly for a short distance from their apex, 
while the five petals of the ligulate corolla adhere up to their 
very points, except on the side next the axis of inflorescence, 
where two of them are altogether distinct except at their base. 
M. Leopold von Buch explains this circumstance in the follow- 
ing manner. He states that these ligulate corollas when vmex- 
panded bear at their point a little, white, and very viscid body 
or gland, which is a peculiar secretion that dries up when it 
comes in contact with the atmosphere. The adhesion of this 
gland is too powerful to be overcome by the force of the style 
and stamens pressing against it from within. The corollas, which 
are gradually curved outwards by the growth of those in the 
centre of the inflorescence, at the same time bend down the 
style, which consequently presses up against the line of union 
of the two petals nearest the axis : although the style cannot 
overcome the adhesion of the viscid gland at the point of the 
corolla, it is able in time to destroy the union of the two inte- 
rior petals, which finally give way and allow the stamens and 
style to escape. As soon as this takes place, the corolla can no 
longer remain erect, but falls back toAvards the circumference 
of the capitulum, and thus contributes to the radiating character 
of this sort of inflorescence. When the viscid body is either 
not at all, or very imperfectly produced at the point of the co- 
rolla, as sometimes happens in the genus Hieracium, especially 
H. bifurcum, tubular corollas are produced instead of ligulate 

The ovulum is the organ where the greatest difficulty has 
occurred in reducing the structure to anything analogous to 
that of other parts. It is true that Du Petit Thouars regarded 
it as analogous to a leaf bud ; but his view appears to have been 
purely hypothetical, for I am not aware that he had any distinct 
evidence of the fact. Some years ago M. Turpin, in showing 
the great similarity that exists between the convolute bracteas 
of certain Marcgraviacece and the exterior envelope of the 
ovulum, took the first step towards proving that the hypothesis 

* Transactions of the Philosophical Society of Cambridge, vol, v. Part I. 

E 2 

52 THIRD REPORT — 1833. 

of Du Petit Thouars was susceptible of demonstration ; it was 
more distinctly shown by the interesting discovery of Professor 
Henslow, that the leaves of Malaxis paludosa had on their 
margins what no doubt must be considered buds, but what in 
structure are an intermediate state between buds and ovula ; 
and it has been recently asserted by Engelmann*, still, how- 
ever, without the production of any proof, that "ovula are buds 
of.a higher kind, their integuments leaves, and their funiculus 
the axis, all which, in cases of retrograde metamorphosis, are in 
fact converted into stem and gi-een leaves." The nearest ap- 
proach to a demonstration that has yet been afforded of ovula 
being buds is in a valuable paper by Professor Henslow, just 
printed in the Transactions of the Philosophical SocietT/ of Cam- 
bridge]-, in which it is shown that in the Mignonette the ovula 
are in fact transformed occasionally into leaves, either solitary 
or rolled together round an axis, of which the nucleus is the 

M. Dumortier has endeavoured to prove | that the embryo 
itself is essentially the same as a single internodium of the stem 
with its vital point or rudimentary bud attached to it. Although 
the author's demonstration is a failure, and his paper a series 
of confused and illogical reasoning, yet there can be little doubt 
that the hypothesis itself is a close approximation to the truth. 

Dr. George Engelmann has recently attempted § to classify 
the aberrations from normal structure, which throw so much 
light upon the real origin and nature of the organs of plants. 
He has collected a very considerable number of cases under 
the following heads. 1 . Retrograde metamorphosis (Regressus), 
when organs assume the state of some of those on the outside 
of them, as when carpella change to stamens or petals, hypo- 
gynous scales to stamens, stamens to petals or sepals, sepals to 
ordinary leaves, irregular structure to regular, and the like. 

2. Foliaceous metamorphosis {Virescentia), when all the parts 
of a flower assume more or less completely the state of leaves. 

3. Disunion {Disjunctio), when the parts that usually cohere 
are separated, as the carpella of a syncarpous pistillum, the 
filaments of monadelphous stamens, the petals of a monopeta- 
lous corolla, &c. 4. Dislocation {Apostasis) ; in this case the 
whorls of the flower are broken up by the extension of the 
axis, 5. Viviparousness {Diaphysis), when the axis is not only 
elongated, but continues to grow and form new parts, as in those 

* De Antholysi Prodromus, p. 61. t vol. v. Part I. 

I Nova Acta Academice Naturee Curiosorum, vol. xvi. p. 245. 
^ De Antholysi Prodromus. 


instances where one flower grows from within another. And 
finally, 6. Proliferousness {Ecblastesis), when buds are deve- 
loped in the axillae of the floral organs, so as to convert a sim- 
ple flower into a mass of inflorescence. A very considerable 
number of instances are adduced in illustration of these divi- 
sions, and the work will be found highly useful as a collection 
of curious or important facts. 

The doctrines of morphology, and the evidence in support 
of them, may now be considered so far settled as to require but 
little further illustration for the present. This is, however, 
only true of flowering plants : in the whole division of flower- 
less plants there has been scarcely any attempt to discover the 
analogy of organs, and to reduce their structure to a correspond- 
ing state of identification. I some time since * endeavoured to 
excite attention to this subject, by hazarding some speculations 
which had at least the merit of novelty to recommend them ; 
but I cannot discover that any one has since turned his atten- 
tion to the inquiry, although it must be confessed that the com- 
parative anatomy of flowerless plants is among the most inter- 
esting topics still remaining for discussion, and that it is rather 
discreditable to Cryptogamic botanists that the elucidation of 
so very curious a matter should be postponed to the compara- 
tively unimportant business of distinguishing or dividing genera 
and species. 

Gradual Development. — The theory of the gradual deve- 
lopment of the highest class of organic bodies, in consequence 
of a combination and complication of the phaenomena attendant 
upon the development of the lowest classes, has acquired so 
great a degree of probability among animals, that it has become 
a question of no small interest whether traces of the same, or a 
similar law, cannot be found among plants. In an inquiry of 
such a nature, it seems obvious that attention should in the first 
instance be dii'ected to a search after positive and incontestable 
facts, and that mere hypotheses should in the beginning be to- 
tally rejected. The only circumstances that occur to me as 
bearing directly upon this point are the following. It has been 
ascertained by M. Mirbel, in his memoir on the Marchantia, 
that the sporule of that very simple plant is a single vesicle, 
which, when it begins to grow, produces other vesicles on its 
surface, which go on propagating in the same manner, every 
new vesicle engendering others ; and that different modifica- 
tions of this process produce the different parts that the per- 
fect plant finally develops. 

* Outlines of the First Principles of Botany, p. 533, &c. Introduction to the 
Natural System of Dot any, p. 313, &c. 

54 THIRD REPORT — '1833. 

The same principle of growth appears to obtain in Confervas, 
and probably is found in other vegetables of the lowest grade. 

This is analogous to what takes place in the formation of the 
embryo of Vasciilares. In the opinion of Dr. Brown and of 
Mirbel, the first rudiment of a plant far more comphcated than 
Marchaniia, consists also of a vesicle, but suspended by a 
thread to the summit of the cavity of the ovulum ; and the dif- 
ference between the one case and the other is, that while in the 
Marchaniia the original vesicle, " as soon as it is formed, pos- 
sesses all the conditions requisite for developing a complete 
plant on the surface of the soil ; on the other hand, that of 
flowering plants must, on pain of death, commence its deve- 
lopment in the interior of the ovulum, and cannot continue it 
further until it has produced the rudiments of root, stem, and 

Beyond this I do not think that any attempt has been made 
to elucidate the question. 

Irritability. — I3r. Dutrochet has published f the result of 
some experiments v,ith the air-pump upon the pneumatic system 
of plants. Independently of confirming the fact, already gene^ 
rally known, of plants having the means of containing a large 
quantity of air, he arrived at the unexpected result, that the 
sleep of plants and their irritability are certainly dependent upon 
the presence of air within them. A sensitive plant, left in the 
vacuum of an air-pump for eighteen hours, indicated no sign 
whatever of the accustomed collapse of its leaflets on the ap- 
proach of night, nor when it was restored to the air could it be 
stimulated by the smartest shocks ; but in time it recovered its 
irritabihty. When flowers that usually close at night were 
placed in a vacuum while expanded, they would not close ; and 
when flowers already closed were placed in the same situation, 
they would not unfold at the return of morning ; whence Dr. 
Dutrochet infers that the internal air of plants is indispensably 
necessary to the exercise of their alternate motions of sleeping 
and waking, and in general to the existence of the faculty they 
possess of indicating by their movements the influence of ex- 
ternal exciting causes. 

Action of Coloured Light. — Professor Morren, of Ghent, lias 
mentioned j the result of some experiments upon the action of 
the coloured rays upon germination ; and he has fovmd that 
while those rays in which the illuminating power is the most 
feeble were, as might have been expected, the most favourable 
to germination, their power of decomposing carbonic acid, and 

* Archives de Botanique, vol. i. t ■•'innalcs des Sciences, vol.xxv. p. 243. 
J Annates des Sciences, vol. xxvii, p. 201, 


producing a green deposit in the parenchyma, is in proportion 
to their illuminating property ; that no decomposed rays effect 
this so rapidly as white light; and that the yellow ray possesses 
the greening power in the highest degree, the orange in a very 
slight degree, and violet, red and purple not at all. 

Colours. — Nothing can be named in the whole range of bo- 
tany upon which information is so much wanted as the cause of 
the various colours of plants. It was, indeed, long since sus- 
pected by Lamarck that the autumnal colouring of leaves and 
fruits was a morbid condition of those parts ; and it has subse- 
quently been ascertained that all colours are owing to the pre- 
sence of a substance, called chromiile by De CandoUe, which 
fills the parenchyma, assuming different tints. Green has also 
been clearly made out to be connected with exposure to light, 
and has been considered to be in all probability owing to the 
deposition of the carbon left upon the decomposition of car- 
bonic acid. Some botanists have also observed the connexion 
of red colour with acidity ; but still we had scarcely any positive 
knowledge of the cause of the production of any colour except 
green, till M. Macaire of Geneva* remarked, that just before 
leaves begin to change colour in the autumn, they cease parting 
with oxygen in the day, although they go on absorbing it at 
night ; whence he concluded that their chromule is oxygenated, 
by which a yellow colour is first caused, and then a red, — for he 
found that in all cases a change to red is preceded by a change 
to yellow. He also ascertained that the chromule of the red 
bracteae and calyx of Salvia splendens is chemically the same 
as that of autumnal leaves. Coupling this with the fact that 
petals do not part with oxygen, it would seem as if their colour, 
if yellow or red, may also be owing to a kind of oxygenation. 
But according to M, Theodore de Saussure f , coloured fruits 
part with their oxygen ; so that, if this be true, red and yellow 
cannot always be ascribed to such a cause. M. De CandoUe J 
has some excellent observations upon this subject in his recent 
admirable digest of the laws of vegetable physiology ; in which 
he concludes, from the inquiries hitherto instituted, that all co- 
lours depend upon the degree of oxygenation. When oxygen 
is in excess, the colour seems to tend to yellow or red ; and 
when it is deficient, or when the chromule is more carbonized, 
which is the same thing, it has a tendency to blue. Local ad- 
ditions of alkaline matters are also called in aid of an explanation 
of the various shades of colour that flowers and fruits present. 

* Mcmoirrs de la Socictc Physique de Geneve, vol. iv. p. 50. 
t Ibid. vol. i. p. 284. % Phjsiologie Vegetale, p. 906. 

56 THIRD KEPORT — 1833. 

Dr. Dutrocliet is of opinion * that the whitish spots we some- 
times see in leaves, and the paler tint that generally character- 
izes the under side of the same organs, are owing to the presence 
of air beneath the cuticle. He finds that the arrow-head shaped 
blotch on the upper side of the leaf of Tnfolium pratense, and 
the whitish spots on Pulmonaria officinalis, disappear when the 
leaves are plunged in water beneath the exhausted receiver of 
the air-pump, and that the lower surface of leaves acquii'es the 
same depth of colour as the upper under similar circumstances. 
This he ascribes to the air naturally found in the leaves being 
abstracted, and its place supplied with water ; a conclusion 
which agrees with what might be inferred from the anatomical 
structure of the parts in question. 

Excretions. — It has long been known that some plants are 
incapable of growing, or at least of remaining in a healthy state, 
in soil in which the same species has previously been cultivated. 
For instance, a new apple orchard cannot be made to succeed 
on the site of an old apple orchard, unless some years inter- 
vene between the destruction of the one and the planting of 
the other : in gardens, no quantity of manui'e will enable one 
kind of fruit-tree to flourish on a spot from which another tree 
of the same species has been recently removed ; and all farmers 
practically evince, by the rotation of their crops, their expe- 
rience of the existence of this law. 

Exhaustion of the soil is evidently not the cause of this, for 
abundant manuring will not supersede the necessity of the usual 
rotation. The celebrated Duhamel long ago remarked, that the 
Elm parts by its roots with an unctuous dark-coloured substance ; 
and, according to De CandoUe, both Humboldt and Plenck 
suspected that some poisonous matter is secreted by roots ; 
but it is to M. Macaire, who at the instance of the first of these 
three botanists undertook to inquire experimentally into the 
subject, that we owe the discovery of the suspicion above al- 
luded to being well founded. He ascertained f that all plants 
part with a kind of faecal matter by their roots, that the nature 
of such excretions varies with species or large natural orders : 
in Cichoracecs and Papaveracece he found that the matter was 
analogous to opium, and in Leguminosce to gum ; in Graminece 
it consists of alkaline and earthy alkalies and carbonates, and 
in Euphorbiacecs of an acrid gum-resinous substance. These 
excretions are evidently thrown off by the roots on account of 
their presence in the system being deleterious ; and it was found 
by experiment, that plants artificially poisoned parted with the 

* Aiinales des Sciences, vol. xxv. p. 216. 
f De CandoUe, Physiologic Vegetale, p. 249. 


poisonous matter by their roots. For instance, a plant of Mer- 
curialis had its roots divided into two parcels, of which one was 
immersed in the neck of a bottle filled with a weak solution of 
acetate of lead, and the other parcel was plunged into the neck 
of a corresponding bottle filled with pure water. In a few days 
the pure water had become sensibly impregnated with acetate 
of lead. This, coupled with the well known fact that plants, 
although they generate poisonous secretions, yet cannot absorb 
them by their roots without death, as, for instance, is the case 
with Atropa Belladonna, seems to prove that the necessity of 
the rotation of crops is more dependent upon the soil being 
poisoned than upon its being exhausted. 

This is a part of vegetable physiology of vast importance to 
an agricultural country like England, and may possibly cause 
a total revolution in our system of husbandry. 

All that M. Macaire can be said as yet to have done, is to 
have discovered the fact and to have pointed out certain strong 
examples of it ; but if the discovery is to be converted to a 
practically useful purpose, we require positive information upon 
the following points : — 

1. The nature of the faecal excretions of every plant culti- 
vated by the farmer. 

2. The nature of the same excretions of the common weeds 
of agriculture. 

3. The degree in which such excretions are poisonous to the 
plants that yield them, or to others. 

4. The most ready means of decomposing those excretions 
by manures or other means. 

It would be superfluous to point out what the application 
would be of such information as thi;?, ; but I cannot forbear ex- 
pressing a hope that a question upon which so many deep inter- 
ests are involved may be among the first to occupy the atten- 
tion of the chemists of the British Association. 

[ 59 1 

Report on the Physiology of the Nervous System. By Wil- 
liam Charles Henry, M.D., Physician to the Manchester 
Royal Infirmary, 

Introduction. — Tme science of Physiology has for its object to 
ascertain, to analyse, and to classify the qualities and actions 
which are peculiar to living bodies. These vital properties re- 
side exclusively in organized matter, which is characterized by 
a molecular arrangement, not producible by ordinary physical 
attractions and laws. Matter thus organized consists essen- 
tially of solids, so disposed into an irregular network of laminae 
and filaments, as to leave spaces occupied by fluids of various 
natures. 'Texture' or 'tissue' is the anatomical term by which 
such assemblages are distinguished. Of these the cellular, or 
tela cellulosa, is most elementary, being the sole constituent of 
several, and a partial component of all tissues and systems. 
Thus the membranes and vessels consist entirely of condensed 
cellular substance ; and even muscle and nerve are resolvable, 
by microscopic analysis, into globules deposited in attenuated 
cellular element. 

But though the phenomena, which are designated as vital, 
are never found apart from organization, and have even by 
some naturalists been regarded as identical with it, yet in the 
order of succession vital actions seem necessarily to stand to 
organized structures in the relation of antecedents ; for the 
production of even the most rudimentary forms and textures 
implies the previous operation of combining tendencies or 'vital 
affinities'. The origin and early development of these vital 
tendencies, and of organized structures, are beyond the pale of 
exact or even of approximative knowledge. But it is matter of 
certainty, that life is the product only of life ; that every new 
plant or animal proceeds from some pre-existent being of the 
same form and character ; and thus that the image of the great 
Epicurean poet, " Quasi cursores vital lampada tradunt," pos- 
sesses a compass and force of illustration which, as a supporter 
of the doctrine of fortuitous production, he could not have him- 
self contemplated. 

The popular notions I'especting life are obscure and indeter- 
minate ; nor are the opinions even of philosophers characterized 
by much greater distinctness or mutual accordance. Like other 
complex terms, ' life ' can obviously be defined only by an cnume- 

60 THIRD REPORT — 1833. 

ration of the phenomena which it associates. This enumera- 
tion will comprehend a greater or a smaller number of particu- 
lars, according to the station in the scale of living beings which 
is occupied by the object of survey. In its simplest manifesta- 
tion, the principle of life may be resolved into the functions of 
nutrition, secretion and absorption. It consists, according to 
Cuvier, of the faculty possessed by certain combinations of 
matter, of existing for a certain time and under a determinate 
form, by attracting unceasingly into their composition a part of 
surrounding substances, and by restoring portions of their own 
substance to the elements. This definition comprehends all 
the essential phenomena of vegetable life. Nutritive matter is 
drawn from the soil by the spreading fibres of the root, through 
the instrumentality of spongioles or minute turgid bodies at 
their extremities, which act, according to Dutrochet, by a power 
which he has called 'endosmosis.* The same agency raises the 
nutrient fluid through the lymphatic tubes to the leaves, where 
it seems to undergo a kind of respiratory process, and becomes 
fit for assimilation. These changes, and the subsequent pro- 
pulsion of the sap to the different parts and textures, plainly 
indicate independent fibrillary movements, which are repre- 
sented in animal life by what Bichat has termed ' the pheno- 
mena of organic contractility'. The power residing in each 
part of detecting in the circulating fluid, and of appropriating, 
matters fitted to renovate its specific structure, is designated in 
the same system by the term ' organic sensibility'. 

Ascending from the vegetable to the animal kingdom, the 
term ' life' advances greatly in comprehensiveness. The exist- 
ence of a plant is limited to that portion of space in which acci- 
dent or design has inserted its germ; while animals are for 
the most part gifted with the faculties of changing their place, 
and of receiving from the external world various impressions. 
Along with the general nutritive functions, the higher attri- 
butes of locomotion and sensation are therefore comprised in 
the extended compass of meaning which the term ' life' acquires 
with the prefix ' animal'. The nutritive functions, too, emerging 
from their original simplicity, are accomplished by a more com- 
plex mechanism, and by agencies further removed from those 
which govern the inanimate world. 

Locomotion is effected either by means of a contractile tissue, 
or of distinct muscular fibres. These fibres have been said to 
consist of globules resembling, and equal in magnitude to, those 
of the blood, disposed in lines, in the elementally cellulosity , which 
by an extension of the analogy is compared to serum. But the 
latest microscopical observations of Dr. Hodgkin are opposed 


to this globular constitution of the contractile fibre. " Innu- 
merable very minute but clear and fine parallel lines or striae 
may be distinctly perceived, transversely marking the fibrillas." 
Irritability, or the faculty of contracting on the application of a 
stimulant, is a property inherent in the living fibre. It is an 
essential element of all vital operations, except of those which 
have their seat in the nervous system, such as sensation, voli- 
tion, the intellectual states, and moral affections. All the phe- 
nomena of life, in the higher animals, may then be ultimately 
resolved into the single or combined action of these two ele- 
mentary properties, — irritability and nervous influence, each 
residing in its appropriate texture and system. 

These preliminary remarks are designed to unfold the prin- 
ciples to be followed in classifying the vital functions. In ge- 
neral or comparative physiology, a strictly scientific ari'angement 
would contemplate first the phenomena of the most elementary 
life, and would successively trace the more perfect development 
of those simple actions and their gradual transition into more 
complex processes, as well as the new functions, superadded 
in the ascending scale of endowment. But such a mode of 
classification is wholly inapplicable to the particular physiology 
of man and of the more perfect animals, viewed by itself and 
without reference to inferior orders of beings ; for the nutri- 
tive functions of this class, which correspond with the elemen- 
tary actions of the simplest vegetable life, are effected by a 
complex system of vessels and surfaces, deriving their vital 
powers from contractile fibres, and controlled, if not wholly 
governed, by nervous influence. It is then manifest, that in 
the higher physiology the general lav/s of contractility and ' in- 
nervation' must precede the description of the several functions, 
which all depend on their single or imited agency. The parti- 
cular functions will afterwards be classed, as they stand in more 
immediate relation to one or other of the two essential princi- 
ples of life. 

In the present state of physiological knowledge, it is impos- 
sible to determine absolutely, and without an opening to con- 
troversy, whether the functions of muscle or those of nerve are 
entitled to precedency. If each were equally independent of 
the other in the performance of their several offices, the question 
of priority would resolve itself into one of simple convenience. 
The actions of the nervous system, if contemplated for the short 
interval of time during which they are capable of persisting 
without renovation of tissue, are entirely independent of the 
contractile fibre. But it is certain that the cooperation of 
nerve is required in most, if not in all, the actions of the mus- 

62 THIRD REPORT — 1833. 

cular system. Tims the voluntary muscles in all their natural 
and sympathetic contractions receive the stimulant impulse of 
volition through the medium of nerve ; and though the mode, 
in which the motive impression is communicated to the invo- 
luntary muscles, is still matter of controversy, there seems suffi- 
cient evidence* to sanction the conclusion that nerve is in this 
case also the channel of transmission ; — " that the immediate 
antecedent of the contraction of the muscular fibre is univer- 
sally a change in the ultimate nervous filament distributed to 
that fibre." If this be correct, the physiological history of 
muscle cannot be rendered complete without reference to that 
of nerve. 

In the higher manifestations of life, nervous matter is in- 
vested with the most eminently vital attributes. It is the ex- 
clusive seat of the various modes of sensation, and of all the 
intellectual operations ; or, rather, it is the point of transition, 
where the physical conditions of the organs, which are induced 
by external objects, pass into states of mind, becoming per- 
ceptions ; and where the mental act of volition first impresses 
a change on living matter. These two offices of conducting 
motive impressions from the central seat of the will to the mus"- 
cles, and of propagating sensations from the surface of the body 
and the external organs of sense to the sensorium commune, 
have been of late years shown to reside in distinct portions of 
nervous substance. 

The honour of this discovery, doubtless the most important 
accession to physiological knowledge since the time of Harvey, 
belongs exclusively to Sir Charles Bell. It constitutes, more- 
over, only a part of the new truths, which his researches have 
unveiled, regarding the general laws of nervous action, and the 
offices of individual nerves. His successive experiments on 
function, guided always by strong anatomical analogies in struc- 
ture, in origin, or in distribution, have led to the entire remo- 
deUing of nervous physiology, and to the formation of a system 
of arrangement, based on essential affinities and on parity of 
intimate composition, instead of on apparent sequence or prox- 
imity of origin. Among the continental anatomists, MM. Ma- 
gendie and Flourens have contributed most largely to our 
knowledge of this part of physiology ; the former by repeating 
and confirming the experiments of Bell, as well as by various 
original inquiries ; the latter by his important researches into 
the vital offices of the brain and its appendages. Much light, 

* See " A Critical and Experimental Enquiry into the Relations subsisting 
between Nerve and Muscle," in the 37th vol. of the JUdinbure/h Medical and 
Surgical Journal. 


too, has been thrown on the functions of several of the ence- 
phalic nerves, and especially of those supplying the face and 
its connected cavities, by Mr. Herbert Mayo, who has analysed 
their anatomical composition, and pursued their course with 
singular precision, and has thus been enabled to correct some 
errors of detail in the system of Sir Charles Bell. 

Nervous System. — In man, and in other vertebrated animals, 
the nervous system consists of the cerebrum, cerebellum, me- 
dulla oblongata, medulla spinalis, and of the encephalic, spinal, 
and ganglionic nerves. It seems most natural to observe this 
order of anatomical sequence in recording what is known of 
nervous functions. 

Cerebrum, or Brain-proper. — The physiology of the brain 
has received of late years very considerable accessions, and its 
vital offices, viewed as an entire organ, have now probably been 
ascertained with sufficient precision. Some portion of this newly 
acquired knowledge has been gathered from experiments on 
living animals, but the greater and more valuable part has 
flowed from the study of comparative development. In this 
latter field of inquiry, Tiedemann's elaborate history of the pro- 
gressive evolution of the human brain during the period of 
foetal existence, with reference to the comparative structure of 
that organ in the lower animals, merits an early and detailed 
notice. It had been discovered by Harvey, that the foetus in 
the human species, as well as in inferior animals, is not a pre- 
cise facsimile of the adult, but that it commences from a form 
infinitely more simple, and passes through several successive 
stages of organization before reaching its perfect development. 
In the circulatory system, these changes have been minutely 
observed and faithfully recorded*. Tiedemann has traced a 
similar progression in the brain and nervous system, and has 
moreover established an exact parallel between the temporary 
states of the foetal brain in the periods of advancing gestation, 
and the jiermanent development of that organ at successive 
points of the animal scale. The first part of his work is simply 
descriptive of the nervous system of the embryo at each suc- 
cessive month of foetal life. It constitutes the anatomical grovmd- 
work upon which are raised the general laws of cerebral forma- 
tion, and the higher philosophy of the science. In the second 
part, Tiedemann has established, by examples drawn from all 
the grand divisions of the animal kingdom, the universality of 
the law of formation, as traced in the nervous system of the 

• See an excellent Essay on the Development of the Vascular System in the 
Foetus of Vertebrated Animals, by Dr, Allen Thomson. 


human foetus, and the existence of one and the same funda- 
mental type in the hrain of man and of the inferior animals. 

The facts which have been unfolded by the industry of Tiede- 
mann, besides leading to the universal law of nervous develop- 
ment, throw important light upon nervous function : for it is 
observed that the successive increments of nervous matter, and 
especially of brain, mark successive advances in the scale of 
being ; and, in general, that the development of the higher in- 
stincts and faculties keeps pace with that of brain. Thus, in 
the zoophyta, and in all living beings destitute of nerves, no- 
thing that resembles an instinct or voluntary act is discovera- 
ble. In fishes the hemispheres of the brain are small, and 
marked with few furrows or eminences. In birds they are 
much more voluminous, more raised and vaulted than in rep- 
tiles ; yet no convolutions or anfractuosities can be perceived 
on any point of their surface, nor are they divided into lobes. 
The brain of the mammalia approaches by successive steps to 
that of man. That of the rodentia is at the lowest point of 
organization. Thus the hemispheres in the mouse, rat, and 
squirrel are smooth and without convolutions. In the carnivo- 
rous and ruminating tribes, the hemispheres are much larger 
and marked by numerous convolutions. In the ape tribe the 
brain is still more capacious and more convex ; it covers the 
cerebellum, and is divided into anterior, middles, and posterior 
lobes. It is in. man that the brain attains its greatest magni- 
tude and most elaborate organization. Sommerring has proved 
that the volume of the brain, referred to that of the spinal mar- 
row as a standard of comparison, is greater in man than in any 
other animal. 

Various attempts have been made of late years, chiefly 
by the French physiologists, to ascertain the functions of the 
brain by actual experiment. It will appear from a detailed 
survey of their labours, that little more than a few general 
facts respecting the function of its larger masses and great na- 
tural divisions have flowed from this mode of research. The 
offices of the smaller parts of cerebral substance cannot with 
any certainty be derived from the phenomena that have been 
hitherto observed to follow the removal of those parts, since 
the most practised vivisectors have obtained conflicting results. 
Nor is it difficult, after having performed or witnessed such 
experiments, to point out many unavoidable sources of fallacy. 
In operations on living animals, and especially on so delicate 
an organ as the brain, it is scarcely possible for the most skilful 
manipulator to preserve exact anatomical boundaries, to restrain 
haemorrhage, or prevent the extension to contiguous parts, of 


the morbid actions consequent upon such serious injuries, and 
to distinguish the secondary and varying phenomena, induced 
by the pressure of extravasated blood, or the spread of an in- 
flammatory process, from those which are essential and pri- 
mary. The ablation of small and completely insulated portions 
of brain must, then, be classed among the " agenda" of experi- 
mental physiology. 

The most decisive researches, that have been hitherto insti- 
tuted on the functions of the brain, are those of M. Flourens. 
His mode of o^ierating was to remove cautiously successive thin 
slices of cerebral matter, and to note the corresponding changes 
of function. He commenced with the hemispheres of the brain, 
which he found might be thus cut away, including the corpora 
striata and thalami optici, without apparently occasioning any 
pain to the animal, and without exciting convulsive motions. 
Entire removal of the cerebrum induces a state resembling coma ; 
the animal appears plunged in a profound sleep, being wholly 
lost to external impressions, and incapable of originating mo- 
tion ; it is deprived, too, according to Flourens, of every mode 
of sensation. Hence the cerebrum is inferred to be the organ in 
which reside the faculties of perception, volition and memory. 
Though not itself sensible, in the ordinary acceptation of the 
word, — that is, capable, on contact or injury, of propagating sen- 
sation, — yet it is the point where impressions made on the ex- 
ternal organs of sense become objects of perception. This ab- 
sence of general sensibihty observed in the brain has also been 
experimentally demonstrated in the nerves dedicated to the func- 
tions of sight, of smell and of hearing, and constitutes, perhaps, 
one of the most remarkable phenomena that have been disclosed 
by interrogating living nature. Flourens appears, however, to 
have failed in proving that all the sensations demand for their 
perception the integrity of the brain. He has himself stated 
that an animal deprived of that organ, when violently struck, 
" has the air of awakening from sleep," and that if pushed for- 
wards, it continues to advance after the impelling force must 
have been wholly expended. Cuvier has therefore concluded, 
in his Report to the Academy of Sciences upon M. Flourens' 
paper, that the cerebral lobes are the receptacle in which the 
impressions made on the organs of sight and hearing only, be- 
come perceptible by the animal, and that probably there too 
all the sensations assume a distinct form, and leave durable im- 
pressions, — that the lobes are, in short, the abode of memory. 
The lobes, too, would seem to be the part in which those mo- 
tions which flow from spontaneous acts of the mind have their 
origin. But a power of effecting regular and combined move- 

1833. p 

66 THIRD REPORT — 1833. 

ments, on external stimulation, evidently survives the destruc- 
tion of the cerebral hemispheres. 

A very elaborate series of experiments on the functions of 
the brain in general, and especially on those of its anterior por- 
tion, have been since performed by M. Bouillaud *. That ob- 
server concurs with Flourens in vievping the cerebral lobes as 
the seat of the remembrance of those sensations which are fur- 
nished to us by sight and hearing, as well as of all the intel- 
lectual operations to which these sensations may be subjected, 
such as comparison, judgment and reasoning. But he proves 
that the ordinary tactual sensibility does not require for its 
manifestation the presence of the brain. For animals entirely 
deprived of brain were awakened by being struck, and gave 
evident indications of suffering when exposed to any cause of 
physical pain. Bouillaud observes, too, that the iris continues 
obedient to the stimulus of light, after ablation of the hemi- 
spheres, and on this ground calls in question the loss of vision 
asserted by Flourens. Nor are the lobes (he contends,) the 
only receptacle of intelligence, of instincts and of volition : for 
to admit this proposition of Flourens would be to grant that an 
animal which retains the power of locomotion, which makes 
every effort to escape from irritation, which preserves its appro- 
priate attitude, and executes the same movements after as be- 
fore mutilation, may perform all those actions without the agency 
of the will or of instinct. Another doctrine of Flourens, which 
has been experimentally refuted by Bouillaud, is, " that the 
cerebral lobes concur as a whole in the full and entire exercise 
of their functions ; that when one sense is lost, all are lost ; 
when one faculty disappears, all disappear ;" in short, that a 
certain amount of cerebral matter may be cut away without ap- 
parent injui'y, but that when this limit is passed, all voluntary 
acts and all perceptions perish simultaneously. Bouillaud, on 
the contrary, has described several experiments which show 
that animals, from whom the anterior or frontal part of the brain 
had been removed, preserved sight and hearing, though de- 
prived of the knowledge of external objects, and of the power 
of seeking their food. 

The second part of M. Bouillaud's researches is entirely de- 
voted to the functions of the anterior lobes of the brain. These 
were either removed by the scalpel, or destroyed by the actual 
cautery, in dogs, rabbits and pigeons. Animals thus mutilated 
feel, see, hear and smell ; are easily alarmed, and execute a 
number of voluntary acts, but cease to recognise the persons 

* Magendie, Journal de Physiolorjie, torn. x. p. 361 


or objects which surround them. They no longer seek food, 
or perform any action announcing a comljination of ideas. Thus 
the most docile and intelhgent dogs lost all power of compre- 
hending signs or words which were before familiar to them, 
became indifferent to menaces or caresses, were no longer 
amenable to authority, and retained no remembrance of places, 
of things, or of persons. They saw distinctly food presented 
to them, but had ceased to associate with its external qualities 
all perception of its relations to themselves as an object of de- 
sire. The anterior or frontal part of the brain is hence inferred 
to be the seat of several intellectual faculties. Its removal oc- 
casions a state resembling idiotism, characterized by loss of the 
power of discriminating external objects, which, however, co- 
exists with the faculties of sensation. 

It will be unnecessary to describe fully in this place the ex- 
periments of Professor Rolando of Turin, performed in 1809, and 
published in Magendie's Journal, tom. iii., 1823, since the more 
important of his facts have reference, not to the brain-proper, 
but to the cerebellum. His paper certainly contains some cu- 
rious anticipations of phenomena, since more accurately ob- 
served by Flourens and Magendie ; yet as regards the brain, 
properly so called, his results are vague and inconclusive. 
Accident, rather than a well matured design, seems to have 
directed what parts of the brain he should remove ; and from 
having comprehended in the same injury totally distinct anato- 
mical divisions, he has rendered it impossible to arrive at the 
precise function of any one part. Thus we are told that injury 
of the thalami optici and tubercula quadrigemina in a dog was 
followed by violent muscular contractions. Now all subsequent 
experimenters agree, that irritation of the thalami is incapable 
of inducing convulsive motions ; and Flourens has proved that 
this property has its beginning in the tubercula, — an important 
fact, which Rolando, with a little more precision in anatomical 
manipulation, could scarcely have failed to discover. 

Magendie has described* some curious experiments on the 
corpora striata, which, though closely analogous in their results 
to those on the cerebellum, have their proper place in this 
section. Removal of one corpus striatum was followed by no 
remarkable change ; but when both had been cut away, the 
animal rushed violently forwards, never deviating from a recti- 
linear course, and striking against any objects in its way. In his 
lecture of February 7, 1828, Magendie, in the presence of his 
class, removed both corpora striata from a rabbit. The anunal 

* Journal de Physiologie, tom. iii. p. 376. 

68 THIRD REPORT — 1833. 

attempted to rush forwards, and, if restrained, appeared rest- 
less, continuing in the attitude of incipient progression. One 
thalamus opticus was then cut away from the same animal. The 
direction of its motion was immediately changed from a straight 
to a curved line. It continued for some time to run round in 
circles, turning towards the injured side. When the other 
thalamus was removed, the animal ceased its motions and re- 
mained perfectly tranquil, with the head inclined backwards. 
These experiments, it may be observed, furnish no support to 
the opinions of MM. Foville and Pinel Grandchamps, who have 
assigned the anterior lobes and corpora striata as the parts 
presiding over the movements of the inferior extremities, and 
the posterior lobes and thalami as regulating the superior. 

Cerebelhim. — It may be regarded as nearly established by 
modern researches, that the cerebellum is more or less directly 
connected with the function of locomotion. The precise nature 
and extent of its control over the actions of the voluntary 
muscles are, however, far from being clearly determined. In 
the higher animals, the mental act of volition probably has its 
commencing point, as productive of a physical change, in the 
brain-proper ; though it must be confessed that some of the 
experiments of Flourens, and all of those of Bouillaud, indicate 
the persistence of many instinctive, and even of some automatic 
motions, after destruction of the brain. But there does appear 
sufficient evidence to prove that those volitions which have 
motion as their effect, whatever be their origin, whether in the 
cerebrum, cerebellum, or medulla oblongata*, require for their 
accomplishment the cooperation of the cerebellum. This evi- 
dence has been mainly supplied by the same inquirers whose 
researches on the cerebrum have been already analysed. 

In the order of time, though not of importance, the experi- 
ments of Professor Rolando stand foremost. Injuries of the 
cerebellum, he observed, were always followed by diminished 
motive power ; and this partial loss of power was always in 
direct proportion to the amount of injui-y. A turtle survived 
upwards of two months the entire removal of the cerebellum, 
continuing sensible to the slightest stimulus ; but when irritants 
were applied, it was totally unable to move from its place. 
M. Flourens has since arrived at similar, but more definitive 
results. He removed in succession thin slices from the cere- 
bellum. After the first two layers had been cut away, a slight 
weakness and want of harmony and system in the automatic 
movements were noticed. When more cerebellic substance had 

* Flourens, Mhnoires de I' Academie, torn. ix. 


been removed, great general agitation became apparent. The 
pigeon which was the subject of operation retained, as at first, 
the senses of sight and hearing, but was capable of executing 
only irregular unconnected muscidar efforts. It lost by degrees 
the power of flying, of walking, and even of standing. Removal 
of the whole cerebellum was followed by the entire disappear- 
ance of motive power. The animal, if laid upon its back, tried 
in vain to turn round ; it perceived and was apprehensive of 
blows, with which it was menaced, heard sounds, seemed aware 
of danger, and made attempts to escape, though ineffectually, 
— in short, while it preserved, uninjured, sensation and the ex- 
ercise of volition, it had lost all power of rendering its muscles 
obedient to the will. The cerebellum is hence supposed by 
Flourens to be invested with the office of " balancing, regu- 
lating or combining separate sets of muscles and limbs, so as 
to bring about those complex movements depending on simvd- 
taneous and conspiring efforts of many muscles, which are ne- 
cessary to the different kinds of progressive motion." Bouil- 
laud, who has successfully disputed several of the opinions of 
Flourens respecting the functions of the cerebrum, fully concurs 
with him as to those of the cerebellum. 

Yet, it must be admitted, that there exists also conflicting 
experimental testimony on this subject. M. Fodera* states 
that he has found the removal of a part of the cerebellum to 
be followed, in all cases, either by motion backwards, or by 
that position of the body which precedes retrograde movement. 
The head is thrown back, the hind legs separated, and the 
fore legs extended forwards, and pressed firmly against the 
ground. More complete destruction of the cerebellum occa- 
sions the animal to fall on its side ; but the head is still inclined 
rigidly backwards, and the anterior extremities agitated with 
convulsive movements, tending to cause retrograde motion of 
the body. Injuries of one side of the cerebellum were observed 
to produce paralysis of the same side of the body ; as might, 
indeed, have been anticipated from the direct course, without 
decussation, of the restiform columns which ascend to form the 
cei'ebellum. Magendie has described f precisely the same re- 
sults. A duck, whose cerebellum had been destroyed, could 
swim only backwards. In the course of his experimental lec- 
tures, Magendie, having removed the cerebellum in several rab- 
bits, demonstrated to his class the phenomena of retrograde 
movement, exactly as they have been recorded by Fodera. It 
is, then, impossible to regard the conclusions of Flourens as 

* Journal de Physique, July 1823. f Ibid. toin. iii. p. 157. 

70 THIRD REPORT — 1883. 

fully established, opposed as they are by those of so skilful an 
experimenter as Magendie. Indeed, while Flourens conceives 
the cei'ebellum to preside over motion, MM. Foville and Pinel 
Grandchamps attribute to it the directly opposite function of 
sensation : and this doctrine seems to derive some support from 
anatomical disposition ; for it has been proved by Tiedemann 
that the cerebellum is nothing more than an expansion or pro- 
longation of the corpora restiformia, and posterior columns of 
the spinal medulla, which columns have been shown by Sir 
Charles Bell to have the office of conveying sensations. But 
it is not the less true that all recent experiments, even those of 
Fodera and Magendie, point to some connexion between the 
cerebellum and the power of voluntary motion. In the present 
state of our knowledge it would be unsafe to contend for more 
than the probable existence of some such general relation. 

This, then, is all that seems deserving of confidence respect- 
ing the functions of the cerebellum itself. But there are some 
singular phenomena which, though residing in other structures 
more or less near to the cerebellum, are so analogous to those 
already described as to call for notice in this place. Magendie 
has described * the results of injury to the crura cerebelli of a 
rabbit. Complete division of the right crus was followed by 
rapid and incessant rotation of the body upon its own axis, from 
left to right. This singular motion having continued two hours, 
Magendie placed the rabbit in a basket containing hay. On 
visiting it the following day he was surprised to find the animal 
still turning round as before, and completely enveloped in hay. 
The eyes were rigidly fixed in different lines; that of the injured 
side being directed forwards and downwards, that of the other 
side backwards and upwards. If both crura were divided, no 
motion followed. Magendie hence concluded that these ner- 
vous cords are the conductors of impulsive forces which coun- 
terbalance one another, and that from the equilibrium of these 
two forces result the power of standing, and even of maintaining 
a state of rest, and of executing the diflTerent voluntary motions. 
The inquiry naturally presented itself, whether these forces 
are inherent in the crura themselves, or emanate from the cere- 
bellum or some other source. To determine this question, 
portions of substance were removed from both sides of the 
cerebellum, but unequally, so as to leave intact | on the left 
side and ^ only on the right. The animal rolled towards the 
right side, and its eyes were fixed in the manner already de- 
scribed. But the left crus being divided, the animal rolled to 

* Journal tie Physiologie, torn. iv. 399. 


the left side. Hence it appears that section of the crus lias 
more influence over the lateral rotation of the hotly than injury 
of the cerebellum itself; and that the impulsive force does not 
belong (at least exclusively) to the cerebellum. Wlien the cere- 
bellum was divided precisely in the median line, the animal 
seemed suspended between two opposing forces, sometimes in- 
clining towards one side, as if about to fall, and again thrown 
suddenly back to the opposite side. Its eyes were singularly 
agitated, and seemed about to start from the orbits. Similar 
movements followed division of the continuous fibres in the 
pons Varolii. Serres has described a case of similar rotatory 
motion occurring in the human subject. A shoemaker ha- 
bituated to excess in alcoholic liquors, after great intemperance 
was seized with an irresistible disposition to turn round upon 
his own axis, and continued to move so till death ensued. On 
inspecting the brain, one of the crura cerebelli was found much 
diseased, and this was the only alteration of structure visible 
in any part of the nervous system. 

M, Flourens has published in a recent volume of the Me- 
moir es de VAcad^mie des Sciences* a description of some 
striking abnormal motions which followed the division of the 
semicircular canals of the ears of birds. Though these organs 
have no anatomical relation to the cerebrum or cerebellum, the 
altered motions resulting from their division are so analogous 
to those observed by Magendie after lesions of the corpora 
striata and crura, that they may be most conveniently described 
in the same section. Two of the semicircular canals are ver- 
tical, and one horizontal. Division of the horizontal canals on 
each side occasioned a rapid horizontal movement of the head 
from right to left, and back again, and loss of the power of 
maintaining an equilibrium, except when standing, or when 
perfectly motionless. There was also the same singular rota- 
tion of the animal round its own axis which follows injury of 
the crura cerebelli. Section of the inferior vertical canal on 
both sides produced violent vertical movements of the head, 
with loss of equilibrium in walking or flying. There was in this 
case no rotation of the body upon itself, but the bird fell back- 
wards, and remained lying on its back. When the superior 
vertical canals were divided, the same phenomena were ob- 
served as in section of the inferior, except that the bird fell 
forward on its head, instead of backward. All the canals, both 
vertical and horizontal, having been divided, in another pigeon, 
violent and iwegular motions in all directions ensued. When, 

* torn. ix. p. 454. 

12 THIRD REPORT — 1833. 

however, the bony canals were so cautiously divided as to leave 
their internal membranous investment uninjured, these ab- 
normal motions were not produced. It is, therefore, in these 
membranes, or rather in the expansion of the acoustic nerve 
which overspreads them, that the cause of this phenomenon 
must reside. No explanation is proposed by Flourens of the 
control thus exercised by a nerve supposed to minister exclu- 
sively to the sense of hearing, over actions so entirely opposite 
in character. It is remarkable that the irregular movements 
should observe the same direction in their course as the canals, 
by the section of which they are induced. Thus the direction 
of the inferior vertical canal is posterior, that of the superior 
is anterior, corresponding perfectly with the directions of the 
abnormal motions. 

Medulla Oblongata. — The medulla oblongata, or "bulbe 
rachidien," is reducible into six columns, or three pairs, viz. 
two anterior or pyramidal, which partially decussate, two mid- 
dle or olivary, and two posterior or restiform, which proceed 
forwards without crossing. It is continuous in structure with the 
spinal marrow, and enjoys, by virtue of this relation, the same 
function of propagating motion and sensation. But it is distin- 
guished from the spinal medulla by special and higher attributes, 
being endowed with the faculty of originating motions, as well 
as with that of regulating and conducting them. The medulla 
oblongata, with the cerebrum and cerebellum, constitute, in short, 
according to Flourens *, those portions of the nervous system 
which exercise their functions "spontaneously or primordial- 
ly," and which originate and preside over the vital actions of 
the subordinate parts. To this latter order of parts, which re- 
quire an exciting or regulating influence, belongs the spinal 
medulla. In the superior class, Flourens seems to assign even 
a higher place to the medulla oblongata than to the cerebrum 
or cerebellum. For the cerebrum, he observes, may act without 
the cerebellum; and this latter organ continues to regulate the 
motions of the body after removal of the cerebrum. But the 
functions of neither cerebrum nor cerebellum survive the destruc- 
tion of the medulla oblongata, which seems to be the common 
bond and central knot combining all the individual parts of the 
nervous system into one whole. 

The medulla oblongata was regarded by Legallois as the 
mainspring or "premier mobile" of the inspiratory movements. 
He repeated before a Commission of the Institute of France the 
leading experiments on which his opinion rested f. In a rabbit 

* Memoires de I'Academie des Sciences, torn. ix. p. 478. 
t CEuvres de Legallois, torn. i. p. 247. 


five or six clays old, the larynx was detached from the os hy- 
oides and the glottis exposed to view. The brain and cere- 
bellum were then extracted without arresting the inspirations, 
which were marked by four simultaneous motions,' — a gaping of 
the lips, an opening of the glottis, the elevation of the ribs, 
and the contraction of the diaphragm. Legallois next removed 
the medulla oblongata, when all these motions ceased together. 
In a second rabbit, instead of extracting at once the entire me- 
dulla, it was cut away in successive thin slices. The four in- 
spiratory movements continued after the removal of the three 
first slices, but ceased after the fourth. It was found that the 
fourth had reached the origin of the eighth pair of nerves. If, 
instead of destroying the part in which this motive influence 
resides, it be simply prevented from communicating with the 
muscles which are subservient to inspiration, a similar effect 
ought to be produced. Now it is obvious that the medulla 
oblongata must transmit its influence to the muscles which 
raise the ribs, through the medium of the intercostal nerves, 
and therefore of the spinal marrow, and to the diaphragm 
through the phrenic nerves, and to these through the spinal 
marrow. In another rabbit, therefore, the medulla spinalis was 
cut across about the level of the seventh cervical vertebra. 
The effect of this operation was to arrest the elevation of the 
ribs, the other three inspiratory motions still continuing. A 
second section was made near the first cervical vertebra, and 
consequently above the origin of the phrenic, with the effect of 
suspending the contraction of the diaphragm. The par vagum 
was next divided in the neck, and the opening of the glottis 
ceased. There remained then, of the four inspiratory move^ 
ments, only the gaping of the lips, which, however, was suflS- 
cient to attest that the medulla oblongata still retained the 
power of producing them all. This power had ceased to call 
forth the other three motions, only because it no longer had 
communication with their organs. 

M. Flourens, in a recent memoir already referred to *, has 
confirmed and extended the views first announced by Legal- 
lois. He has distinctly traced the comparative action of the 
medulla spinalis and oblongata, on respiration, in the four classes 
of vertebrated animals. In birds, he found that all the lumbar 
and the posterior dorsal medulla might be destroyed without 
impeding the respiratory function, though it was arrested by 
removal of the costal medulla. In the mammalia the costal also 

* Memoir cs de V Academie, torn. ix. 1830. 

74 THIRD REPORT — 1833. 

might be removed, for though the raisjng of the ribs ceased, 
the action of the diaphragm continued as long as the origin of 
the phrenic nerve remained uninjured. In frogs, all the spinal 
medulla may be destroyed, except the portion, whence spring the 
nerves supplying the hyoideal apparatus. Every part of the 
spinal marrow may be removed in fishes without affecting re- 
spiration ; for all the nerves distributed to the respiratory organs 
of fishes have their origin in the medulla oblongata. It is 
hence apparent that the spinal marrow exercises only a variable 
and relative action on the I'espiratory function, in the different 
classes of vertebrated animals. In descending from the higher 
to the lower points of this scale, the spinal marrow is seen pro- 
gressively to disengage itself from cooperation in these move- 
ments, while the medulla oblongata tends more and more to 
concentrate them in itself, till in fishes the proper functions of 
the two meduUge show themselves completely distinct, the spinal 
ministering to locomotion and sensation, and the oblongata to re- 
spiration. The medulla oblongata is, then, the "premier mo- 
teur " or the exciting and regulating principle of the inspiratory 
movements in all classes of vertebrated animals ; besides par- 
ticipating, by virtue of its continuity with the spinal marrow, in 
the proper functions of that organ. From a second series of 
experiments, M. Flourens concludes that there exists a point 
in the nervous centres at which the section of those centres 
produces the sudden annihilation of all the inspiratory move- 
ments ; and that this point corresponds with the origin of the 
eighth pair of nerves, commencing immediately above, and ending 
a little below, that origin, — a result precisely agx'eeing with that 
obtained by Legallois. 

Spinal Marrow. — It is apparent, that the functions of the 
three grand divisions of the nervous system, already described, 
have not yet been distinctly and fully ascertained. Our know- 
ledge of those, which next fall under survey, is more definite 
and substantial. The vital offices of the spinal medulla — re- 
garded by Legallois as the mainspring of life, and as alone re- 
gulating the actions of the heart and nobler organs, — are now 
reduced to conveying to the muscles the motive impulse of voli- 
tion, and to propagating to the sensorium commune, impressions 
made on the external senses. It is not invested with the power 
possessed by the cerebrum and cerebellum, and perhaps by the 
medulla oblongata, of spontaneously originating muscular mo- 
tions. It is mainly, if not exclusively, a conductor; a medium 
of communication between the brain and the external instru- 
ments of locomotion and sensation. Flourens, indeed, conjee- 


tures that it also has the office of associating the partial con- 
tractions of individual muscles into "mouvemens d'ensemble," 
necessary to the regular motions of the limbs. 

Before recording what is known of the spinal cord itself, it 
will be proper to advert to some recent experiments of Magen- 
die on the serous fluid in which it is immersed. It would 
appear that a quantity of liquid, varying from two to five 
ounces in the human subject, is always interposed between 
the arachnoid tunic and the pia mater, or proper membrane of 
the cord. The intermembranous bag, occupied by this fluid, 
communicates with the ventricular cavities at the calamus scrip- 
torius by a round aperture, often large and patent in hydroce- 
phalic subjects. Magendie has therefore named this serous 
liquid ' cerebro-spinal '. In living animals, it issues in a stream 
from a puncture of the arachnoid. Its removal occasions great 
nervous agitation, and symptoms resembling those of canine 
madness. The sudden increase of its quantity induces coma. 
Its presence seems essential to the undisturbed and natural ex- 
ercise of the nervous functions ; and this influence pi-obably is 
dependent upon its pressure, temperature and chemical con- 
stitution, since any variation of these conditions is followed by 
the phenomena of nervous disorder. 

The great medullary cord is divided by a double furrow into 
two lateral halves ; and each of these is again subdivided by the 
insertions of the ligamenta dentata into two columns, one pos- 
terior and one anterior. It has been long known that section 
of any part of the spinal marrow excludes from intercourse with 
the brain all those parts of the body, which derive their nerves 
from the cylinder of medulla below the point of injury. The 
muscles, so supplied, are no longer obedient to the control of 
the will, and the tegumentary membranes similarly situated en- 
tirely lose their sensibility. This interruption of the relations 
which subsist between the central seat of volition and sensation, 
and the rest of the body, whether due to direct injury of the 
great nervous masses or communicating nerves, or produced by 
the pressure of extravasated fluids, by morbid growths, or by 
various poisonous matters, constitutes the condition known by 
the name ' pai-alysis'. In cases of this kind it is frequently ob- 
served that the powers of sensation and locomotion are simulta- 
neously impaired or destroyed. But examples are not want- 
ing, even in the earliest clinical records, of the total loss of one 
of those faculties with perfect integrity of the other. Such facts 
naturally suggested the belief that the power of propagating 
sensations, and that of conveying motive impressions, resided 
in distinct portions of the nervous system. This opinion, how- 

76 THIRD REPORT — 1833. 

ever, remained mere matter of conjecture until a recent period, 
when it was unequivocally established by Sir Charles Bell. 
From the original experiments of that most distinguished phy- 
siologist, repeated and confirmed by Magendie, it follows that 
the faculty of conducting sensations resides exclusively in the 
two posterior columns of the medulla, while that of communica- 
ting to the muscular system the motive stimulus impressed by 
volition is the attribute of the two anterior columns. The same 
limitation of function is found in the nervous roots which spring 
from these separate columns. Thus each spinal nerve is fur- 
nished Avitli a double series of roots, one set of which have their 
origin in the anterior medullary column, and one in the pos- 
terior. The spinal nerves are, in consequence of this anatomi- 
cal composition, nerves of twofold function, containing in the 
same sheath distinct continuous filaments from both columns, 
and exercising, in the parts to which they are distributed, the 
double oflice of conductors of motion and sensation. It will 
afterwards appear, in our history of individual nerves, that all 
those which spring from the brain, except the fifth and eighth 
pairs, possess only a single function. 

Sufficient experimental proof of the foregoing propositions 
has been furnished by Sir Charles Bell and by M. Magendie. 
Thus, division of the posterior roots of the spinal nerves is uni- 
formly followed by total absence of feeling in the parts of the 
body to which the injured nerves are distributed, while their 
motive power remains undiminished. Magendie has further 
observed, that if the medullary canal be laid open, and the two 
posterior cords be touched or pricked slightly, there is instant 
expression of intense sufi^ering; whereas, if the same or a greater 
amount of irritation be applied to the anterior columns, there 
are scarcely any signs of excited sensibility. The central parts 
of the medulla seem also nearly impassable *. They may be 
touched, and even lacei*ated, according to Magendie, without 
exciting pain, if precautions are taken to avoid the surrounding 
medullary substance. In general, the properties of the spinal 
marrow, and especially its sensibility, seem to reside mainly on 
its surface ; for slight contact, even of the vascular membranes 
covering the posterior columns, caused acute pain. 

The first experiment of Sir C. Bell consisted in laying open 
the &pinal canal of a living rabbit, and dividing the posterior 
roots of the nerves that supply the lower limbs. The animal 
was able to crawl. In his second trial he first stunned the 
rabbit, and then exposed the spinal marrow. On irritating the 

* Annales de Chimie et de Physique, torn, xxiii. p. 436. 


posterior roots, no motion was induced in any part of the mus- 
cular frame ; but on grasping the anterior roots, each touch of 
the forceps was followed by a corresponding contraction of the 
muscles supplied by the irritated nerve. Magendie has de- 
scribed* the following experiments, which he has since declared 
were made without any knowledge of the prior ones of Sir C. 
Bell. The subjects chosen for the operation were puppies 
about six weeks old; for in these it was easy to cut with a sharp 
scalpel through the vertebrae and to expose the medulla. In the 
first, the posterior roots of the lumbar and sacral nerves were 
divided, and the wound closed : violent pressure, and even prick- 
ing with a shai'p instrument, awakened no sensation in the limb 
supplied by the nerves which had been cut ; but its motive power 
was uninjured. A second and a third trial gave the same re- 
sults. Magendie then divided in another animal, though with 
some difficulty, the anterior roots of the same nerves on one 
side. The hind limb became flaccid and entirely motionless, 
though it preserved its sensibility. Both the anterior and pos- 
terior roots were cut in the same subject with destruction of 
motion and sensation. In a second paperf Magendie has re- 
lated the following additional facts. The introduction of nux 
vomica into the animal economy is well known to give rise to 
violent tetanic convulsions of the whole muscular system. This 
property was made available as a test of the functions of the 
separate orders of nervous I'oots. It was found that, while all 
the other muscles of the body were agitated, when under the 
influence of this poison, by violent spasmodic contractions, the 
limb, supplied by nerves whose anterior roots had been pre- 
viously divided, remained supple and motionless. But when 
the posterior roots only had been cut, the tetanic spasms were 
universal. It would seem, however, that the seats of the two 
faculties of conducting motion and sensation are not strictly 
insulated by exact anatomical lines, but that they rather pass 
into each other with rapidly decreasing intensity. Thus irri- 
tation of the anterior roots, when connected with the medvdla, 
gives birth, along with motive phenomena, to some evidences 
of sensibility ; and, vice versa, stimuli applied to the posterior 
roots, also undivided, occasion slight muscular contractions. 
In this last case it is, indeed, probable that the irritation tra- 
velled from the posterior roots upwards to the brain in the ac- 
customed channel, and gave rise to a perception of pain, which 
prompted the muscular effort. Indeed, after division of the 
posterior nervous roots, ordinary stimulants, applied to the 

• Journal de Physiologie, torn. ii. p. 276. August 1822. f Ibhl torn. ii. p. 366. 

78 THIRD REPORT — 1833. 

ends not connected with the medulla, produced no apparent 
eftects ; though the galvanic fluid directed upon either order 
of roots gave rise to muscular contractions. These were more 
complete and energetic when the anterior roots were the sub- 
jects of the experiment. 

Besides the evidence thus obtained by direct experiments on 
living animals, several important facts have been gathered from 
the pathology of the nervous system in man. These consist of 
cases of insulated paralysis of either motion or feeling, referred 
to the changes in structure obsei'ved after death. Sir Charles 
Bell has himself recorded several examples of this kind strongly 
confirming his experimental results ; and others of similar ten- 
dency are scattered through the successive volumes of Magen- 
die's Journal*. But it must be admitted, that evidence of this 
kind is seldom distinct and conclusive. The structural changes, 
induced by disease, are rarely so circumscribed in seat and 
extent as to represent adequately the operations of the scalpel; 
and often when they are thus isolated within anatomical bound- 
ing lines, they affect, by pressure, or by the spread of the same 
morbid process, in a degree too slight to leave decided traces, 
the functions of contiguous parts, thus clouding the judgments 
of the best pathologists, and invalidating their inferences. 
There is, however, a very I'emarkable case described by Pro- 
fessor Royer CoUard, to which these objections do not apply. 
Sprevale, an invalided soldier, was upwards of seventeen years 
the subject of medical observation in the Maison de Sante of 
Charenton. This individual remained for the last seven years 
of his life with the legs and thighs permanently crossed, and 
totally incapable of motion, though retaining their sensibility. 
On opening after death the spinal canal, there was found the 
pultaceous softening {ramolUssement) of the whole anterior part 
of the medulla, and of almost the whole of the fibrous cords 
which form it. The anterior roots of the spinal nerves had 
also lost their accustomed consistency; while the posterior sur- 
face of the spinal cord, and its investing membrane, were healthy. 
Several of the cases observed by Sir Charles Bell furnish also 
unequivocal proof of the soundness of the views developed by 

There exist, indeed, few truths in physiology established 
on so wide and solid a basis of experimental research and pa- 
thological observation, as those deduced by Sir Charles Bell, 
the original discoverer, and by Magendie, his successor in the 
path of inquiry, respecting the offices of the spinal medulla. 

* See in particular Dr. Rullier's case, torn. iii. p. 1 73 ; and Dr. KorefTs, 
torn. iv. p. 376. 


This organ may now be regarded as mainly, if not solely, a 
medium of intercourse between the external world and the brain, 
and again between the brain and the voluntary muscles, its two 
anterior columns being subservient to motion, its two posterior 
to sensation. In the present state of our knowledge it would 
be fruitless to try to penetrate into the minute philosophy of 
these actions : but it seems probable, from recent discoveries 
on the ultimate anatomy of tissue, that these actions are mole- 
cular, having their place in the globular elements, into which all 
living textures are resolvable by microscopic analysis ; — that 
the physical changes, e. g. impressed by external objects on the 
delicate net-work of nerve which invests the tegumentary mem- 
branes and open cavities, are propagated thence, from particle 
to particle, along the continuous filaments, to their origins in the 
posterior spinal columns, and thence to the central point, where 
they become objects of perception; — and that the motive sti- 
mulus of volition is similarly transmitted down the anterior co- 
lumns and nerves, to the organs of locomotion. Indeed, it is a 
legitimate inference from Sir Charles Bell's discoveries, that a 
simple nervous filament, or medullary column, can only propa- 
gate an impression in one line of direction, viz. either towards 
or from the central seat of perception and of will ; and this cu- 
rious law of nervous actions would seem to point at some in- 
sensible molecular motion as their essential condition. 

It remains to investigate the arguments which have been 
supposed to prove the residence in the spinal marrow of the 
power of originating and controlling the actions of the heart. 
This question has been matter of eager controversy, from its 
bearing vipon the general relations of nerve and muscle. With- 
out prejudging this latter topic, it may simplify its future con- 
sideration, and will at the same time be more consistent with 
strict arrangement, to state here merely the facts which have 
reference to the spinal medulla. 

The woi'k of Legallois, entitled ^^ Experiences sur le Prin- 
cipe de la Vie, notamment sur celni des Mouvemens du Cceur 
et si/r le Siege de ce Principe* ," was the first remarkable essay 
on the relations between the heart and the spinal cord. It will, 
however, be sufficient to allude in general terms to the conclu- 
sions of Legallois, since they have been entirely subverted by 
the subsequent researches of Dr. Wilson Philip and M. Flou- 
rens. Legallois's main doctrine was, that the principle which 
animates each part of the body resides in that part of the spinal 
medulla whence its nerves have their origin; and that it is also 

• CEuvres de Legallois, torn. i. pp. 97, 99, &c. 

80 THIRD REPORT — 1853. 

from the spinal cord that the heart derives the principle of its 
life and its motion*. The experimental proof supposed to 
establish these propositions consisted in destroying in different 
rabbits portions of the cervical, dorsal and lumbar medulla. 
Cessation of the heart's action Avas affirmed to be the constant 
result of the operation ; but even in some of Legallois's own ex- 
periments f, the motions of the heart continued after consider- 
able injury had been inflicted on the spinal cord, and especially 
on its lower divisions. Still more unequivocal is the evidence 
that has been advanced by Dr. Wilson Philip, in his Inquiry 
into the Laivs of the Vital Functions. His experiments, which 
were very numerous and judiciously varied, show that the cir- 
culation continues long after entire removal of the spinal mar- 
row, and that by artificially maintaining respiration, the motions 
of the heart may be almost indefinitely prolonged. Flourens, 
in the 10th vol. of the Mem. de lAcademieX, ^^^^ lately con- 
firmed Dr. Philip's views : he has shown that the circulation is 
entirely independent of the spinal marrow. The influence ap- 
parently exerted is only secondary, being due to the suspension 
of the respiratory movements. Thus all those portions of the 
spinal marrow which can be destroyed in the different classes 
of animals without arresting respiration, may be removed with- 
out affecting the cii'culation. In fishes and frogs the entire 
spinal cord may be destroyed without checking the heart's mo- 
tions, because in these classes the medulla oblongata presides 
exclusively over the respiratory function. 

Nerves. — The classification of nerves, which is most conve- 
nient to the physiologist, is based upon their vital properties or 
functions. Such an arrangement wovdd distribute them into — 
1, nerves of motion; 2, nerves both of motion and sensation; 
3, the nerves ministering to the senses of sight, smell and hear- 
ing ; and 4, the ganglionic system, or, according to Bichat, 
nerves of organic life. Sir Charles Bell has added a fifth class, 
comprising nerves which he supposes are dedicated to the 
respiratory motions. But it will afterwards appear, that the 
existence of an exclusive system of respiratory nerves is not 
supported by sufficient evidence. 

The first class of nerves exercising the single office of con- 
veying motion comprehends the 3rd, 4th, 6th, portio dura of 
the 7th, the 9th, and perhaps two divisions of the 8th, viz. the 
glossopharyngeal and spinal accessory. Mr. H. Mayo's expe- 
riments detailed in his Anatomical and Physiological Commen- 
taries, No. 1 1 . (and Journal de Physique, tom. iii.) throw much 

* p. 259. t pp. 100^ 101, 105. + p. 625. 


light on the functions of several of these nerves. The motions of 
the iris, he shows, require the integrity of the third pair, division 
of these nerves being always followed by full dilatation of the 
pupils, which cease to be obedient to the stimulus of light. If the 
divided end of the nerve communicating with the eye be pinched 
by the forceps, the iris contracts. Hence it is apparent that dimi- 
nution of the aperture of the pupil is the result of action, and 
dilatation of the pupil the result of relaxation, of the iris. Flou- 
rens has shown that complete extirpation of the tubercula quad- 
rigemina also paralyses the iris, and that irritation of those bo- 
dies excites its contractions. The same effect is noticed by Mayo 
to arise from division or irritation of the optic nerve. He divided 
the optic nerves within the cranium of a pigeon immediately 
after decapitation. When the end of the nerve connected with 
the ball of the eye was seized in the forceps, no action ensued; 
but when the end attached to the brain was irritated, the iris 
immediately contracted. These several experiments clearly in- 
dicate the dependence of the iris upon the optic nerve, upon 
the tubercula from which one root of that nerve springs, and 
upon the third pair. The stimulus of light impinges upon the 
retina, is conveyed along the optic nerve through the tubercle 
to the sensorivmi, whence the motive impression is propagated 
to the iris by the third encephalic nerve. 

It is not so easy to define the precise mode of action of the 
pathetici, or fourth pair of nerves. Sir Charles Bell * supposes 
that they are destined " to provide for the insensible and in- 
stinctive rolling of the eyeball, and to associate this motion of 
the eyeball with the winking motions of the eyelids." He even 
conjectures that "the influence of the fourth nerve is, on cer- 
tain occasions, to cause a relaxation of the muscle to which it 
goes." It is certain, however, from its exclusive distribution to 
the superior oblique muscle, that the fourth is a nerve of motion. 
The sixth nerve is also a nerve of voluntary motion, and is sent 
to the rectus externus of the eyeball. 

Sir Charles Bell has placed the portio dura of the seventh 
pair among his respiratory nerves. There is, however, no doubt 
that it is simply a motive nerve, and that it is indeed the only 
nerve of motion, which supplies all the muscles of the face, 
except those of the lower jaw and palate. Division of this 
nerve occasions no expression of pain, according to Bell; but 
Mayo's experience is opposed to this absence of sensibility f. 
"The motion of the nostril of the same side instantly ceased, 

* Natural System of Nerves, p. 3.58. 

t See Mr. H. Mayo's Anatomical and Pliijsioloffical Commentaries, Part I.; 
and Outlines of Human PLy^ioloijij, 2nd edit., p. 3;34, 
1 S.Jo. G 

82 TIIIUD REPORT — 1833. 

after its section in an ass *, and that side of the face remained 
at rest and placid during the highest excitement of the other 
parts of the respiratory organs." These and similar observa- 
tions are all consistent with the opinion, that the seventh is 
simply a nerve of voluntary motion. It will afterwards appear 
that it has no claim to any further endowment. 

Mr. Herbert Mayo infers from his experiments, that the three 
divisions of the eighth pair are all nerves both of motion and 
sensation. Thus the glossopharyngeus is a nerve of motion to 
the pharynx, and perhaps of sensibility to the tongue. He 
observed that " on irritating the glossopharyngeal nerve in an 
animal recently killed, the muscular fibres about the pharynx 
acted, but not those of the tongue f." Irritation of the spinal 
accessory produced both muscular contractions and pain. The 
par vagum, he conceives, bestows sensibility on the membrane 
of the larynx, besides conveying the motive stimulus to its 
muscles. This nerve has been the subject of experiment from 
the earliest times, and Legallois has minutely described the 
results obtained by successive inquirers X- These were singu- 
larly discordant, and gave origin to the most opposite theories 
of the mode of action of the par vagum. In the greater number 
of experiments, section of this nerve was followed, after a longer 
or shorter interval, by death. Piccolhomini contended that the 
division of the nerve was fatal from its arresting the move- 
ments of the heart, and after him Willis supported the same 
doctrine. By Haller, on the contrary, the cause of death was 
sought in disturbance of the digestive fvmctions. Bichat and 
Dupuytren seem to have been the first to obtain a glimpse of the 
true seat of injury. The former remarked that the respiration 
became very laborious after section of the nerve, and Dupuytren 
distinctly traced death to asphyxia. Legallois has established 
by numerous experiments the accuracy of this last view. He 
has shown that in very young animals death is the immediate 
consequence of the operation of cutting either the par vagum 
or its recurrent branch, and that the suddenness of the effect 
is due to the narrowness of the aperture of the glottis in early 
age. In adult animals, the asphyxia is induced by the effusion 
of serous fluids and ropy discoloured mucus into the bronchial 
tubes and air-cells. More recently. Dr. Wilson Philip has prac- 
tised the section of the par vagum with an especial reference 
to its influence upon digestion. He divided the nerve below 
the origin of the inferior laryngeal branch, as in this case the 

« pp. 106, 107. t Outlines of Human Physiology, 2nd edit., p. 337. 

+ (Euvres, p. 1 54 et seq. 


dyspnoea is much less considerable than when the wound is in- 
flicted on the higher portion*. It was found, in all these trials, 
that food introduced into the stomach after the operation re- 
mained wholly undigested. Hence Dr. Philip infers the de- 
pendence of secretion upon nervous influence, a conclusion, it 
has been remarked by Dr. Alison, not logically deducible from 
the experimental dataf. 

The par vagum cannot then, it is obvious, be included in the 
class of nerves subservient solely to motion ; and it is even 
doubtful whether the other two divisions of the eighth pair are 
not also endowed with sensibility. Respecting the function of 
the ninth, or lingvial, there is, however, no place for hesitation. 
It has been experimentally proved by Mr. Mayo to supply the 
muscles of the tongue ; though he also asserts that pinching it 
with the forceps excited pain. Three of these nerves, the third, 
sixth, and ninth, arise, it was first remarked by Sir Charles 
Bell, from a tract of medullary matter continuous with the an- 
terior column of the spinal marrow: and hence their exclusive 
oflice of conducting motive impressions. 

II. There are thirty-two pairs of nerves of similar anatomical 
origin and composition, which possess the twofold office of com- 
municating motion and sensation. Of these, all excepting one 
(the fifth pair of the cerebral nerves) spring from the spinal 
marrow. These thirty-one pairs are precisely analogous in 
formation, being all constituted of two distinct series of roots, 
one from the anterior column, and one from the posterior column 
of the spinal marrow. The posterior funiculi collected together 
form a ganglion, seated just before this root is joined by the 
anterior root. It has been already stated that the power of 
propagating sensation resides in the posterior column, and in 
the nervous roots arising from it, and that the motive faculty 
has its seat in the anterior column and roots. The evidence, 
also, supplied by Bell and Magendie, that the spinal nerves are 
hence nerves of double office, has been fully detailed. It re- 
mains, then, to establish the title of the fifth pair of cerebral 
nerves to be included in the same class with the spinal nerves. 

The analogy in structure and mode of origin between the 
fifth pair and the nerves of the spine has been long matter of 
observation. Prochaska has thus distinctly noticed it in a pas- 
sage of his Essay De Structurd Nervorum, published in 1779, 
first pointed out to me by my friend Dr. Holme : " Quare 
omnium cerebri nervorum, solum quintum par post ortum suum 

* Experimental Inquirt/, 3rd edit., p. 109. 

t Dr. Alison, Journal of Science, vol. ix. p. 106. 


84 THIRD REPORT — 1833. 

more nervorum spinalium, ganglion semilunare dictum, facere 
debet? sub quo peculiaris funiculorum fasciculus ad tevtium 
quinti paris ramum, maxillarem inferiorem dictum, properat, 
insalutato ganglio semilunari, ad similitudinem radicum ante- 
riorum nervorum spinalium ?" Siimmerring has also pointed 
out with equal clearness the resemblance in distribution be- 
tween the smaller root of the fifth and the anterior roots of the 
spinal nerves. But Sir Charles Bell was the first to establish 
the identity of their functions, and to arrange them prominently 
in the same natural division. His experiment consisted in 
exposing the fifth pair at its root, in an ass, the moment the 
animal was killed. " On irritating the nerve, the muscles of the 
jaw acted, and the jaw was closed with a snap. On dividing 
the root of the nerve in a living animal, the jaw fell relaxed." 
In another experiment the superior maxillary branch of the 
fifth nerve was exposed. " Touching this nerve gave acute 

pain ; the side of the lip was observed to hang low, and 

it was dragged to the other side." Sir Charles Bell concluded 
that the fifth nerve and its branches are endowed with the attri- 
butes of motion and sensation. This, though correct as regards 
the nerve itself, viewed as a whole, is strictly true only of the 
lowest of its three divisions, viz. the inferior maxillary. The 
ophthalmic and the superior maxillary, the subject of the last 
experiment, are nerves simply of sensation. Mr. Herbert Mayo 
in the Essay already referred to, has pointed out this error, 
and has defined with minute precision the relative offices of the 
fifth and seventh nerves. By a careful dissection of the fifth 
nerve he found that the anterior branch, or smaller root, which 
goes, as Prochaska was aware, entirely to the inferior maxillary, 
is distributed exclusively to the circumflexus palati, the ptery- 
goids, and temporal and masseter muscles. He observed that sec- 
tion of the supra and infra orbitar branches, and of the inferior 
maxillary, near the foramina, whence they emerge, induces loss 
of sensation in the corresponding parts of the face. It may then 
be regarded as fully proved that the trigeminus or fifth pair is 
the nerve which bestows sensation on the face and its appen- 
dages, and motion only on the muscles connected with the lower 
jaw. The other muscles of the face derive their motive power 
from the portio dura of the seventh nerve. 

M. Magendie has also published several memoirs on the 
functions of the fifth pair. In these he attempts to prove 
that the olfactory nerve is not the nerve of smell ; that the op- 
tic is but partially the nerve of vision ; and that the auditory is 
not the principal nerve of hearing. It is in the fifth pair that 
he supposes all these distinct and special endowments to reside. 


But the experimental proof will be found to be singularly in- 
conclusive. The olfactory nerves were entirely destroyed in a 
dog. After the operation it continued sensible to strong odours, 
as of ammonia, acetic acid, or essential oil of lavender ; and the 
introduction of a probe into the nasal cavity excited the same 
motions and pain as in an unmutilated dog. The fifth pair was 
then divided in several young animals, the olfactory being left 
entire. All signs of the perception of strongly odorous sub- 
stances, as sneezing, rubbing the nose, or turning away the head, 
entirely disappeared. From these facts Magendie infers that 
the seat of the sensations of smell is in the fifth, and not in the 
first pair of nerves. It is obvious that Magendie has con- 
founded two modes of sensation, which are essentially distinct 
in their nature and in their organic seat, viz. the true percep- 
tions of smell, and the common sensibility of the nasal passages. 
The phenomena, which he observed to cease after the section 
of the fifth nerve, are the results of simple irritation of the pi- 
tuitary membrane, and are manifestly wholly unconnected with 
the sense of smelUng, since they are producible by all powerful 
chemical agents, even though inodorous, as, for example, by 
sulphuric acid. No proof has been given that the true olfac- 
tory perceptions do not survive the destruction of the fifth pair. 
Indeed, in a subsequent paper, Magendie confesses that the 
loss of sensibility in the nasal membrane, after section of the 
fifth, does not prove the residence of the sense of smell in the 
branches of that nerve ; but merely that the olfactory nerve re- 
quires, for its perfect action, the cooperation of the fifth pair, 
and that it possesses only a special sensibility to odorous parti- 

There is even less ground for supposing that the fifth pair is 
in any degree subservient to the senses of sight and hearing. 
After cutting this nerve on one side, the flame of a torch was 
suddenly brought near the eye, without inducing contraction of 
the pupil ; but the direct light of the sun caused the animal to 
close its eyelids. Thus the sensibility of the retina, though 
somewhat impaired, was not destroyed by division of the fifth 
pair. But section of the optic nerves was immediately followed 
by total blindness. In another rabbit Magendie divided the 
fifth pair on one side, and the optic nerve on the other. The 
animal, he states, was completely deprived of sight, though the 
eye, in which the fifth pair only had been cut, remained suscep- 
tible to the action of the solar rays. No evidence, however, is 
offered to show that the animal was entirely blind : on the con- 
trary, the only change observed, on approaching a torch to an 
vninjiired eye, was contraction of the iris ; and this we are told 

86 THIRD REPORT 1833. 

was actually observed in the eye of the side, on which the fifth 
nerve had been divided. 

Magendie has assigned another singular function to the fifth 
pair, viz. to preside over the nutrition of the eye. Twenty-four 
hours after section of this nerve, incipient opacity of the cornea 
was observed, which gradually increased till the cornea became 
as white as alabaster. There was also great vascularity of the 
conjunctiva extending to the iris, with secretion of pus, and for- 
mation of false membranes in the anterior chamber. About the 
eighth day, the cornea began to detach itself from the sclerotica, 
the centre ulcerated, and the humours of the eye finally escaped, 
leaving only a small tubercle in the orbit. In this experiment, 
the nerve had been divided in the temporal fossa, but when cut 
immediately after leaving the pons Varolii, the morbid changes 
were less marked, the movements of the globe of the eye were 
preserved, the inflammation was limited to the superior part of 
the eye, and the opacity occupied only a small segment of the 
circumference of the cornea. After division of the nerve near 
its origin in the medulla, no traces of disease were discoverable 
in the eye till the seventh day, and these symptoms never be- 
came very prominent. Several cases have been since recorded 
of structural disease of this nerve in the human subject, with 
the concomitant symptoms. That of Laine, described by Serres 
in the 4th vol. of Magendie's Journal, furnishes strong support 
to the views of Magendie *, 

A different explanation of this fact and of others which have 
a tendency to refer secretion and nutrition to the control of 
the nervous system has been proposed by Dr. Alison. Mucous 
surfaces are protected from the contact of air and foreign bo- 
dies by a copious secretion, which is evidently regulated in 
amount by their sensibility, since it is increased by any unusual 
iri'itation. This is especially true of the membrane of the eye. 
Now section of the fifth pair is known to paralyse the sensibi- 
lity of that organ, and the contact of air or other irritating body 
upon the insensible membrane, instead of inducing an aug- 
mented mucous discharge, will excite the inflammatory process 
described by Magendie. The disorder of the digestive func- 
tion f, which followed division of the par vagum in the experi- 
ments of Dr. Wilson Philip, and the ulceration of the coats of 
the bladder after injury of the lower part of the spinal marrow, 
are attributed by Dr. Alison to the same cause. 

The class of nerves which comprehends the fifth pair and 

* See also a case of destruction of the olfactory nerves, lom. v. 
•f Outlines of Phyniolngy, p. 71. 


the thirty-one pairs of spinal nerves, becomes, after the vniion 
of their roots, invested with a twofold endowment, and conti- 
nues so throughout their entire course and final distribution to 
the muscular tissue. It would appear, indeed, from a later 
paper of Sir Charles Bell*, that nerves of sensation, as well as 
of motion, are necessary to the perfect action of the voluntary 
muscles. "Between the brain and the muscles there is a circle 
of nerves ; one nerve conveys the influence from the brain to 
the muscle, another gives the sense of the condition of the 
muscle to the brain." In the case of the spinal nerves this 
circle of intercourse is at least probable ; but proof of its ne- 
cessity must be obtained, from observing the habitudes of those 
encephalic nerves, which minister exclusively to motion. Now 
it is found, on minute dissection, that the muscles of the eye- 
ball, which are supplied by the third, fourth and sixth motive 
nerves, also receive sensitive filaments from the ophthalmic 
branch of the fifth ; and that the muscles of the face, to which 
the portio dura is distributed, are also furnished with branches 
of sensation from the fifth. Sir Charles Bell has further shown 
that the muscles of the lower jaw, to which the motive im- 
pression is propagated by the muscular branch of the inferior 
maxillary, draw nervous supplies also from the ganglionic or 
sensitive branch of that division of the fifth pair. This com- 
plicated provision has its origin, he supposes, in its being " ne- 
cessary to the governance of the muscular frame that there 
should be consciousness of the state or degree of action of the 

III. The olfactory, auditory and optic nerves are gifted with 
a special sensibility to the otajects of the external senses, to 
which they respectively minister, Magendie seems to have 
been the first to prove, experimentally, that they do not also 
share the common or tactile sensibility. He exposed the olfac- 
tory nerves, and found that, like the hemispheres of the brain 
from which they spring, they are insensible to pressure, prick- 
ing, or even laceration. Strong ammonia was dropped upon 
them without eliciting any signs of feeling. The optic nerve, 
and its expansion on the retina, participate with the olfactory 
in this insensibility to stimulants. This was proved by Ma- 
gendie in the human subject as well as in animals. In perform- 
ing the operation of depressing the opaque lens, he repeatedly 
touched the retina in two diflPerent individuals without awaken- 
ing the slightest sensation. The portio mollis, or acoustic nerve, 
was also touched, pressed, and even torn without causing pain. 

• Philosophical Transactions, 1826, p. 163. 


IV. The functions of the ganglia, of the great sympatlietic 
nerve, and its intricate plexuses and anastomotic connexions, are 
matter, at present, of conjectui'e. Dr. Johnstone, in an Essay 
on the Use of the Ganglions, published in 1771, has described 
a few inconclusive experiments on the cardiac nerves. He 
supposes that " ganglions are the instruments by which the 
motions of the heart and intestines are rendered uniformly in- 
voluntary," — a notion which Sir Charles Bell has shown to be to- 
tally unsound. The best history of opinions, to which indeed 
our knowledge reduces itself, will be found in the physiological 
section of Lobstein's work, De Nervi Sympathetici Fabrica, 
Usu, et Morbis*. 

In the earliest of his communications to the Royal Society, 
as well as in his last work on the Nervous System ■!-, Sir Charles 
Bell has maintained the existence of a separate class of nerves, 
subservient to the regular and the associated actions of respira- 
tion. The origins of these nerves J "are in a line or series, and 
from a distinct column of the spinal marrow. Behind the corpus 
olivare, and anterior to that process, which descends from the 
cerebellum, called sometimes the corpus restiforme, a convex 
strip of medullary matter may be observed. From this tract of 
medullary matter, on the side of the medulla oblongata, arise, in 
succession from above downwards, the portio dura of the seventh 
nerve, the glossopharyngeus nerve, the nerve of the par vagum, 
the nervus ad par vagum accessorius, and, as I imagine, the 
phrenic and the external respiratory nerves." The fourth pair 
is also received into the same class. 

This doctrine of an exclusive system of respiratory nerves, 
associated in function by virtue of an anatomical relation of 
their roots, has not, as Sir Charles Bell seems himself aware §, 
received the concurrence of many intelligent physiologists of 
this counti'y or of the Continent. Mr. Herbert Mayo, in the ad- 
mirable Essay already referred to, was the first to point out the 
true relations of the fifth and seventh nerves. He has shown 
that the muscles of the face, excepting those already enumer- 
ated, which elevate the lower jaw, receive their motive nerves 
exclusively from the seventh, and consequently that this nerve 
must govern all their motions, voluntary as well as respiratory. 
But Dr. Alison, in his very elaborate paper || " On the Physiolo- 
gical Principle of Sympathy," has cast considerable doubts on 

* Paris 1823. t 4to, 1830. 

J The Nervous System of the Human Body, p. 129. 'Ito, 1830. 
§ Op. cit., p. 11;'). 

II Transacliom of Oie Medko-chirurykal Socidy of Edinburgh, 1826, vol. ii. 
p. 165. 


the soundness of this part of Sir Charles Bell's arrangement, as 
respects not only the individual nerves thus classed together, but 
even the general principle on which the entire system rests. 
The reasoning of Dr. Alison consists, first, in referring the 
phenomena of natural and excited respiration to the compre- 
hensive order of sympathetic actions. In these " the pheno- 
menon observed is, that on an irritation or stimulus being 
applied to one part of the body, the voluntary muscles of an- 
other, and often distant part, are thrown into action." Now 
it has been long since fully established by Dr. Whytt, that 
these associations in function cannot be referred to any con- 
nexions, either in origin or in course, of the nerves supplying 
remote organs so sympathizing; and that a sensation is the 
necessary antecedent of the resulting muscular aetion. Thus 
it is known that these actions cease in the state of coma ; 
are not excited when the mind is strongly impressed by any 
other sensation or thought ; and that the same muscular con- 
tractions may be induced by the irritation of different parts of 
the body, provided the same sensation be excited. Dr. Alison 
has, however, failed to show* that the essential acts of inspira- 
tion, viz. the contractions of the diaphragm and intercostals, 
require the intervention of a sensation. Their continuance in 
the state of coma, and in the experiments of Legallois and 
Flourens after the entire removal of the brain, and their di- 
.stinct reference by these two inquirers to the medulla oblon- 
gata, which has never been supposed to be the seat of sensa- 
tion, prove them to be independent of the will and of perception. 
But this is true only of the essential, not of the associated 
respiratory phenomena. 

Dr. Alison proceeds to show that there is equal reason for 
classing almost all the nerves of the brain, and many more of 
the spinal nerves, with those exclusively named respiratory by 
♦Sir Charles Bell. Thus the lingual nerve governs an infinite 
number of motions strictly associated with respiration : the in- 
ferior maxillary " moves the muscles of the lower jaw in the 
action of sucking, — an action clearly instinctive when first per- 
formed by the infant, frequently repeated voluntarily during 
life, and always in connexion with the act of respiration." 
Again, the sensitive branches of the fifth pair cooperate in the 
act of sneezing. But if these nerves be admitted into the 
system, the fundamental principle of that system, viz. origin 
in a line or series, is at once violated. Nor is this connexion 
in origin more than matter of conjecture, as regards two of the 

* p. 1 76, and note. 

90 THIRD REPORT 1833. 

most important of the nerves, classed by Sir Charles Bell himself 
as respiratory, — the phrenic and the external respiratory. These 
two nerves branch from the cervical or regular double-rooted 
series. Moreover, the circumstance of rising in linear suc- 
cession is not found to associate nerves in function. " Be- 
tween the roots of the phrenic nerve and those of the inter- 
costals, there intervene in the same series the origins of the 
three lowest cervical nerves, and the first dorsal, which go 
chiefly to the axillary plexus and to the arm, and which are 
not respiratory nerves." 

In recapitulation, the following facts are among the most 
important that have been fully ascertained in the physiology of 
the nervous system. 

1. One universal type has been followed in the formation of 
the nervous system in vertebrated animals. The brain of the 
human foetus is gradually evolved in the successive months of 
uterine existence ; and these stages of progressive develop- 
ment strictly correspond with permanent states of the adult 
brain at inferior degrees of the animal scale. 

2. These successive increments of cerebral matter are found 
to be accompanied by parallel advances in the manifestation of 
the higher instincts and of the mental faculties. 

3. That the brain is the material organ of all intellectual 
states and operations, is proved by observation on comparative 
development, as well as by experiments on living animals, and 
by the study of human pathology. But there does not exist 
any conclusive evidence for referring separate faculties, or 
moral affections, to distinct portions of brain. 

4. Certain irregular movements are produced by injuries of 
the corpora striata, thalami optici, crura cerebelli, and semi- 
circular canals of the internal ear. 

5. The tubercula quadrigemina preside over the motions 
of the iris, and their integrity seems essential even to the func- 
tions of the retina. They are also, according to Flourens, the 
points, at which irritation first begins to excite pain and mus- 
cular contractions. 

6. The cerebellum appears to exercise some degree of con- 
ti'ol over the instruments of locomotion ; but the precise na- 
ture and amount of this influence cannot be distinctly defined. 

7. The cerebrum, cerebellum and medulla oblongata possess 
the faculty of acting primordially, or spontaneously, without 
requiring foreign excitation. The spinal cord and the nerves 
are not endowed with spontaneity of action, and are therefore 
termed subordinate parts. 


8. The medulla oblongata exercises the office of originating 
and regulating the motions essential to the act of respiration. 
By virtue of its continuity with the spinal marrow, it also par- 
ticipates in the functions of that division of nervous matter. 

9. The function of the spinal cord is simply that of a con- 
ductor of motive impulses, from the brain to the nerves supply- 
ing the muscles, and of sensitive impressions from the surface 
of the body to the sensorium commune. These two vital 
offices reside in distinct portions of the spinal medulla, — the 
propagation of motion in its anterior columns, the transmission 
of sensations in its posterior columns. There is no necessary 
dependence of the motions of the heart, and the other invo- 
luntary muscles, on the spinal marrow. 

10. The nerves are comprehended in the four following 
classes : — I. Nerves simply of motion ; II. Of motion and sen- 
sation ; III. Of three of the senses ; IV. The ganglionic sy- 

I. The nerves of motion are the third, fourth, sixth, portio 
dura of the seventh, and the ninth. It is not ascertained whe- 
ther the glossopharyngeal and spinal accessory nerves belong to 
this or to the second class. 

11. The function of ministering both to motion and sensation 
is possessed by the fifth pair of cerebral nerves, and by the 
spinal nerves, which agree precisely in anatomical composition. 
The par vagum, however, which is one of the irregular nerves, 
has also a twofold endowment. 

III. This division comprises the first and second pairs, and 
the portio mollis of the seventh pair. These nerves are insen- 
sible to ordinary stimulants, and possess an exclusive sensibility 
to their respective objects, — viz. odorous matter, light, and aei-ial 

IV. The system of the great sympathetic nerve, and its as- 
sociated plexuses and gangUa. 

[ 93 ] 

Report on the present State of our Knowledge respecting the 
Strength of Materials. By Peter BarloWj Esq., F.R.S., 
Corr. Memb. Inst. France, 8fc. Sfc. 

The theory of the strength of materials, considered merely as 
a branch of mechanical or physical science, must be admitted 
to hold only a very subordinate rank ; but in a covmtry in which 
machinery and works of every description are carried to a great 
extent, it certainly becomes a subject of much practical im- 
portance ; and it was no doubt viewing it in this light which led 
the Committee of the British Association, at their last Meeting, 
to do me the honour to request me to furnish them with a 
communication on the subject. In drawing my attention to this 
inquiry, the Committee have subdivided it into the following 
heads : — 1. Whether, from the experiments of different authors, 
we have arrived at any general principles ? 2. What those 
principles are ? 3. How modified in their application to dif- 
ferent substances ? And what are the differences of opinion 
which at present prevail on those subjects ? 

To these questions, without a formal division of the Essay, 
I shall endeavour to reply in the following pages, by drawing 
a concise sketch of the experimental and theoretical researches 
which have been undertaken with reference to these inquiries. 

The subject of the sti*ength of materials, from its great prac- 
tical importance, has engaged the attention of several able 
men, both theoretical and practical, and much useful informa- 
tion has been thereby obtained. As far as relates to the me- 
chanical effects of different strains, everything that can be 
desired has been effected ; but the uncertain nature of mate- 
rials generally, will not admit of our drawing from experiment 
such determinate data as could be wished. Two trees of the 
same wood, grown in the same field, having pieces selected 
from the same parts, will frequently differ from each other very 
considerably in strength, when submitted to precisely the same 
strain. The like may be said of two bars of iron from the 
same ore, the same furnace, and from the same rollers, and 
even of different parts of the same bar ; and so likewise of 
two ropes, two cables, &c. We must not, therefore, in ques- 
tions of this kind, expect to arrive at data so fixed and deter- 
minate as in many other practical cases ; but still, within cer- 
tain limits, much important information has been obtained for 


the guidance of practical men ; and by tabulating such results 
in a subsequent part of this article, I shall endeavour to answer 
the leading questions of the Committee of the British Associa- 
tion, as far, at least, as relates to experimental results. In re- 
ference to theory, it must also be admitted that some uncer- 
tainty still remains ; but this likewise is in a great measure to 
be referred to the nature of the materials, which is such as to 
offer resistances by no means consistent with any fixed and 
determinate laws. 

Hence some authors have assumed the fibres or crystals 
composing a body to be perfectly incompressible, and others 
as perfectly elastic ; whereas it is known that they are strictly 
neither one nor the other, the law of resistance being differ- 
ently modified in nearly every different substance ; and as it is 
requisite theoretically to assume some determinate law of action, 
it necessarily follows that some doubt must also hang over this 
branch of the subject. It is, however, fortunate that whatever 
may be the uncertainty on these points, the relative strength 
of different beams or bolts of the same material, of similar forms 
and submitted to similar strains, is not thereby affected ; so that 
whatever may be the law which the fibres or particles of a 
body observe in their resistance to compression or extension, 
still, from the result of a well conducted series of experiments, 
the absolute resisting force of beams of similar forms, of the 
same materials, of any dimensions, submitted to similar strains, 
may, as far as the mean strength can be depended upon, be 
satisfactorily deduced. An examination of these different views 
taken of the subject by different writers will, it is hoped, be 
found to furnish a reply to the other queries of the Committee. 
The first writer who endeavoured to connect this inquiry with 
geometry, and thereby to submit it to calculations, was the ve- 
nerable Galileo, in his Dialogues, published in 1633. He there 
considers solid bodies as being made up of numerous small 
fibres placed parallel to each other, and their resistance to se- 
paration to a force applied parallel to their length, to be pro- 
portional to their transverse area, — an assumption at once ob- 
vious and indisputable, abstracting from the defects and irre- 
gularities of the materials themselves. He next inquired in 
what manner these fibres would resist a force applied perpen- 
dicularly to their length : and here he assumed that they were 
wholly incompressible ; that the fibres under every degree of 
tension resisted with the same force, and, consequently, that 
when a beam was fixed solidly in a horizontal position, with one 
end in a wall or other immoveable mass, the resistance of the 
integrant fibres was equal to the sum of their direct resistances 


multiplied by the distance of the centre of gravity of their sec- 
tion from the lowest point ; abovit which point, according to 
this hypothesis, the motion must necessarily take place. 

The fallacy of these assumptions was noticed, but not cor- 
rected, by several subsequent authors. Leibnitz objected to 
the doctrine of the fibres resisting eqvially under all degrees 
of tension, but admitted their incompressibility, thereby still 
making the motion take place about the lowest point of the sec- 
tion; but he assumed for the law of resistance to extension that 
it was always proportional to the quantity of extension. Ac- 
cordingly as the one or the other of these hypotheses was 
adopted, the computed transverse resistance of a beam, as de- 
pending on the absolute strength of its fibres, varied in the 
I'atio of 3 to 2 ; and many fanciful conclusions have been drawn 
by different authors relative to the strength of differently formed 
beams, founded upon the one or the other of these assumptions, 
which, however, it will be unnecessary to refer to more parti- 
cularly in this article. 

We have seen that each of these distinguished philosophers 
supposed the incompressibility of the fibres ; but James Ber- 
noulli rejected this part of Leibnitz's hypothesis, and considered 
the fibres as both compressible and extensible, and that the 
resistance to each force was proportional to the degree of ex- 
tension or compression. Consequently, the motion instead of 
taking place, as hitherto considered, about the lowest point of 
the section, was now necessarily about a point vi^ithin it ; and 
his conclusion was, that whatever be the position of the axis of 
motion, or, as it is now commonly called, the neutral axis, the 
same force applied to the same arm of a lever will always pro- 
duce the same effect, whether all the fibres act by extension 
or by compression, or whether only a part of them be extended, 
and a part compressed. Dr. Robison, in an elaborate article 
on this subject, also assumes the compressibility and exten- 
sibility of the fibres, and as a consequence, assumes the centre 
of compression as a fulcrum, about which the forces to exten- 
sion are exerted, and the resistance of both forces to be directly 
proportional to the degree of compression or extension to which 
they are exposed ; that is, he assumed each force, although 
not necessarily offering equal power of resistance, to be indivi- 
dually subject to the law of action appertaining to perfectly 
elastic bodies. In carrying on the experiments which laid the 
foundation of my Essay on the Strength of Timber, Sfc, in 1817, 
I v/as led by several circumstances I had observed to doubt 
whether, in the case of timber, this assumption of perfect elas- 
ticity was admissible. And as some of the specimens used in 

96 thiiId report — 1833. 

my experiments showed very distinctly after the fracture the 
line about which the fracture took place, I thought of availing 
myself of this datum, and that which gave the strength of direct 
cohesion, in order to deduce the law of resistance from actual 
experiment, instead of using any assumed law whatever. 

The result of this investigation implied that the resistance 
was nearly as first assumed by Galileo, and although very dif- 
ferent from what I had anticipated, yet, as an experimental re- 
sult, I felt bound to abide by it, attributing the discrepance to 
the imperfect elastic properties of the material. Mr. Hodgkin- 
son, however, in a very ingenious paper read at the Manchester 
Philosophical Society in 1822, has pointed out an error in_ 
my investigation, by my having assumed the momentum of 
the forces on each side the neutral axis as equal to each other, 
instead of the forces themselves ; consequently the above de- 
duction in favour of the Galilean hypothesis fails. This paper 
did not come to my knowledge till the third edition of my Essay 
was nearly printed off, and the correction could not then be 
made ; but being made, it proves that the law of actual resistance 
approaches much nearer to that of perfect elasticity than from 
the nature of the materials there could be any reason to expect ; 
so that in cases where the position of the neutral axis is known, 
and also its resistance to direct cohesion, a tolerably close ap- 
proximation may be made to the transverse strength of a beam 
of any form, by assuming the resistance to extension to be pro- 
portional to the quantity of extension, and the centre of com- 
pression as the fulcrum about which that resistance is exerted. 
But I have before observed, and beg again to repeat, that by 
far the most satisfactory data will always be obtained by ex- 
periments on beams of the like form (however small the scale,) 
and of the same material as those to be employed, because then 
the law of resistance forms no part of the inquiry, and does not 
necessarily enter into the calculation, the ultimate strengths 
being dependent on the dimensions only, whatever may be the 
absolute or relative resistance of the fibres to the two forces 
we have been considering. 

At present I have only considered the resistance of a beam 
to a transverse strain ; but there is another mode of application, 
in which, again, the law of resistance necessarily enters, and 
has led to many curious and even mysterious conclusions. This 
is when a force of compression is applied parallel to the length. 
In the case of short blocks, the resistance of the material to a 
crushing force is all that is necessary to be known ; and in the 
Philosophical Transactions for 1818 we have a highly valuable 
table of experimental results on a great variety of materials, by 


George Rennie, Esq., which contains nearly all the information 
on this subject that can be desired. But when a beam is of con- 
siderable length in comparison with its section, it is no longer 
the crushing force that is to be considered : the beam will bend 
and be ultimately destroyed by an operation very similar to 
that which breaks it transversely ; and the investigation of these 
circumstances has called forth the efforts of Euler, Lagrange, 
and some other distinguished mathematicians. 

When a cylindric body considered as an aggregate of pa- 
rallel fibres is pressed vertically in the direction of its length, 
it is difficult to fix on data to determine the point of flexure, 
since no reason can be assigned why it should bend in one way 
rather than in another ; still, however, we know that practically 
such bending will take place. And it is made to appear, by the 
investigations of Euler and Lagrange, that with a certain weight 
this ought theoretically to be the case, but that with a less 
weight no such an effect is produced, — an apparent interruption 
of the law of continuity not easily explained, which exhibits 
itself, however, analytically, by the expression for the ordinate 
of greatest inflection being imaginary till the weight or pressure 
amounts to a certain quantity. Another mysterious result from 
these investigations is, that while the column has any definite di- 
mensions, and is loaded with a certain weight, inflection as above 
stated takes place ; but if the column be supposed infinitely 
thin, then it will not bend till the weight is infinitely great. 
These investigations of two such distinguished geometers are 
highly interesting as analytical processes, but the hypothesis 
on which they are founded, namely, that of the perfect elas- 
ticity of the materials, is inconsistent with the nature of bodies 
employed in practice : they form, therefore, rather an exercise 
of analytical skill than of useful practical deductions. There 
is, however, one useful result to be drawn from these processes, 
which is, that the weight under which a given column begins 
to bend is directly as its absolute elasticity ; so that, having de- 
termined experimentally the weight which a column of given 
elasticity will support safely, or that at which inflection would 
commence, we may determine the weight which another column 
of the same dimensions, but of different elasticity, may be 
charged with without danger. 

M. Gerard, a member of the Institute of France, aware of 
the little practical information to be drawn from investigations 
wholly hypothetical, has given the detail of a great number of 
actual experimental results connected with this subject on oak 
and fir beams of considerable dimensions, carried on at the ex- 

1833. H 

98 THIRD REPORT — 1883. 

pense of the French Government, from which he has drawn the 
following empirical formulae, viz.^ 

1. In oak bea,„s Ul = ■1784451 (/+ -03). «■ 

bo V'o 

2. In fir beams ?/! = 8161 128 a /r^ 


where P = half the weight in kilogrammes, a the less, and h 
the greater sides of the section, /half the length of the column, 
and b the versed sine of inflection, the dimensions being all in 
metres *. 

How far these formulae are to be trusted in practical con- 
structions is, however, I consider, rather doubtful, because they 
are drawn from a number of results which differ very greatly 
from each other ; and in one case in particular the result, as 
referred to the deflection of beams, has been satisfactorily shown 
to be erroneous by Baron Charles Dupin, in vol. x. of the Jour- 
nal de VE'cole Polytechnique, as also by a carefully conducted 
series of experiments in my Essay on the Strength of Timber, 
^•c. I conceive it, therefore, to be very desirable that a set of 
experiments on this application of a straining force on vertical 
columns should be undertaken, and it is, perhaps the only 
branch of the inquiry connected with the strength of materials 
in which there is a marked deficiency of practical data ; at the 
same time it is one in which both timber and iron are being con- 
stantly employed. We see every day in the metropolis houses 
of immense height and weight being built, the whole fronts of 
which, from the first floors, are supported entirely by iron or 
wooden columns ; and all this is done without any practical rule 
that can be depended upon for determining whether or not 
these columns are equal to the duty they have to perform. 

I say this with a full knowledge that Mr, Tredgold has fur- 
nished an approximate rule for this purpose ; but the principle 
on which it is founded has no substantial basis. The extra- 
ordinary skill which Mr. Tredgold possessed in every branch 
of this subject, and the great ingenuity he has displayed in in- 
vestigating and simplifying every calculation connected with 
architectural and mechanical construction, certainly entitle his 
opinion to high consideration ; but still on a subject of such 
high importance, it would be much more satisfactory to be pos- 
sessed of actual experimental data. The supposition he ad- 
vanced was made entirely as a matter of necessity, and I am 

* See Traite Analytique de la Rhistance des Solidei. 


confident that no one would have been more happy than him- 
self to have been enabled to substitute fact for hypothesis, had 
he possessed the means of adopting the former. But unfor- 
tunately such a series of experiments are too expensive and 
laborious to be undertaken by an individual situated as he was, 
having a family to maintain by his industry, and whose close 
and unremitting application to these and similar inquiries, in all 
probability shortened his valuable life*. 

At present I have referred principally to experiments made 
with a view of determining the ultimate strength of materials ; 
and with data thus obtained practical men have been enabled 
to pursue their operations with safety, by keeping sufficiently 
within the limits of the ultimate strain the materials would bear, 
or rather with which they would just break, some working to a 
third, others to a fourth, &c., of the ultimate strength, according 
to the nature of the construction, or the opinion of the con- 

But it is to be observed, that although we may thus ensure 
perfect safety as far as relates to absolute strength, there are 
many cases in which a certain degree of deflection would be 
very injurious. It is therefore highly necessary to attend also 
to this subject, particularly as the deflection of beams and their 
ultimate strength depend upon different principles, or are at 
least subject to different laws. Hence most writers of late date 
give two series of values, one exhibiting the absolute or relative 
strength, and the other the absolute or relative elasticities. 
These values will of course be found to differ in different au- 
thors, on account of the uncertainty in the strength of the ma- 
terials already referred to, but amongst recent experiments the 
difference is not important : they will also be found differently 
expressed, in consequence of some authors deducing these 
numbers from experiments differently made. Some, for ex- 
ample, have drawn their formulae for absolute strength from 
experiments made on beams fixed at one end and loaded at the 
other, using the whole length ; some, again, from experiments 
on beams supported at each end and loaded in the middle, 
using the half length. Some take the length in feet, and the 
section in inches ; others all the dimension in inches ; and a 
similar variety occurs in estimating the elasticity. Also, in the 
latter case, some authors employ what is denominated the mo- 
dulus of elasticity, in which latter case the weight of the beam 

* Mr. Tredgold's Principles of Carpentry, and his Treatise on the Strength 
of Iron, ought to be in the possession of every practical builder; besides which 
two works, he published many separate articles on the same subject in different 
numbers of the Philosophical Magaziiie. 

H 2 

100 THIRD REPORT — 1833. 

itself, and consequently its specific gravity, enters. These va- 
rieties of expressions, however, are not to be understood as 
arising from any difference of opinion amongst the authors from 
whom they proceed, but merely as different modes of expressing 
the same principles : indeed, in reply to that inquiry of the 
Committee with reference to this point, I may, I think, venture 
to say there is not at present any difference of opinion on any 
of the leading principles connected with the strength of mate- 
rials, with the excejDtion of such as are dependent entirely upon 
the imperfect nature of the materials themselves, and which, as 
we have seen, will give rise to different results in the hands of 
the same experimenter and under circumstances in every re- 
spect similar. 

As I distinguish the doctrine of the absolute resistance or 
strength of materials, which is founded on experiment, from 
that which relates to the amount and resolution of the forces or 
strains to which they are exposed, which is geometrical ; and 
as I confine myself to the former subject only in this Essay, it 
is not, I conceive, necessary to extend the preceding remarks 
to any greater length. I shall therefore conclude by giving 
a table of the absolute and relative values of the ultimate 
strength and elasticity of various species of timber and other 
materials, selected from those results in which I conceive the 
greatest reliance may be placed. 

Formula; relating to the ultimate Strength of Materials in 
cases of Transverse Strain. — Let I, b, d, denote the length, 
breadth and depth in inches in any beam, w the experimental 


breaking weight in pounds, then will j— r-^ = S be a constant 

quantity for the same material, and for the same manner of ap- 
plying the straining force ; but this constant is different in dif- 
ferent modes of application. Or, making S constant in all cases 
for the same material, the above expression must be prefixed 
by a coefficient, according to the mode of fixing and straining. 

1. When the beam is fixed at one end, and loaded at the 
other, , 


2. When fixed the same, but uniformly loaded, 

1 lie _ ^ 

3. When supported at both ends, and loaded in the middle, 

1 Itv _ ^ 
4 ^ ZTrf^^ ~ ^' 



4. Supported the same, and uniformly loaded, 



Iw „ 
^ bd^-^' 

Fixed at b 

oth ends. 

and loaded in 

the middle, 



Iw ^ 

Fixed the 

same, bui 

. uniformly loaded, 



Iw ^, 
^ bd^~ 

7. Supported at the ends, and loaded at a point not in the 
middle. Then, n m being the division of the beam at the point 
of application, 

11711 I HI _ „ 

Some authors state the coefficients for cases 5 and 6 as -rj- 

1 . ^ 

and jp, but both theory and practice have shovpn these numbers 

to be erroneous. 

By means of these formulse, and the value of S, given in the 
following Table, the strength of any given beam, or the beam 
requisite to bear a given load, may be computed. This column, 
however, it must be remembered, gives the ultimate strength, 
and not more than one third of this ought to be depended upon 
for any permanent construction. 

Formulce relating to the Deflection of Beams in cases of 
Transverse Strains. — Retaining the same notation, but repre- 
senting the constant by E, and the deflection in inches by 8, 
we shall have. 

Case 1. 


32., Pw _ 



5 Piv 

8 ^ bdn~ ^ 

1 ^ bdH-^- 

12 Pw 
1 ^ bdH- ^• 


2 Pw 

3 ^ bdH"^ 

1 Ptv „ 
1 ^ bdH-^- 


5 Pio ^ 
12 ^ bdH-^ 

Hence, again, from the column marked E in the following 
Ttble, the deflection a given load will produce in any cate may be 
computed; or, the deflection being fixed, the dimension of the 
beam may be found. Some authors, instead of this measure of 

102 THIRD REPORT — 1833. 


elasticity, deduce it immediately from the formula . .^ ^ = E, 

substituting for w the height in inches of a column of the ma- 
terial, having the section of the beam for its base, which is equal 
to the weight w, and this is then denominated the modulus of 
elasticity. It is useful in showing the relation between the 
weight and elasticity of different materials, and is accordingly 
introduced into the following Table. 

The above formulae embrace all those cases most commonly 
employed in practice. There are, of course, other strains con- 
nected with this inquiry, as in the case of torsion in the axles 
and shafts of wheels, mills, &c., the tension of bars in suspen- 
sion bridges, and those arising from internal pressure in cylin- 
ders, as in guns, water-pipes, hydraulic presses, &c. ; but these 
fall rather under the head of the resolution of forces than that 
of direct strength. It may just be observed, that the equation 
due to the latter strain is 

t{c — n) ■= n R, 

where t is the thickness of metal in inches, c the cohesive power 
in pounds of a square inch rod of the given material, n the 
pressure on a square inch of the fluid in pounds, and R the in- 
terior radius of the cylinder in inches. Our column marked C 
will apply to this case, but here again not more than one third 
the tabular value can be depended upon in practice. 



Table of the Mean Strength and Elasticity of various Materials, as 
deduced from the most accurate Experiments. 

Names of Materials. 





Birch, Common 

, American Black 


Bullet Tree 


Deal, Christiana , 
, Memel .... 


Fir, New England 

— .Riga 

— , Mar Forest ..., 

Green heart , 

Larch, Scotch 

Locust Tree 


Norway Spars 

Oak, English | j^°'" 

, African 

, Adriatic 

, Canadian ... 

, Dantzic 

Pear Tree 


Pine, Pitch 
, Red 

Teak , 

Tonquin Bean . 


Iron, Cast < , 

, Malleable 

, Wire 












strength of 
cohesion on 
an inch sec- 










36000 ] 

for trans- 






for deflec- 







Modulus of 









69120000 5530000 
91440000 6770000 


of English growth. 







America, South. 

Results very va- 

East Indies. 

East Indies. 

[Mean of English 
f and Foreign. 

[ 105 ] 

Report on the State of our Knowledge respecting the Magnetism 
of the Earth. By S. Hunter Christie, Esq., M.A., F.R.S. 
M.C.P.S., Corr. Memh. Philom. Sac. Paris, Hon. Memb. 
Yorkshire Phil. Soc.; of the Royal Military Academy ; and 
Member of Tritiity College, Cambridge. 

Had the discovery of the loadstone's du-ective power been made 
by a philosopher who at the same time pointed out its import- 
ance to the purposes of navigation, we might expect that his 
name would have been handed down to posterity as one of the 
greatest benefactors of mankind. The discovery was, however, 
most likely made by one so engaged in maritime enterprise that, 
in his eyes, this application constituted its whole value ; and it 
is not improbable that, being for some time kept secret, it may 
have been the principal cause of the success of many enterprises 
attributed to the superior skill and bravery of the leaders. The 
knowledge of this property of the magnet, though gradually 
diffused, would long be guarded with jealousy by those who 
justly viewed it as of the highest advantage in their predatory 
or commercial excursions ; and this is, perhaps, the cause of the 
obscurity in which the subject is veiled. If the discovery is 
European, there is no people, from the character of their early 
enterprises, and, I may add, from the nature of the rocks of 
their country, more likely to have made it than the early Nor- 
wegians ; and as there is reason for believing that they were 
acquainted with the directive property of the loadstone at least 
half a century earlier than its use is supposed to have been 
known in other parts of Europe, it may be but justice to allow 
them the honour of having been the discoverers. Whether the 
discovery was made in Asia or in Europe, in the North or in 
the South, I am not, however, now called upon to decide, but 
to point out the consequences which have followed that disco- 
very by unveiling gradually phgenomena, though less striking, 
yet equally interesting, and some even more difficult of expla- 

These phasnomena are, the variation of the magnetic needle, 
with its annual and diurnal changes ; the dip of the needle ; and 
the intensity of the magnetic force of the earth ; which are, how- 
ever, all comprised under two heads, — The Direction and the 
Ijitensity of the terrestrial magnetic force. 

106 THIRD REPORT — 1833. 

I. The Direction of the Terrestrial Magnetic Force. 

1. The Variation of the Needle. — For some centuries after 
the directive property of the loadstone was discovered, it was 
generally supposed that the needle pointed correctly towards 
the pole of the heavens. It has however been said, on the 
authority of a letter by Peter Adsiger, that the variation of 
the needle was known as early as 1269; and if we fully admit 
the authenticity of this letter, we must allow that the writer 
was at that date not only aware of the fact, but that he had 
observed the extent of the deviation of the needle from the 
meridian*. It is possible that such an observation as this 
may have been made at this early period by an individual de- 
voting his time to the examination of magnetical phaenomena; 

* This curious and highly interesting letter, dated the 8th of August 1269, 
is contained in a volume of manuscripts in the Library of the University of 
Leyden, and we are indebted to Cavallo for having published extracts from it. 
The variation is thus referred to : " Take notice that the magnet (stone), as well 
as the needle that has been touched (rubbed) by it, does not point exactly to 
the poles ; but that part of it which is reckoned to point to the south declines a 
little to the west, and that part which looks towards the north inclines as much 
to the east. The exact quantity of this declination I have found, after numer- 
ous experiments, to be five degrees. However, this declination is no obstacle 
to our guidance, because we make the needle itself decline from the true south 
by nearly one point and an half towards the west. A point, then, contains five 
degrees." (Letter of Peter Adsiger, Cavallo On Magnetism, London 1 800, p. 317.) 
It is certainly extraordinary, if so clear an account of the deviation of the needle 
from the meridian as this, was communicated to any one by the person who had 
himself observed that deviation, that for more than two centuries afterwards we 
should have no record of a second observation of the fact. This alone would 
throw doubt on the authenticity of the letter, and the estimate given of the 
variation may appear to confirm these doubts ; for, according to the period of 
change which best agrees with the observations during more than two hundred 
years, the variation, if observed, would have been found to be westerly instead 
of easterly in 1269. It may however be urged, that as the whole period of 
change has not yet elapsed since observations were made, we are not in pos- 
session of a sufficient number of facts to authorize us to draw conclusions re- 
specting the variation at such an early date ; and also, that if the letter be spu- 
rious, or the original date have been altered to that which it bears, this or the 
fabrication can only have been for the purpose of founding claims in consequence 
of the contents of this letter ; and as no such claims have been advanced, there 
appears no motive either for fabrication or alteration. In a preceding part of 
the letter the author gives methods for finding the poles of a loadstone ; and 
certainly the direction of the axis could not be determined to within five degrees 
by either of these ; so that, as regards the loadstone, we may, I think, conclude 
that the author did not make the observation. As a matter of curious history 
connected with magnetism, it is desirable that either the authenticity of this 
letter should be clearly established, or reasons given for doubting it, by those 
who have an opportunity of cousulting the original. 


and as it is probable that for some time subsequent to the dis- 
covery of the directive property of the needle the deviation in 
Europe was not of sufficient magnitude to have been easily de- 
tected by means of the rude instruments then in use, it may 
very likely be owing to this circumstance that we have not 
earlier records of the variation*. That Columbus, the most 
scientific navigator of his age, when he commenced his career 
of discovery, and undertook to show the western route to India, 
was not aware of it, is clear, since the discovery during his first 
voyage has been attributed to him. However, although Co- 
lumbus may have noticed that the needle did not in every situa- 
tion point due north, and Adsiger, long before him, may even 
have rudely obtained the amount of its deviation, the first ob- 
servations of the variation on which any reliance can be placed 
appear to have been made about the middle of the sixteenth 
century, and shortly afterwards it was well known that the va- 
riation is not the same in all places f . 

2. Change in the Direction of the Needle. — When it was 
first determined by observation, about 1541, that the needle 
did not point to the pole of the earth, it was found that this vari- 
ation from the meridian, at Paris, was about 7° or 8° towards 
the east. In 1550 it was observed 8° or 9° east; and in 
1580, 11 1° east. Norman appears to have been the first who 
observed the variation with any degree of accuracy in Lon- 
don. He states that he observed it to be 11° 15' east J, but he 
was not aware that it does not remain constant in the same 
place §. In 1580, Burough found the variation at Limehouse 
to be 11^° or 11^° east||, and his observations appear to be 

• Another reason why the variation was not earlier observed may be that the 
natural magnet was first used for the purposes of navigation, and its directive 
line was that which pointed to the pole star. As it was therefore considered 
that the natural magnet indicated the direction of the meridian, and it was 
found that a needle touched by it had the directive power, when the needle was 
introduced it was assumed that this also pointed in the meridian. 

t The Netv Attractive, by Robert Norman, chap. ix. London 1596. 

X Ibid. No date is given for this observation ; but from the circumstance of 
Burough referring to Norman's book in the preface to his Discourse of the Va- 
riation of the Compasse, dated 1581, (the copy of this to which I have access 
was printed in 1596, but the Bodleian Library contains one printed in 1581,) 
it would appear that there must have been an earlier edition of Norman's book 
than that of 1596, and that his observations must have been made before 1581. 
Bond, Philosophical Transactions, vol. viii. p. 6066, gives 1576 as the date of 
Norman's observations. 

§ " And although this variation of the needle be found in travaile to be divers 
and changeable, yet at anie land or fixed place assigned, it remaineth alwaies 
one, still permanent and abiding." New Attractive, chap. ix. 

II The mean of his observations, which do not differ 20', is 11° 19' east. 

108 THIRD REPORT — 1833. 

entitled to much confidence ; but he was of the same opinion 
as Norman with respect to the constancy of the variation*. 
Gunter, in 1612, found the variation in London to be 5° 36' 
east; and Gellibrand, in 1633, observed it 4° 4' east. Dr. Wal- 
lis considers Gellibrand to have been the discoverer of " the 
variation of the variation f ; " but if Gunter had any confidence 
in his own observations and those of Burough, he must have 
been aware of the change in the variation. In 1630, Petit 
found the variation at Paris to be 4^° east, but suspected, at 
the time, that the earlier observations there had been incorrect; 
and it was not until 1660, when he found the variation to be 
only 10' east, that he was satisfied of the change of the varia- 
tion. About ten years later, Azaut, at Rome, where the va- 
riation had been observed 8° east, found it to be more than 2° 
west; and Hevelius, who at Dantzick in 1642 had found it to 
be 3° 5' west, now found it to be 7° 20' west. 

3. Diurnal Change in the Variation. — This was discovered 
in 1722 by Graham, to whose talents and mechanical skill 
science is so deeply indebted. He found that with several 
needles, on the construction of which much care had been be- 
stowed, the variation was not always the same ; and at length 
determined that the variation was different at different hours 
in the day, the greatest westerly variation occurring between 
noon and four hours after, and the least about six or seven 
o'clock in the evening f. Wargentin at Stockholm in 1750, 
and Canton in London from 1756 to 1759, moi-e particularly 
observed this phsenomenon; and the latter determined that 
the time of minimum westerly variation in London was between 
eight and nine in the morning, and the time of maximum be- 
tween one and two in the afternoon. Canton likewise deter- 
mined in 1759, that the daily variation was different at different 
times in the year, the maximum change occurring about the end 
of June, and the minimum in December §. Cassini, during more 
than five years and a half, namely, from May 1783 to January 
1789, carefully observed, at particular hours, the direction of a 
needle suspended in the Observatory at Paris ; and although 
he does not correctly state the course of the daily variation, 
overlooking altogether the second maximum west, and the pro- 
gress of the needle towards the east in the early part of the 

* " For considering it remayneth alwaies constant without alteration in every 
severall place, there is hope it may be reduced into method and rule." Dis- 
course, chap. X. 

f Philosophical Transactions, 1701, vol. xxii. p. 10-36. 

+ Ibid. 1724, vol. xxxiii. p. 96. 

§ Ibid. 1759, vol. xli. p. JWS. 



morning *, yet his observations and remarks are of great value 
as pointing out the annual oscillations of the needle f. Since 
this, the diui-nal variation has been very generally observed, 
but by no one with greater care and perseverance than by the 
late Colonel Beaufoy ;}:. 

In order to determine whether the course of the diurnal va- 
riation is influenced by the elevation of the place of observation, 
the zealous and indefatigable De Saussure undertook a series 
of observations on the Col du Geant, nearly 11,300 feet above 
the level of the sea. This series, after incurring much personal 
inconvenience and even risk in that region of snow and of storms, 
he completed ; and he has compared the results with observa- 
tions which he made immediately before and after at Chamouni 
and Geneva. From this comparison it appears that the course 
of the diurnal variation was nearly the same on one of the 
highest mountains, in a deep and narrow valley at its foot, and 
in the middle of a plain or of a large valley. The times of the 
maxima, east and west, are in each case nearly those previously 
determined by Canton, these maxima occurring rather later on 
the Col du Geant than at the other stations. Excluding in all 
cases the results where extraordinary causes appear to have 
operated, the extent of the diurnal variation at Chamouni ex- 
ceeds that at Geneva and also that on the Col, the two latter 
being very nearly the same. The observations, however, are, 
as Saussure very justly remarks, nuich too limited to give cor- 
rect means §. 

5. The Dip of the Magnetic Needle. — Norman having found 
with different needles, and with one in particular on the con- 
struction of which he had bestowed much pains, that although 
perfectly balanced on the centre previously to being touched 
by the magnet, after this operation the north end always de- 
clined below the horizon, devised an instrument by which he 

• Journal de Physique, Mai 1792, torn. xl. p. 345. f Ibid. p. 348. 

Many of the results of Colonel Beaufoy 's observations are published in the 

Edinburgh PhilosophicalJoumal, vols. i. ii. iii. iv. and vii. 

§ Saussure, Voyayes dans les Alpes, torn. iv. p. 302 au p. 312. 
does not give the mean results, I insert them here. 

As Saussure 

Time of absolute maximum. Time of second maximum. 


Cliamouni ... 
Col du Geant 


h m 

7 56 A.M. 

7 34 

8 09 


h m 

1 09 P.M. 

1 41 

2 00 


6 26 P.M. 

7 44 
5 51 


h m 
11 17 P.M 

10 46 
10 17 

Extent of Elevation 
diurnal above the 
change. sea. 

15 42 
17 06 
15 43 





110 THIRD REPORT — 1833. 

could determine the inclination of the needle to the plane of the 
horizon*. The figure given of the instrument is sufficiently 
I'ude, but the principles of its construction, as stated by Nor- 
man, are correct. With this instrument he found the inclina- 
tion of the needle to the horizon in London to be about 71° 50', 
but gives no date to the observation, though Bond assigns 1576 
as the time f . Although in a theoretical point of view it would 
be desirable to have so early a record of the dip, particularly as 
subsequent observations lead us to suppose that the dip attained 
its maximum after this time, yet, considering the uncertainty 
attending such observations, even with the present improved 
instruments, we cannot place much confidence in this result, 
however we may rely upon the author having used every pre- 
caution in his power to ensure accuracy. Having determined 
the dip of the needle in London, Norman states that this de- 
clining of the needle will be found to be different at different 
places on the earth X, though he does not take a correct view 
of the subject, for he considers that the needle will always be 
directed towards a fixed point. 

5. Variation of the Dip. — Subsequent observations by Bond, 
Graham, Cavendish, and Gilpin, and the more recent ones in 
our own time, have shown that the inclination of the needle to 
the horizon at the same place, like the angle which it makes 
with the meridian, is subject to change ; but the diurnal oscil- 
lations of the direction are of too minute a character to have 
been ascertained with the imperfect instruments which we 

This is an outline of the phsenomena hitherto observed, de- 
pending upon the direction of the forces acting upon the needle. 
Various attempts have been made to account for those obser- 
vable at fixed points on the earth's surface at different periods, 
and also to connect those depending on the different positions 
of the places of observation, but hitherto with only very partial 
success. It is not my intention to enter into a detailed history 
of these attempts, but I may briefly notice some of the most 

To Gilbert we are indebted not only for the first clear views 
of the principles of magnetism, but of their application to the 
phsenomenon of the directive power of the needle ; and indeed 
we may say that, with the exception of the recent discoveries, 
all that has been done since, in magnetism, has for its foundation 
the principles which he established by experiment §. He con- 

* New Attractive, chap. iii. iv. 

t Philosophical Transactions, 1673, vol. viii. p. 6066. 

X New Attractive, chap. vii. § Gilbert, De Magtiete, ^c, Lond. 1600.. 


sidered that the earth acts upon a magnetized bai% and upon 
iron, like a magnet, the directive power of the needle being due 
to the action of magnetism of a contrary kind to that at the 
end of the needle directed towards the pole of the earth. He 
applied the term "pole" to the ends of the needle directed 
towards the poles of the earth, according to the view he had 
taken of terrestrial magnetism, designating the end pointing 
towards the north, as the south pole of the needle, and that point- 
ing towards the south, as its north pole*. It is to be regretted 
that some English philosophers, guided by less correct views, 
have since his time applied these terms in the reverse sense, 
which occasionally introduces some ambiguity, though now they 
are used in this country, as on the Continent, in the sense ori- 
ginally given to them by Gilbert. 

In 1668 Bond published a Table of computed variations in 
London, for every year, from that time to the year 1716 f. The 
variations in this Table agree nearly with those afterwards ob- 
served for about twenty- five years, beyond which time they 
differ very widely ; and I only notice this Table as the first em- 
pirical attempt at the solution of a problem which is, as yet, 
unsolved. Bond afterwards proposed to account for the change 
in the variation and dip of the needle by the motion of two 
magnetic poles about the poles of the earth. He professed not 
only to give the period of this motion, but to be able to point 
out its cause, and even proposed to determine the longitude 
by means of the dip J. He, however, did not make public either 
his methods or his views ; but with regard to the longitude, it 
is probable they were the same as those afterwards adopted by 

Halley considered that the direction of the needle at different 
places on the earth's surface might be explained on the suppo- 
sition that the earth had four magnetic poles §, and that the 
change in the direction at the same place was due to the motion 
of two of these poles about the axis of the earth, the other two 
being fixed. He does not enter into any calculations to show 
the accordance of the phaenomena with such an hypothesis, but 
conjectures that the period of revolution of these poles is about 
700 years li. 

Since this time, calculations have been made by various au- 
thors, both on the hypothesis of two magnetic poles and on 
that of four, with the view of comparing the results of these 

, * Gilbert, De Magnete, Sfc, lib. i. cap. iv. 

t Philosophical Transaction)!, 1668, vol. iii. p. 789. 

X Ibid. 1673, vol. viii. p. 6065. § Ibid. 1683, vol. xiii. p. 208. 

II /6irf. 1692, vol. xvii. p. 563. 

lis THIRD REPORT — 1833. 

hypotheses with actual observation. The most recent attempt 
of this kind is that by Professor Hansteen. He adopts Halley's 
hypothesis of four magnetical poles, but considers that they all 
revolve, and in different periods, the northern poles from west 
to east, and the southern ones from east to west. The results 
calculated on this hypothesis agree pretty nearly with the ob- 
servations with which they are compared ; but as considerable 
uncertainty attends magnetical observations, excepting those 
of the variation made at fixed observatories, and especially the 
early ones of the dip and variation, on which the periods of the 
poles and their intensities must so much depend, it would cer- 
tainly be premature to say that such an hypothesis satisfactorily 
explains the phaenomena of terrestrial magnetism. If we admit 
that the progressive changes which take place in the direction 
of the needle are due to the rotation of these poles, we must 
look to the oscillations of the same poles for the cause of the 
diurnal oscillation of the needle. Any hypothesis which by 
means of two or more magnetic poles will thus connect the 
phaenomena of magnetism, is of great advantage, however un- 
able we may be to give a reason for the particular positions of 
the poles, or for their revolution. Hansteen refers these to 
the agency of the sun and moon. 

Without assigning any cause either for the direction of the 
needle, or for the progressive change of that direction, attempts 
have been made to account for its diurnal oscillations. But 
before taking a review of these, it is necessary that I should 
state more particularly the precise nature of the phsenomenon. 
This I cannot do better than by referring to the results de- 
duced from Canton's observations*. From these it appears 
that in London, during the twenty-four hours, a double oscilla- 
tion of the needle takes place, the absolute maximum west 
happening about half-past one in the afternoon, and the abso- 
lute maximum east, that is, the minimum west, about nine in 
the morning; besides which there was another maximum east 
about nine in the evening, and a maximum west near midnight 
or very early in the morning, the two latter maxima being small 
compared with the absolute maxima. Colonel Beaufoy's very 
extensive series of observations, made when the variation was 
between 24° and 25° west, (Canton's having been made when 
it was 19°,) give nearly the same results, the absolute maxima 
happening somewhat earlier, and the second maxima west 
about eleven in the evening. 

Canton explained the westerly motion of the needle in the 

» Philosophical Transactions, 1759, p. 398, and 1827, pp. 333, 334. 


latter part of the morning, and the subsequent easterly motion, 
by supposing that the heat of the sun acted upon the northern 
parts of the earth as upon a magnet, by weakening their in- 
fluence, but offered no explanation of the morning easterly mo- 
tion of the needle. 

Oersted's discovery of the influence of the closed voltaic 
circuit upon the magnetic needle, and the consequent discoveries 
of Davy, Ampere and Arago, immediately led to the considera- 
tion, whether all the phaenomena of terrestrial magnetism were 
not due to electric currents ; and the discovery of Seebeck, that 
electric currents are excited when metals having different 
powers of conducting heat are in contact, — which discovery 
with but few holds the rank to which it is eminently entitled, — 
pointed to a probable source for the existence of such currents. 
At the conclusion of a highly interesting paper on the develop- 
ment of electro-magnetism by heat, Professor Gumming re- 
marks that "magnetism, and that to a considerable extent, it 
appears, is excited by the unequal distribution of heat amongst 
metallic, and possibly amongst other bodies. Is it improbable 
that the diurnal variation of the needle, which follows the 
course of the sun, and therefore seems to depend upon heat, 
may result from the metals, and other substances which com- 
pose the surface of the earth, being unequally heated, and con- 
sequently suffering a change in their magnetic influence ? " And 
in the second part of a paper, detailing some thermo-magnetical 
experiments, read before the Royal Society of Edinbui-gh, 
Dr. Traill considers "that the disturbance of the equihbrium 
of the temperature of our planet, by the continual action of the 
sun's rays on its intertropical regions, and of the polar ices, 
must convert the earth into a vast thermo-magnetic apparatus : " 
and "that the disturbance of the equihbrium of temperature, 
even in stony strata, may elicit some degree of magnetism*." 
Previous to this, I had adopted the opinion that temperature, 
if not the only cause, is the principal one of the daily variation f. 
It did not, however, appear to me, that any of the experiments 
hitherto made bore directly on the subject, since the metals 
producing electric currents by their unequal conduction of heat 
were only in contact at particular parts, and in no case had 
such currents been excited by different metals having their 
surfaces symmetrically united throughout. I in consequence 
instituted a series of experiments with two metals so united, 
and found that electric currents were still excited on the 

• Transactions nf the Philosophieal Society of Cambridge, vol. ii. p. 64. 
+ Pliihtxophtenl Transartions, 1823, p. 392. 

18,'Jrj, 1 

114 ' THIRD REPORT — 1833. 

application of heat, the phaenomena corresponding to magnetic 
polarization in a particular direction with reference to the place 
of greatest heat*. From these experiments I drew the con- 
clusion that one part of the earth, with the atmosphere, being 
more heated than another, two magnetic poles, or rather elec- 
tric currents producing effects referrible to such poles, would 
be formed on each side of the equator, poles of difterent names 
being opposed to each other on the contrary sides of the equa- 
tor; and that different points in the earth's equator becoming 
successively those of greatest heat, these poles would be carried 
round the axis of the earth, and would necessarily cause a de- 
viation in the horizontal needle f. On comparing experimentally 
the effects that would result from the revolution of such poles 
with the diurnal deviations at London, as observed by Canton 
and Beaufoy, also with those observed by Lieut. Hood at Fort 
Enterprise, and finally with the late Captain Foster's at Port 
Bowen, I found a close agreement in all cases in the general 
character of the pliEenomena, and that the times of the maxima 
east and west did not differ greatly in the several cases. The 
double oscillation of the needle, to which I have referred in 
Canton's and Beaufoy's observations, clearly resulted from this 
view of the subject. Some of the experiments to which I have 
referred showed that when heat was applied to a globe, the 
electric currents excited were such, that on contrary sides of 
the equator the deviations of the end of the needle of the 
same name as the latitude were at the same time always in the 
same direction, either both towards east or both towards west. 
No observations having at that time been made on the diurnal 
variation of the needle in a high southern latitude, I considered 
*' that the agreement of the theoretical results with such ob- 
servations would be almost decisive of the correctness of the 
theory." Captain Foster's observations at Cape Horn, South 
Shetland, and the Cape of Good Hope, show most decidedly 
that in the southern hemisphere the diurnal deviations of the 
south end of the needle correspond very precisely with those of 
the north end in the northern hemisphere ; and most fully bear 
me out in the view which I had taken. These valuable obser- 
vations have been placed in my hands by His Royal Highness 
the President, and the Council of the Royal Society, and I in- 
tend, when I have sufficient leisure, rigidly to compare them, 
and likewise those to which I have already referred in the 
northern hemisphere, with the diurnal deviations that would 

• "Theory of the Diurnal Variation of the Magnetic Needle," Philosoj)Mcal 
Tramactions, 1827, pp. 321, 326. 
I- Ibid. pp. 327, 328. 


result at the corresponding places on the earth's surface, on 
the supposition that such electric currents as I have supposed 
are excited on contrary sides of the equator, in consequence of 
different parts on the earth's surface becoming successively the 
places of greatest heat, during its revolution upon its axis. 
Should there be found in these results that accordance w^hich I 
have reason to expect, there will, I think, be no doubt that the 
diurnal deviation of the needle is due to electric currents excited 
by the heat of the sun. 

I have already adverted to the hypotheses of two or more 
poles, by means of which attempts have been made to explain 
the phaenomena of terrestrial magnetism, and I may now re- 
mark, that if we admit the existence of such poles, we must be 
careful not to consider the magnetic meridians as great circles : 
they are unquestionably curves of double curvature. Nor must 
we consider these poles to be, like the poles of a magnet, cen- 
tres of force not far removed from the surface. If such centres 
of force exist for the whole surface of the earth, the experi- 
mental determinations of the magnetic force at different places, 
to which I shall shortly advert, at least show that they cannot 
be far removed from the centre of figure. 

In the delineation of magnetic charts, more attention has 
hitherto been paid to the Halleyan lines, or lines of equal varia- 
tion, than to any others ; and I am not disposed to undervalue 
charts where such lines alone are exhibited : to the navigator they 
are of the greatest value ; but they throw little light on the phae- 
nomena in general. If the meridians wei'e correctly represented, 
they would at least indicate clearly their points of convergence, if 
such in all cases exist ; but the lines that would be most likely to 
guide us to a true theory of terrestrial magnetism, are the nor- 
mals to the direction of the needle. If, as is highly probable, 
the direction of the needle is due to electric currents circulating 
either in the interior or near the surface of the earth, these 
normals would represent the intersection of the planes of the 
currents with the surface of the earth ; and, by their delineation, 
we should have exhibited in one view the course of the currents 
and the physical features by which that course may be modified, 
so that any striking correspondences which may exist, would 
be immediately seized, and lead to important conclusions. 
Changes of temperature I consider to be the principal cause of 
the diurnal changes in the direction of the needle : and if any 
connexion exist between these electric currents and climate, 
we are to expect that the curvature of these normal lines will 
be influenced by the forms, the extent and direction of the con- 
tinents or seas over which they pass, and also by the height, 


11(> THIRD REPORT — 1833. 

direction and extent of chains of movmtains, and probably by 
their geological structure. 

These normal lines may, to a certain extent, agree with the 
lines of equal dip, which have already been delineated upon 
some charts. In Churchman's charts they are represented in 
the positions they would have on Euler's hypothesis of the earth 
having two magnetic poles. The only use, however, of such 
hypothetical representations is, that by comparison with actual 
observation they become tests of the correctness of the theory, 
or they may point out the modifications which it requires, in 
order that it may accord with observation. In Professor Hans- 
teen's chart the hnes of equal dip are projected from observa- 
tions reduced to the year 1780. Considering how very deficient 
we are, even now, in correct observations of the dip, I should 
not be disposed to place much reliance upon the accuracy of 
these lines, particularly where they cross great extents of sea 
aflbrding no points of land necessary for observations of the dip. 
Of these lines of equal dip the most important is the magnetic 
equator, or that line on the earth at which the dipping needle 
would be horizontal. The observations eivina^ this result can 
of course be but few, and are therefore very inadequate for the 
correct representation of this line. In order to obviate this 
difficulty, M. Morlet made use of all observations not very re- 
mote from the equator, determining the distance of that line 
from the place of observation by means of the law, that the 
tangent of the magnetic latitude is half the tangent of the dip, 
which is derived from the hypothesis of two magnetic poles near 
to the centre of the earth. By this means the position of the 
equator was determined throughout its whole extent; and a 
surprising agreement was fovind between the determinations of 
each point by means of different observations, which shows 
that, within certain limits near the equator, the hypothesis very 
correctly represents the observations. This line exhibits in- 
flections in its course which have been attributed, and probably 
with justice, to the physical constitution of the surface in their 
vicinity*. It has been considered also that a general resem- 
blance exists between the isothermal lines and the lines of equal 
dip on the surface of the earth f. 

All the lines to which I have here referred have been liitherto 
represented on a plane, either on the stereographical, the glo- 
bular, or Mercator's projection. Mr. Barlow has, however, 
very lately represented the lines of equal variation on a globe, 
from a great mass of the most recent documents connected with 

» Biot, Traite de Physique. 

f Hansteen, Edinhtrgh fhilosophical Journal, vol, iii. p. 127. 


the variation, furnished to him by the Admiralty, the East India 
Company, and from other sources. If to the hues of equal .va- 
riation were added the magnetic meridians and their normals, 
the isodynamic lines, with those of equal dip, such a globe would 
form the most complete representation of facts connected with 
terrestrial magnetism that has ever been exhibited, and might 
indicate relations which have hitherto been overlooked. 

Having discovered that a peculiar polarity is imparted to iron 
by the simple act of rotation, I was led to consider whether the 
principal phsenomenon of terrestrial magnetism is not, in a great 
measvu-e, due to its rotation. The subsequent discovery by 
Arago, that analogous effects take place during the rotation of 
all metals, and Faraday's more recent discovery, that electrical 
currents are not only excited during the motion of metals, but 
that such currents are transmitted by them, render such an 
opinion not improbable. It is, however, to be remarked, that, 
in all these cases, motion alone is not the cause of the effects 
produced ; but that these effects are due to electricity induced 
in the body by its motion in the neighbourhood of a magnetized 
body. If, then, electrical currents aie excited in the earth in 
consequence of its rotation, we must look to some body exterior 
to the earth for the inducing cause. The magnetic influence 
attributed by Morichini and Mrs. Somerville to the violet ray, 
and the effect M'hich I found to be produced on a magnetized 
needle when vibrated in sunshine, and which appeared not to 
admit of explanation without attributing such influence to the 
sun's rays, might appear to point to the sun as the inducing 
body. The experiments, however, of Morichini and Mrs. So- 
merville, have not succeeded on repetition ; and in a recent re- 
petition of my own experiments, in a vacuum, by Mr. Snow 
Harris, the effects which I observed were not detected. I had 
found that the effects produced on an unmagnetized steel needle 
differed from those produced on a similar needle when magnet- 
ized, and therefore considered that the idea of these effects 
being independent of magnetism was precluded ; but Mr. Har- 
ris's results may possibly be considered to indicate that they 
were due solely to currents of air excited by the sun's rays. 
These circumstances render it doubtful whether the sun's rays 
possess any magnetic influence independent of their heating 
power ; but besides this, supposing such influence to exist, if 
electric currents were induced in the earth during its rotation, 
they would be nearly at right angles to the equator, and would 
therefore cause a magnetized needle to place itself nearly per- 
pendicular to the meridians, or parallel to the equator. 

Altiiough it Avould therefore appear that the rotation of the 

118 THIRD REPORT — lSo3. 

earth is not the cause of its magnetism, yet it is highly pro- 
bable, from Mr. Faraday's experiments *, that, magnetism ex- 
isting in the earth independently of it, electrical currents may 
be produced, not only by the earth's rotation, but by the motion 
of the waters on its surface, and even by that of the atmosphere ; 
so that the direction and intensity of the magnetic forces would 
be modified by the influence of these currents. 

This subject is at present involved in obscurity : still, consi- 
dering how many have been the discoveries made within a few 
years, — all bearing more or less directly upon it, though none 
afford a complete explanation of the phaenomena, — it does not 
appear unreasonable to expect that we are not far removed from 
a point where a few steps shall place us beyond the mist in which 
we are now enveloped. 

Mr. Fox, having observed effects attributable to the electri- 
city of metalliferous veins, appears disposed to refer some of 
the phaenomena of terrestrial magnetism to electrical cur- 
rents existing in these veins -f ; but although we should not be 
jvarranted in denying the existence of these currents, indepen- 
dently of the wires made use of in Mr. Fox's experiments, or 
even their influence on the needle, yet I think we should be 
cautious in drawing conclusions from these experiments J. 

II. Intensity of the Terrestrial Magnetic Force. 

I have as yet said little on the intensity of the terrestrial mag- 
netic forces. Graham, after having discovered the daily varia- 
tion of the needle, suspected that the force which urges it varies 
not only in direction, but also in intensity. He made a great 
variety of observations with a dipping needle, but drew no ge- 
neral conclusion from his results. Indeed, with the instruments 
then in use, he was not likely to determine that which has al- 
most escaped detection with instruments of more accurate con- 
struction, for the diurnal variation of the whole magnetic force 
may perhaps still be considered doubtful. Later observations, 
particvilarly those of Professor Hansteen, have shown that the 
time of vibration of a horizontal needle varies during the day, 
from which it was inferred that the horizontal force also varies. 
Professor Hansteen, by this means, found that the horizontal 
intensity of terrestrial magnetism has a diurnal variation, de- 


Philosophical Transactions, 1832, p. 176. f ^l>'^- 1830, .p 407. 

X Mr. Henwood informs me that he has repeated the experiments of Mr. 
Fox ill from forty to fifty places not before experimented on, and that he pro- 
poses greatly extending them. i\s far as he can yet see, he considers that his 
results go to confirm Mr. Fox's deductions, — I suppose with regard to the elec- 
tricity of metalliferous veins. 


creasing, at Christiana, until ten or eleven o'clock in the morn- 
ing, when it attains its minimum, and then increases until four 
or five o'clock after noon, when it appeared to reach its maxi- 
mum*. By observing, at different times of the day, the direc- 
tion of a horizontal needle thrown nearly at right angles to the 
meridian, by the action of two powerful magnets, placed in the 
meridian, passing through its centre, after correcting the ob- 
servations for the effect of changes of temperature on the in- 
tensity of the force of the magnets, I found that at Woolwich 
the terrestrial horizontal intensity decreased until lO** 30" a.m., 
when it reached its minimum, and increasing from that time, 
attained its maximum about 1^ 30™ p.m. f . This agreement, in 
results obtained by totally independent methods, removes all 
doubt respecting the diurnal variation of the horizontal force. 
The difference in the time of the maximum in the two cases 
may be accounted for, independently of the difference in the 
variation at the two places of observation, by the circumstance 
that no correction for the effect of temperature on the time of 
vibration is made in Professor Hansteen's observation. As no 
such correction had hitherto been made, it must have been con- 
sidered that differences in the temperature at which observations 
were made had little influence on the intensity of the vibrating 
needle ; but in the communication containing these observations, 
I pointed out the necessity of such a correction J; and since 
then, in deducing the terrestrial intensity from the times of vi- 

• Edinburgh Philosophical Journal, vol. iv. p. 297. 

+ Philosophical Transactions, 1825, pp. 50 & 57. An inconvenience attending 
the method which I employed is, that the observations require a correction for 
temperature which is not very readily applied, as will T)e seen by referring to my 
paper ; but this might in a great measure be obviated, by rendering the tempera- 
ture of the magnets employed always the same previous to observation. If, how- 
ever, in order to retain the needle in its position nearly at right angles to the me- 
ridian, torsion were applied instead of the repulsive forces of magnets, the correc- 
tion for temperature would be nearly reduced to that due to the eifects produced 
on the intensity of the needle itself by changes of temperature. But even this 
method is not without objection ; for the sensibility of the needle depending 
upon the number of circles of torsion requisite to bring it into the pi-oper posi- 
tion, if a wire were employed, unless very long, its elasticity would be impaired 
by more than two or three turns ; and it is doubtful whether a filament of glass 
of moderate length would bear more than this without fractiu-e. I had pro- 
posed to the late Captain Foster, previous to his last voyage, that he should de- 
termine the horizontal intensity at different stations, and also its diurnal changes 
by this method, and had a balance of torsion constructed for him for the purpose ; 
but as the instrument is extremely troublesome in its adjustments, I consider 
that the many other observations which he had to make did not allow him time 
for the extensive use of this instrument which he had proposed. It is, however, 
very desirable that it should be ascertained how far this method is applicable. 

X Philosophical l^ransaclions, 1825. 

l20 THIRD REPORT— 18S3. 

bration of a needle, it has been customary to apply a correction 
for differences in the temperatures at which the observations 
may have been made. 

The horizontal intensity varying during the day, it becomes 
a question whether this arises from a change alone in the direc- 
tion of the force, or whether this change of direction is not 
accompanied by a change in the intensity of the whole force. 
In a communication to the Philosophical Society of Cambridge *, 
I suggested that deviations, fi'om whatever cause, in the direc- 
tion of the horizontal needle, were referrible to the deviations 
which, under the same circumstances, would take place in the 
direction of the dipping needle. Adopting these views. Captain 
Foster infers, from observations made by him at Port Bowen, 
on the corresponding times of vibration of a dipping needle, 
supported on its axis and suspended horizontally, that the diur- 
nal change in the horizontal intensity is due principally, if not 
wholly, to a small change in the amount of the dip. The observa- 
tions, however, do not indicate that the force in the direction of 
the dip is constant. Captain Foster's obsei'vations at Spitzber- 
genf show, more decidedly, the diurnal variation of this force : 
there, its maximum intensity appears to have occun*ed at about 
3^ 30"" A.M., and the minimum at 2^ 47™ p.m. ; its greatest change 
amountino; to -J^ of its mean value. The maximum horizontal 
intensity appears to have occurred a little after noon, and the 
minimum nearly an hour after midnight ; but there is consider-r 
able irregularity in the changes which it undergoes. It would, 
however, appear, from these observations, that the variations hi 
the absolute intensity were in opposition to those in the hori- 
zontal resolved part of it ; so that the principal cause of the 
latter variations must have been a change in the dip itself. 
Captain Foster considers " that the times of the day when these 
changes are the greatest and least, point clearly to the sun as 
the primary agent in the production of them ; and that this 
agency is such as to produce a constant inflection of the pole 
towards the sun during the twenty-four hours." This is in per- 
fect accordance with the conclusions I had previously drawn 
from the experiments on A'-hich I founded the theory of the di- 
urnal variation of the needle :j:, as I had shown that if the diur- 
nal variation of the needle arise from the cause which I have 
assigned for it, the dip ought to be a maximum, in northern la- 
titudes, nearly when the sun is on the south magnetic meridian, 
and a minimum when it has passed it about 130°. 

* Transactions of the Philosophical Society of Cambridffe, 1820. 

t Phitosophical Transactions, 1828. ' J Ibid. 1827, pp. 345, 349. 


Humboldt was the first who determined that the intensity of 
the whole magnetic force is different at different positions on 
the earth's surface. Having made observations on the times of 
vibration of the same dipping needle, at various stations in the 
vicinity of the equator, and approaching to the northern pole, 
he found that the intensity of the terrestrial force decreases in 
approaching the equator ; but no precise law, according to which 
the intensity depends upon the distance from the equator, can 
be determined from these observations. Numberless observa- 
tions have since been made in both hemispheres, with every 
precaution to ensure accuracy in the results, but they do not in 
general accord with the theoretical formulse with which they 
have been compared. 

On the hypothesis of two magnetic poles not far removed 
from the centre of the earth, if 8 represent the dip, A the mag- 
netic latitude of the place of observation, 1 the intensity of the 
force in the direction of the dip, and m a constant, then 

iind therefore. 

-v/ (4 - 3 sin2 8)' 
tan S = 2 tan K ; 

I = |V(3sin^A+l); 

or if i is the angular distance from the magnetic pole, or the 
complement of the latitude, 


By comparing his own observations with the first of these 
formulae, Captain Sabine came to the conclusion that they were 
" decisive against the supposed relation of the force to the ob- 
served dip, and equally so against any other relation whatso- 
ever, in which the respective phaenomena might be supposed to 
vary in correspondence with each other." Comparing them, 
however, with the last formula, he concludes that "the accord- 
ance of the experimental results with the general law proposed 
for their representation, cannot be contemplated as otherwise 
than most striking and remarkable." How the same set of 
observations should be in remarkable accordance with the one 
formvda and at variance with the other, when these formulae are 
dependent on each other, it is difficult to conceive ; but the 
conclusion drawn by Captain Sabine from his observations, at 
least shows the danger of relying upon any single set of obser- 
vations as confirmatory or subversive of theoretical views. I 

122 THIRD RKPOKT — lS3o. 

have not yet compared with these results of theory the numer- 
ous observations made by Captain Foster, both in the northern 
and in the southern hemispheres ; but it is my intention to do 
this as soon as I can determine what correction ought to be 
made for the differences of temperature at the several stations: 
I do not, however, anticipate any very close accordance. 

In Captain Sabine's observations, the observed intensities, 
compared with those deduced from the preceding formulas, are 
in excess near the equator, and in defect near the pole ; and it 
is not improbable that, as Mr. Barlow has suggested, this in- 
crease of magnetic action near the equator above that which 
the theory gives, is due to the higher temperature in the equa- 
torial regions *. I am, however, disposed to assign even a more 
powerful influence than this to difference of temperature ; for I 
think it very possible, and indeed not improbable, that this may 
be the primary cavise of the polarity of the earth, although its 
influence may be much modified by other circumstances. At 
the conclusion of the paper on the diurnal variation f, to which 
I have already referred, I have suggested an experiment which 
I think might throw much light on this subject. I have pro- 
posed that a large copper sphere, of uniform thickness, should 
be filled with bismuth, the two metals being in perfect contact 
throvighout, and that experiments should be made with it simi- 
lar to those which I had made with one of smaller dimensions, 
but from which I was unable to obtain any very definite results, 
in consequence of the want of uniformity in the thickness of the 
copper and in the contact of the two metals. On heating the 
equator of such a sphere, the parts round the poles being cooled 
by caps of ice — which might not unaptly represent the polar ices, 
— we may expect that currents of electricity would be excited ; 
in which case the direction of those currents would decide whe- 
ther the experiment wei'e illustrative of the principal phaenome- 
non of terrestrial magnetism, or not. Should these currents of 
electricity be in the direction of the meridians, — which is impro- 
bable, since in this case opposing currents would meet at the 
poles, and there would be no means of discharge for them, — I 
think we might then conclude that the magnetism of the earth 
cannot be due to the difference in the temperature of its polar 
and equatorial regions ; but if, on the contrary, the currents 
should be in a direction parallel to the equator,^ — in which case 
their action upon a magnetized needle would be to urge it in 
the direction of the meridians, — I should then say that, in order 
to account for the terrestrial magnetic forces, and the diurnal 

* Edinburgh New Philosophical Journal, July 1 827. 
f Philosophical Transaclions, 1827, p. 354. 

REPORT On'tHE magnetism OF THE EARTH. 123 

changes in their direction and intensity, it would only be re- 
quired to show, that electrical phaenomena may be excited, in 
such bodies as the earth and the atmosphere, by a disturbance 
in their temperature when in contact. As I consider that if 
such an experiment were carefully made it must give conclusive 
results, I would strongly suggest to the Council of the British 
Association the importance of having it made. 

It has been a question whether the intensity of terrestrial 
magnetism is the same at the surface of the sea and at heights 
above that surface to which we can attain. MM. Gay-Lussac 
and Biot, in their aerostatic ascent, could detect no difference 
at the height of 4000 metres *. Saussure had, however, con- 
cluded, from the observations which he made at Geneva, Cha- 
mouni, and on the Col du Geant, that the intensity was consi- 
derably less at the latter station than at either of the former, 
the difference in the levels being in the one case about 10,000 
feet, in the other about 7800 f. 

M. Kupffer X also considers that his observations in the vi- 
cinity of Elbours, in which the difference of elevation of his two 
stations was 4500 feet, show clearly that the horizontal intensity 
decreases as we ascend above the surface ; and he accounts for 
this decrease not having been observed by MM. Biot and Gay- 

* Biot, Traite de Physique. 

f Voyages dans les Alpes, torn. iv. p. 313. — I take for granted that, admit- 
ting the accuracy of Saussure's observations, they warranted the conclusions 
he drew from them ; but some unaccountable errors must have crept in, either in 
transcribing or in printing them ; for not only the means which he deduces do not 
result from the observations, but the numbers which he employs contradict his 
conclusions. 1 transcribe the passage from the only edition I can consult, pub- 
lished at Neufchatel, 1796. " A' Geneve ces vingt oscillations employ erent 
5m 2'; 4™ 50'; 5"*; 4™ 40'; dont la moyenne etoit 5™ 0*"4; le thermometre 
6tant a 6 degres. A* Chamouni S"" 33' ; 5" 34' ; moyenne 5" 33''5 ; thermometre 
12 deg. Au Col du Geant 5™ 30"-3 ; S-" 30'-5 ; 5" 3r-4; 5^ 34'-6, moyenne 
5m 32"-45; thermometre 12-4 degres." 

" Or les forces magnetiques sont inversement comme les quarres des tems. 
Mais, a Geneve, le tems etoit ■')'" 0'-4 ou 300'.4, dont le quarre = 111155-56 ; 
a Chamouni 5" 33'-5 = 333'-5, dont le quarre = 111223. Au Geant 5" 
32'-45 = 332'-45, dont le quarre = 11523-0025 ; d'ou il suivroit que la plus 
grande force etoit dans laplaine, et la plus petite sur la plus haute montagne, 
a pen pres d'une cinquieme : observation bien importante, si elle etoit confirmee 
par des experiences repetees, et faites a la meme temperature." 

The means of the above observations are 4™ 53* = 293', 5'" 33'-5 = 333'-5, 
andS"! 31'-7 = 331*-7; and the squares of these numbers are 85849, 111222-25, 
110024-89. So that, according to this, the force was greatest at Geneva, and 
least at Chamouni. Taking Saussure's numbers, 300' -4, 333''-5, 332'-45, their 
squares are 90240-16, 111222-25, 110523-0025; so that still the general con- 
clusions are the same. 

t Voyage dans les Environs du Mont Elbronz. Rapport fait a I'Acadimie 
Jmpiriale des Sciences de St. Petersbourg, p. 88. 

124 THIRD REPORT— 1838. 

Lussac, by its having been counteracted by the increase of in- 
tensity, arising from the diminution of temperature. Mr. Hen- 
wood informs me that he has made corresponding observations, 
consisting of two series, each of 3900 vibrations at each place ; 
on Cairn Brea Hill, 710 feet above the level of the sea ; at the 
surface of Dolcoath mine, 370 feet above the sea ; and at a depth 
of 1320 feet beneath the surface in Dolcoath mine, or 950 feet 
below the level of the sea ; and that, after clearing the results 
from the effects of temperature, the differences are so minute 
that he cannot yet venture to say he has detected any difference 
in the magnetic intensity at these stations. If, notwithstanding 
these results, we are to admit the correctness of M. Kupffer's 
conclusions, I think we must infer that the diminution of hori- 
zontal intensity at his higher station was due to an increase in 
the dip, which element would not probably be so much affected 
by a change of elevation in a comparatively level country, like 
Cornwall, as on the flank of such a mountain mass as Elbours. 

Before dismissing the subject of the terrestrial intensity, I 
should mention that attempts have been made to delineate on 
charts the course of isodynamic lines. Professor Hansteen has 
published a chart in which this is done for the year 1824. Of 
all observations, however, requisite for graphic exhibitions con- 
nected with tei'restrial magnetism, those on the authority of 
which such lines must be drawn are fewest in number and least 
satisfactory in their results ; we should, therefore, be very cau- 
tious in drawing conclusions from such delineations. 

Hitherto I have only referred to such changes in the direction 
of the magnetic force, and in its intensity, as appear to depend 
upon general causes ; but, besides these, sudden and sometimes 
considerable irregular changes occur. These have very gene- 
rally been attributed to the influence of the aurora borealis, 
whether visible or not at the place of observation ; and I think 
it not improbable that some may be due to a peculiar electrical 
state of the atmosphere, independent of that meteor. The in- 
fluence of the aurora borealis on the magnetic needle has, how- 
ever, been denied by some, principally because, during the 
occurrence of that meteor at Port Bowen, Captain Foster did 
not observe peculiar changes in the direction of the needle, al- 
though, from his proximity to the magnetic pole, the diurnal 
change sometimes amounted to 4° or 5° ; and, under such cir- 
cumstances, it was considered that these changes ought to have 
been particularly conspicuous. In a paper inserted in the se- 
cond volume of the Journal of the Royal Institution, I have, 
however, shown that Captain Foster's Port Bowen observations 
<lo not warrant the conclusions which have been drawn from 


them, and have pointed out cu'cumstances which may, in this 
case, have rendered the effect of the aurora upon the horizon- 
tal needle less sensible than might have been expected. That 
changes in the direction and intensity of the terrestrial forces 
are simultaneous with the aurora borealis I feel no doubt, for I 
have seen the changes in the direction of the needle to accord 
so perfectly with the occurrence of this meteor, and to such an 
extent, that in my mind the connexion of the phgenomena be- 
came unquestionable*. As, however, the magnetic influence 
of the aurora boreahs has been doubted, I shall here point out 
the manner in which I consider the effects may be best ob- 

If the magnetic forces brought into action during an aurora 
are in the direction of the magnetic meridian, they will affect a 
dipping needle adjusted to the plane of that meridian, but the 
direction of an horizontal needle will remain unchanged : on 
the other hand, if the resultant of these forces makes an angle 
with the meridian, the direction of the horizontal needle will be 
changed, but the dipping needle may not be affected. In order 
to determine correctly the magnetic influence of the aurora by 
means of an horizontal needle, it is therefore necessary not only 
to have regard to those forces which influence its direction, but 
likewise to those which affect the horizontal intensity. The 
effects of the former are the objects of direct observation, but 
those of the latter are not so immediately observable. As, du- 
ring an aurora, the intensity may vai'y at every instant, — and it is 
these changes which are to be detected, — the method of deter- 
mining the intensity by the time of vibration of the needle can- 
not hei'e be applied, and other means must be adopted. The 
best method -appears to me to be that which I employed for 
determining the diurnal variation of the horizontal intensity, 
* the needle being retained nearly at right angles to the meridian 
by the repulsive force of a magnet, or by the torsion of a fine 
wire or thread of glass. For the purpose, then, of detecting 
in all cases the magnetic influence of the aurora, I consider that 
two horizontal needles should be employed ; one, adjusted in 
the meridian, for determining the changes which may take place 
in the direction of the horizontal force, and the other at right 
angles to the meridian, to determine the changes in the inten- 
sity of that force, arising principally from new forces in the 
plane of the meridian, and which would affect the direction of 
the dipping needle alone. Both these needles should be deli- 

* For the ohsei vations to which I here particularly refer, see the Journal of 
the Royal Inslitufiu/i, vol. ii. p. 272. 

126 THIRD REPORT 1833. 

cately suspended, either by very fine wire, or by untwisted 
fibres of silk. In order to render the changes in the direction 
of the needle in the meridian more sensible, its directive force 
should be diminished by means of two magnets north and south 
of it, and having their axes in the meridian. These magnets 
should be made to approach the needle until it points about 
30° on either side of the meridian, and they should be so ad- 
justed that the forces acting upon the needle will retain it i?i 
equiUbrio with its marked end at about 30° to the east and 30° 
to the west of north, and also at south. The needle is to be 
left with its marked end pointing south, for the purpose of ob- 
serving the changes occurring in its direction. If magnets are 
employed to retain the second needle nearly at right angles to 
the meridian, they should be made to approach its centre until 
the points of equilibrium are at about 80° east, 80° west and 
south, the observations being made with the needle at 80° east 
or 80° west. An objection to this method of adjusting this 
needle by means of magnets, and to which I have already re- 
ferred in a note, is that any change in their temperature will 
have a very sensible effect on the direction of the needle in this 
position ; and should such change take place dui'ing the ob- 
servations, corrections must be applied to the results before 
any accurate conclusions can be drawn from them. As, how- 
ever, an aurora is not generally of long continuance, any change 
in the temperature of the magnets during the observations is 
much more easily guarded against than where the observations 
have to be continued during successive days and at different 
seasons of the year. I have before remarked that this incon- 
venience will be, in a great measure, obviated by employing the 
torsion of a fine wire, or a very fine filament of glass, to retain 
the needle at about 80° from the meridian. In this case, the 
ratio of the force of torsion to the terrestrial force acting upon 
the needle having been determined, a measure will be obtained 
of the changes which take place in the intensity of the terres- 
trial force during the occurrence of an aurora. It is very de- 
sirable that it should be ascertained whether the effects on the 
needle are simultaneous with any particular class of phaenomena 
connected with the aurora ; whether these effects are dependent 
on the production of beams and corruscations, or on the forma- 
tion of luminous arches ; or whether any difference exists in the 
effects produced by these. In order to determine this, it is ne- 
cessary that the times of the occurrence of the different phae- 
nomena, and also of the changes in the directions of the needles, 
should be accurately noted ; and for such observations, three 
observers appear to be indispensable. 


Whether the direction of the needle may be influenced by 
the electrical state of the clouds, is much more doubtful than 
the influence of the aurora. I am not aware of any extended 
series of observations made with a view to determine this point. 
Having adjusted, in a particular manner, a needle between two 
magnets, so that the directive force was considerably diminished, 
I found that the changes in the positions of electric clouds was 
accompanied by changes in the direction of the needle ; but, 
although the observations indicate that the needle was thus 
affected, they are of too limited a nature to draw any general 
conclusion from*. Some observations of Captain Sir Kverard 
Home, however, indicate the same kind of influence. In a con- 
versation which I had with him last year, having referred to the 
effect I had observed to be produced by the sun's rays, of bring- 
ing a vibrating needle to rest, it brought to his mind a similar 
effect which he observed during a thunder-storm. He has fa- 
voured me with his observations, and from these it appears 
that, in two instances, a needle came sooner to rest during a 
thunder-storm than it had previous or subsequent to it. I'he 
arc at which the vibrations ceased to be counted is not recorded, 
but the number of vibrations was reduced in one case from 100 
to 40, and in another from 200 to 120. I have, in consequence 
of these observations, requested Lieutenant Barnett of the Royal 
Navy, who is engaged in the survey of the southern coast of the 
Gulf of Mexico, to make similar observations, should he have 
an opportunity; and as thundei'-storms are so frequent, and of 
svich intensity on that coast, I think he may obtain some im- 
portant results as connected with the influence of the electric 
state of the atmosphere upon the vibrations and direction of 
the needle. 

Upon a review of all the phasnomena of terrestrial magnetism, 
and considering the intimate relation which has been established 
between magnetism and electricity, by which it appears that, if 
not identical, they are only different modifications of the same 
principle, there can, I think, be little doubt that they are due 
to electric curx*ents circulating round the earth. How these 
currents are excited, whether by heat, by the action of another 
body, or in consequence of rotation, we are not at present able 
to determine ; but however excited, they must, though not 
wholly dependent upon them, be greatly modified by the phy- 
sical constitution of the earth's surface. We are, therefore, not 
to expect that symmetry in their course which would be the 

• Philosophical Transaction!!, 1823, p. 364. The arrangements which I have 
just described for determining the influence of the aurora borealis arc well 
adapted for deciding this point. 

128 THIRD REPORT — 1833. 

consequence of a symmetrical constitution of that surface. But 
even if such symmetry did exist, the action of all the currents 
at different stations on the surface could scarcely be referred 
to the same two points as centres of force ; and without this 
symmetry, it would be absurd to expect it. The hypothesis, 
therefore, of only two poles, as explanatory of the phaenomena, 
must be rejected ; and if we are to refer these phasnomena to 
centres of action, we must, besides two principal ones, admit 
the existence of others depending, upon local causes. 

It has been said that if we refer the magnetism of the earth 
to another body, we only remove the difficulty, and gain little 
by the supposition*. It, however, appears to me, that if we 
could show that the magnetism of the earth is due to the action 
of the sun, independent of its heat, — which, however, I think 
the more probable cause, — the problem would be reduced to 
the same class as that of accounting for the light of the sun, the 
heating and chemical properties of its rays : we only know the 
fects, and are not likely to know more. 

If difficulties meet us at every step when we attempt to ex- 
plain the general phaenomena of terrestrial magnetism, these 
difficulties become absolutely insurmountable when we come to 
the cause of their progressive changes. Here, at least, we 
must for the present be satisfied with endeavouring to discover 
whether these changes are governed by any general laws : 
should they be so, their cause may possibly be discovered. 
Diligent and careful observation is the only means by which 
we can hope to attain this end, and indeed is that on which we 
must principally rely for gaining a more correct knowledge of 
all the phaenomena, and of their causes ; and, consequently, im- 
provements in the methods of observation, and in the instrvi- 
ments to be employed, become of the highest importance. 

This Report has already so far exceeded the limits within 
which I wished to have confined it, that I must restrict the re- 
marks on this part of the subject to a few points. 

In the observations of Humboldt, in those of M. Rossel, of 
Captain Sabine, and of Captain Foster, the terresti'ial magnetic 
intensity had been determined by the vibrations of a dipping 
needle in the plane of the magnetic meridian ; but as there is 
by this means, in consequence of the friction upon the axis, a 
difficulty in obtaining a sufficient number of vibrations to ensure 
accuracy, and a dipping instrument is besides ill adapted for 
carriage. Professor Hanstepn proposed to determine the same 
by means of a small needle suspended horizontally by a few 

• Hansteen's Inquiries concerning the Magnetism of the Earth. 


untwisted fibres of silk. The advantages, however, attending 
this method of Professor Hansteen, I consider to be more ap- 
parent than real ; for without determining the dip, the hori- 
zontal force, deduced from the vibrations of the horizontal 
needle, cannot be reduced to the force in the direction of the 
dip ; and if the dip is determined, two instruments become ne- 
cessary where, before, only one was requisite. 

In order to obviate the inconveniences attending each of these 
methods, I have proposed a construction for a dipping needle, 
by means of which the observations which determine the di- 
rection of the terrestrial force will also give a measure of its 
intensity. The general principle of the construction is simply, 
that the centre of gravity of the needle should not be in its 
centre of figure, but in a line drawn from that centre at right 
angles, both to its axis of motion and to its magnetic axis ; so 
that, by two observations, one with the centre of gravity up- 
wards, and the other with it downwards, the dip, and likewise 
the relation which the static momentum of its weight bears to 
that of the terrestrial magnetic force acting upon the magnetism 
of the needle, may be determined. The principles on which 
these determinations depend, and the advantages which I pro- 
pose from the adoption of this construction, are fully described 
in a paper read before the Royal Society, and which will appear 
in the Philosophical Transactions of this year. 

Professor Gauss has proposed a method of determining the 
intensity and the changes it undergoes, by v/hich he hopes to 
reduce magnetical observations to the accuracy of astronomical 
ones. By the vibrations of a magnetized bar he determines the 
product of the terrestrial magnetic intensity by the static mo- 
mentum of its free magnetism. By introducing a second bar, 
and by observing at different distances the joint effects of the 
first, and of the terrestrial magnetism on this, he determines 
the ratio of the terrestrial intensity to the static momentum of 
the free magnetism of the first. Eliminating this last from the 
two equations, he obtains an absolute measure of the terres- 
trial magnetic intensity, independent of the magnetism of the 
bar. This is a most important result, for we shall thus be en- 
abled to determine the changes which the terrestrial intensity 
undergoes in long intervals of time. It is, however, to be ob- 
served, that it is only the horizontal intensity which is thus 
determined, and that, in order to determine the intensity of the 
whole force, another element, namely, the dip, must also be ob- 
served ; and I fear much that the introduction of this element 
will, in a great measure, counteract that accuracy of which the 
methods proposed for determining the times of vibration appear 

1833. K 

130 THIRD REPORT — 1833. 

capable. This must be an objection, even where the observa- 
tions are made in a fixed observatory ; but where an apparatus 
has to be moved from one station to another, I think the method 
could scarcely be applied successfully, principally on account 
of the delicacy of the preliminary observations, and of the time 
requisite for making them, in addition to that required for the 
observations by which the terrestrial intensity and its variations 
are to be determined. However greatly I may admire the saga- 
city which Professor Gauss has shown in devising means for 
the determination of an absolute measure of the horizontal in- 
tensity, I cannot avoid seeing the difficulties which may occur 
in its practical application. 

The method which Professor Gauss proposes, and has prac- 
tised, of observing the course of the daily variation, and of de- 
termining the time of vibration, by means of a plane mirror 
fixed on the end of the needle, perpendicularly to its axis, and 
observing the reflected image of the divisions of a scale by 
means of a theodolite fixed at a distance, appears to admit of 
the greatest possible precision, and will probably supersede 
other methods of observing the daily variation. 

I have adverted to the necessity of careful and dihgent ob- 
servation of all the phaenomena of terrestrial magnetism, as the 
surest means of arriving at a knowledge of their causes : it is 
with i-eluctance I state it, but I believe it to be a fact, that this 
is the only country in Europe in which such observations are not 
regularly carried on in a national observatory. Such an omission 
is the more to be regretted, seeing that no one has, I believe, 
cai'ried on a regular series of observations on the diurnal va- 
riation, since the valuable ones by Colonel Beaufoy were inter- 
rupted by his death, this interruption happening at a time when 
it was peculiarly desirable that the series should be unbroken. 
At this time the needle near London had begun to show a re- 
turn towards the true meridian ; but whether this was one of 
those oscillations which have occasionally been observed, or 
that, having really attained its maximum of westerly deviation, 
it was returning in the contrary direction, is, I believe, undecided 
at the present moment. Of all the data requisite for deter- 
mining the laws which govern the phaenomenon of the variation, 
the time of the maxima and their magnitude are the most im- 
portant. I trust that ere long the important desideratum will 
be supplied of a regular series of magnetical observations in 
the national Observatory of Great Britain. 

Royal Military Academy, 
22nd June, 1833. 

[ liJl ] 

Report on the present State of the Analytical Theory of Hydro- 
statics and Hydrodynamics. By the Rev. J. Challis, late 
Fellow of Trinity College Cambridge. 

The problems relating to fluids, which have engaged the atten- 
tion of mathematicians, may be classed under two heads, — those 
which involve the consideration of the attractions of the con- 
stituent molecules, and the repulsion of their caloric ; and those 
in which these forces are not explicitly taken account of. In 
the latter class the reasoning is made to depend on some pro- 
perty derived from observation. For instance, water is observed 
to be very difficult of compression ; and this has led to the 
assumption of iabsolute incompressibilityj as the basis of the 
mathematical reasoning : air at rest, and under a given state of 
temperature, is observed to maintain a certain relation between 
the pressure and the density ; hence the fundamental property 
of the fluid which is the subject of calculation is assumed to be 
the constancy of this relation, to the exclusion of all the circum- 
stances which may cause it to vary. The fluids treated of in 
this kind of problems are rather hypothetical than real, yet not 
so different from real fluids but that the mathematical deduc- 
tions obtained respecting them admit of having the test of ex- 
periment applied. I propose in this Report to confine myself 
entirely to problems of the second class, — those in the common 
theory of fluids. The reasons for making this limitation ai'e, 
that both kinds together would afford too ample matter for one 
Report, and that those which I have selected are distinguished 
from the others by the different purpose in regard to science 
which correct solutions of them would answer : for the treat- 
ment of any hydrostatical or hydrodynamical questions which 
involve the consideration of molecular attraction and the repul^ 
sion of heat, must proceed upon certain hypotheses respecting 
the mode of action of these forses, and the intei-ior constitution 
of the fluid, as these are circumstances which from their nature 
cannot be data of observation ; and hence, assuming the ma- 
thematical reasoning founded on the hypotheses to be correct, 
a satisfactory comparison of the theoretical deductions with 
facts must serve principally to establish the truth of the hypo- 
theses, and so to let us into secrets of natui'e which probably 
could never be known by any other process. But when the 


132 THIRD REPORT— 1833. 

basis of calculation, as in the questions that will come before 
us, is some observed and acknowledged fact, solutions which 
satisfy experiments will first of all serve to confirm the truth of 
the mathematical reasoning, and then give us confidence in the 
theoretical results, which, as often happens, cannot readily re- 
ceive the test of experiment. Calculations of this kind do not 
add much to our conviction that the facts applied as the test of 
the theory are really consequences of those which are the basis 
of it. For instance, we feel satisfied, independently of any ma- 
thematical reasoning, that the motions of waves on the surface 
of water are consequences of the incompressibility of the fluid, 
and the law of equal pressure. But the purpose which these 
calculations answer of confirming methods of applying analysis 
is very important, particularly in regard to the higher class of 
physical questions, which M. Poisson has proposed to refer to 
a distinct department of science, under the title of Math^ma- 
tique Physique, viz. those that require in their theoretical treat- 
ment some hypotheses respecting the interior constitution of 
bodies, and the laws of corpuscular action : for in questions of 
this nature, as well as in problems in the common theory of 
fluids, the mathematical reasoning conducts to partial differen- 
tial equations ; and if the method of treating these, and of 
drawing inferences from their integrals, be established in one 
kind, it may be a guide to the method to be adopted in the 
other. It is plainly, then, desirable that the mathematical pro- 
cesses be first confirmed in the cases in which the basis of rea- 
soning is an observed fact, that the reasoning may proceed with 
certainty in those cases where it is based on an hypothesis, the 
truth of which it proposes to ascertain. 

The subjects of this Report may now be stated to be, the 
leading hydrostatical and hydrodynamical problems recently 
discussed, which proceed upon the supposition of an incom- 
pressible fluid, or of a fluid in which the quotient of the pres- 
sure divided by the density is a constant ; and the end it has 
in view is, to ascertain to what extent, and with what success, 
analysis has been employed as an instrument of inquiry in these 
problems. I am desirous it should be understood that I have 
not attempted to make a complete enumeration either of the 
questions that have been discussed in this department of science, 
or of the labours of mathematicians in those which have come 
under notice. It has rather been my endeavour to give some 
idea of the most approved methods of treating the leading 
problems, and the possible sources of error or defect in the 
solutions. In taking this course I hope I may be considered to 
.have acted sufficiently in accordance with the recommendation 


of the Committee for Mathematics, which was the occasion of 
my receiving the honour of a request to take this Report in 

With the hmitation above stated as to the subjects our Re- 
port is to embrace, we shall have scarcely anything to say on 
the analytical theory of hydrostatics. The problems of interest 
in this department were early'solved, and present no difficulty 
in principle, and little in the detail of calculation. The deter- 
mination of the height of mountains by the barometer is a 
hydrostatical question, the difficulty of which does not consist 
in the analytical calculation, but only in ascertaining the law of 
the distribution of the atmospheric temperature. We shall not 
have to speak of the theories that have been invented to over- 
come this difficulty. Neither does it fall within the scope of this 
Report to notice the very valuable memoir of M. Poisson on 
the equilibrium of fluids *, which has for its object the deriva- 
tion of the general equations of equilibrium from a consideration 
of molecular attraction and the repulsion of caloric, and seems 
to have been composed in immediate reference to the theory 
of capillai'y attraction, which the author svibsequently pub- 
lished. With regard to the problem of capillary attraction, we 
may remark, that it is not possible by any supposition respect- 
ing the forces which sustain or depress the fluid in the tube, 
to solve it as a question in the common theory of hydrostatics. 
M. Poisson has shown the insufficiency of Laplace's theory, 
and by taking into account the molecular forces and the eiFect 
of heat, has proved that the explanation of the phasnomenon is 
essentially dependent on a modification of the property which 
is the basis of the common theory, viz, the incompressibility of 
the fluid. It does not fall within our province to say more on 
the celebrated theory of M. Poisson. 

One improvement I consider to have been recently made in 
the common theory of fluids. It has been usual to take the law 
of equal pressure as a datum of observation. Professor Airy, 
in his Lectures in the University of Cambridge, has shown that 
this property may be derived, by reasoning according to esta- 
blished mechanical principles, from another of a simpler kind, 
the notion of which may be gathered from observation, viz. 
that the division of a perfect fluid may be effected without the 
application of sensible force ; from which it immediately follows 
that the state of equilibrium or motion of a fluid mass is not 
altered by mere separation of its parts by an indefinitely thin 
partition. A definition of fluids founded on this principle, and 

• Memoircs de I' Academic des Sciences, Paris, torn. ix. 1830. 

134 THIRD REPORT — 1833. 

a proof of the law of equal pressure, are given at the beginning 
of the Elements of Hydrostatics and Hydrodynamics of Pro- 
fessor Miller *. Several advantages attend this mode of com- 
mencing the mathematical treatment of fluids. The principle 
is one which perfectly characterizes fluids, as distinguished in 
the internal arrangement of their particles from solids. It may 
be rendered familiar to the senses. It is, I think, necessary for 
the solutions of some hydrostatical and hydrodynamical pro- 
blems, particularly those of reflection f. Lastly, in reference 
to the department of science proposed to be called Physical 
Mathematics, the propositions of the common theory ought to 
be placed on the simplest possible basis, because the questions 
of most interest in that department are those which have in 
view the explanation of the phasnomena that are the founda- 
tions of the reasoning in the other kind. The solution of one 
such question is a great step in scientific generalization. It is 
plainly, therefore, of importance that the fact proposed for ex- 
planation should be the simplest that direct observation can 
come at. 

The analytical theory of hydrodynamics is of a much more 
difficult nature than that of hydrostatics. The assumptions it is 
necessary to make to obtain even approximate solutions of the 
simplest problems of fluid motion betray the difficulty and im- 
perfection of this part of science. There are cases, however, of 
steady motion, that is, of motion which has arrived at a pei'ma- 
nent state, so that the velocity is constantly the same in quantity 
and direction at the same point, which require a much more 
simple analysis than those which do not satisfy this condition. It 
does not appear that the equations applicable to this kind of mo- 
tion were obtained in any general manner till they were given in 
an Elementary Treatise on Hydrostatics and Hydrodynaynics 
by Mr. Moseley %, who has derived them from a principle of so 
simple a nature, that, as it can be stated in a few words, it may 
be mentioned here. When the motion is steady, each particle 
in passing from one point to another, passes successively through 
the states of motion of all the particles which at any instant lie 
on its path. This principle is valuable for its generality : it is 
equally applicable to all kinds of fluids, and will be true, whe- 
ther or not the efl^ect of heat be taken into accoimt, if only the 
condition of steadiness remains. The equations of motion are 
readily derived from it, because it enables us to consider the 

.* Cambridge 183 J. 
t Dr. Young employed an equivalent principle to determine the manner of 
the reflection of waves of water. See his Natural Philosophy, vol. ii. p. 64. 
t Cambridge 1830. 


motion of a single particle, in the place of the motion of an 
aggregate of particles. Though this mode of deriving them is 
the best possible on account of its simplicity, it was yet de- 
sirable to know how they may be obtained from the general 
equations of fluid motion. In a paper contained in the Trans- 
actions of the Philosophical Society of Cambridge *, the author 
of this Report has given a method of doing this, both for incom- 
pressible and elastic fluids, and has shown that a term in the 
general formulae which gives rise to the complexity common to 
most hydrodynamical questions, disappears for this kind of 
motion. Euler had already done the same for incompressible 
fluids f . The instances in nature of fluid motion of the steady 
kind are far from uncommon ; and it is probable that when the 
equations applicable to them are better known, and studied 
longer, they may be employed in very interesting researches. 
The motion of the atmosphere, as affected by the rotation of 
the earth, and a given distribution of the temperature due to 
solar heat, seems to be an instance of this kind. 

We will now proceed to consider in order the principal hydro- 
dynamical problems that have recently engaged the attention 
of mathematicians. For convenience we shall class them as 
follows : — I. Motion in pipes and vessels. II. The velocity of 
propagation in elastic fluids. III. Musical vibrations in tubes. 
IV. Waves at the surface of water. V. The resistance to the 
motion of a ball-pendulum. 

I. The motion of fluids in pipes and vessels has not been 
treated with any success, except in the cases in which the con- 
dition of steadiness is fulfilled. The paper above alluded to, 
in the Transactions of the Philosophical Society of Catnbridge, 
contains some applications of the equation of steady motion for 
incompressible fluids, to determine the velocity of water issuing 
from different kinds of adjutages in vessels of any shape : also 
a theoretical explanation of a phsenomenon which a short while 
ago excited some attention, — that of the attraction of a disc to 
an orifice through which a steady current either of water or air 
is issuing. 

In the Memoirs of the Paris Academy of Sciences X there is 
an Essay by M. Navier on the motion of elastic fluids in ves- 
sels, and through different kinds of adjutages into the sur- 
rounding air, or from one vessel into another. For the sake of 
simplicity the author considers the fluid to be subject to a con- 
stant pressure, and consequently the motion to have arrived at 
a state of permanence. His calculations are founded upon the 

• Vol. iii. Part III. f Memoires de I'/lcadimie de Berlin, 1755, p. 344. 
X Tom. ix. 1830. 

136 THIRD REPORT— 1883. 

hypothesis of parallel slices, which assumes the velocity to be 
the same, and in the same direction, and the density to be the 
same at all points of any section transverse to the axis of the 
vessel or pipe. This hypothesis is one of those that the theory 
of hydrodynamics has borrowed from experience to supply its 
defects. Lagrange has, however, shown theoretically* that it 
always furnishes a first approximation, the breadth of the ves- 
sel being considered a quantity of the first order, and the effect 
of the adhesion of the fluid to the sides of the vessel being neg- 
lected. It is right to observe, that in the problems M. Navier 
has considered, this hypothesis might have been in a great 
measure dispensed with : the expression he has given, — more 
correct than that commonly adopted for the velocity of issuing 
through a small aperture by which airs of different densities 
communicate, — ^might have been obtained by employing the equa- 
tion above mentioned of steady motion, as, in fact, Mr. Moseley 
has done f. This would be a preferable mode of treating such 
questions, because in every instance in which these auxiliary 
hypotheses are got rid of, something is gained on the side of 
theory. This memoir contains another hypothesis, which can- 
not be so readily dispensed with. Theory is at present quite 
inadequate to determine the retardation in the flow of fluids 
occasioned by sudden contractions or widenings in the bore of 
the pipe. It is found by experiments with water, that the re- 
tardation is sufficiently represented by taking account of the 
loss of vis viva which, on the hypothesis of parallel slices, will 
result from the sudden changes of velocity which must be sup- 
posed to take place at the abrupt changes in the bore of the 
pipe. M. Navier extends these considerations to elastic fluids. 
The theory manifests a sufficient agreement with the experi- 
ments it is compared with, and is valuable on account of the 
applications it may receive. 

II. The most interesting class of problems in hydrodynamics 
are perhaps those which relate to small oscillations. Newton 
was the first to submit the vibrations of the air to mathema- 
tical calculation. The propositions in the second book of the 
Principia, devoted to this subject, and to the determination of 
the velocity of sound, may be ranked among the highest pro- 
ductions of his genius. He has assumed that the vibratory 
motion of the particles follows the law of the motion of an oscil- 
lating pendulum. It was soon discovered that many other 
assumed laws of vibration would, by the same mode of reasoning, 

* Mecanique Analytique, P&tt II. § xi. art. 34. 
t Elementary Treatise, p. 204. 


conduct to the same velocity of propagation. This, which was 
thought to be an objection to the reasoning, is an evidence of 
its correctness : for the plain consequence is, that the velocity 
of propagation is independent of the kind of vibration which 
we may arbitrarily impress on the fluid ; — and so experience 
finds it to be. 

When the partial differential equation, which applies equally 
to the vibrations of the air and those of an elastic chord, had 
been formed and integrated, a celebrated discussion arose 
between Euler and D'Alembert as to the extent to which the 
integral could be applied ; whether only to cases in which the 
motion was defined by a continuous curve, or also to motion 
defined by a broken and discontinuous line. It is well known 
that the question was set at rest by Lagrange, in two Disser- 
tations published in vol. i. and vol. ii. of the Miscellanea Tau- 
rinensia. The difficulty that arrested the attention of these 
eminent mathematicians was one of a novel kind, and peculiar 
to physical questions that require for their solution the integrals 
of partial differential equations. The difficulty of integration, 
which is the obstacle in most instances, had been overcome by 
D'Alembert. It remained to draw inferences from the inte- 
gral, — to interpret the language of analysis. When an aggre- 
gate of points, as a mass of fluid or an elastic chord, receives 
an arbitrary and irregular impulse, any point not immediately 
acted upon may have a correspondent irregular movement after 
the initial disturbance has ceased. This is a matter of experi- 
ence. Was it possible, then, that these irregular impulses, and 
the consequent motions, were embraced by the analytical calcu- 
lation ? From Lagrange's researches it follows that the func- 
tions introduced by integration are arbitrary to the same degree 
that the motion is so practically, and that they will therefore 
apply to discontinuous motions. (Of course we must except 
the practical disturbances which the limitations of the calcula- 
tion exclude, — those which are very abrupt, or very large.) 
This has been a great advance made in the application of ana- 
lysis to physical questions. Had a diff'erent conclusion been 
arrived at, many facts of nature could never have come under 
the power of calculation. The Researches of Lagrange, which 
will ever form an epoch in the science of applied mathematics, 
estabUsh two points principally : First, That the arbitrary func- 
tions, as we have been just saying, are not necessarily conti- 
nuous : Secondly, That (in the instance he considered) they are 
equivalent to an infinite series of terms having arbitrary con- 
stants for coefficients, and proceeding according to the sines of 
multiple arcs. This latter result, which appears to be true for 

138 THIRD REPORT — 1833. 

all linear partial differential equations of the second order, with 
constant coefficients, is valuable as presenting an analogy be- 
tween arbitrary constants and arbitrary functions. 

But the way in which Lagrange, after establishing these two 
points, proceeds to find the velocity of propagation, does not 
appear to me equally satisfactory with the rest of his reasoning. 
His method seems to be a departure from the principle which 
may be gathered from that of Newton. For, as was mentioned 
above, the reasoning of the Principia shows that the velocity of 
propagation is independent of all that is arbitrary. It seems 
important to the truth of the analytic reasoning, that it should 
not only obtain a constant velocity of propagation, but arrive at 
it by a process which is independent of the arbitrary nature of 
the functions ; whereas the method which the name of La- 
grange has sanctioned, is essentially dependent on the discon- 
tinuity of the functions, that is, on their being arbitrary. With 
a view of calling attention to this difficulty, and as far as possi- 
ble removing it, the author of this Report read a paper before 
the Philosophical Society of Cambridge, Avhich is published in 
Vol. iii. Part I. of their Transactions. I am far from assert- 
ing that that Essay has been successful; but some service, I 
think, will be done to science if it should lead mathematicians to 
a reconsideration of the mode of mathematical reasoning to be 
employed in regard to the ajjplications of arbitrary ftinctions. 
If the determination of the velocity of propagation in elastic 
fluids were the only problem affected by this treatment of arbi- 
trary functions, it would not be worth while to raise a question 
respecting the principle of the received method, as no doubt 
attaches to the result obtained by it ; but there are other pro- 
blems, (one we shall have to consider,) the correct solutions of 
which mainly depend on the construction to be put upon these 
functions. The difficulty I am speaking of, which is one of a 
delicate and abstx*act nature, will perhaps be best understood 
by the following queries, which seem calculated to bring the 
point to an issue : — Can the arbitrary functions be immediately 
applied to any but the parts of the fluid immediately acted upon 
by the arbitrary disturbance, and to parts indefinitely near to 
these ? To apply them to parts more remote, is it not necessary 
first to obtain the law of propagation ? And do not the arbi- 
trary functions themselves, by the quantities they involve, fur- 
nish us with means of ascertaining the law of propagation, 
independently of any consideration of discontinuity ? 

Euler and Lagrange determined the velocity of propagation 
in having regard to the three dimensions of the fluid, on the li- 
mited supposition that the initial disturbance is the same as to 


density and velocity, at the same distance in every direction 
from a fixed point, which is the centre of it. Laplace first dis- 
pensed with this limitation in the case in which two dimensions 
only of the fluid are taken account of*. The principal cha- 
racter of his analysis is a new method of employing definite 
integrals. Finally, M. Poisson solved the same problem for 
three dimensions of the fluid f. This memoir deserves to be 
particularly mentioned for the interesting matter it contains. 
The object of the author is to demonstrate, in a more general 
manner than had been before done, some circumstances of the 
motions of elastic fluids which are independent of the particular 
motions of the fluid particles, such as propagation and reflection. 
The general problem of propagation just mentioned he solves 
by developing the integral of the partial differential equation of 
the second order in x, y, z, and t, applicable to this case, in a 
series proceeding according to decreasing powers of the di- 
stance from the centre of disturbance, as it cannot be obtained 
in finite terms, and then transforming the series into a definite 
integral, — a method which has of late been extensively em- 
ployed. The crossing of waves simultaneously produced by 
disturbances at several centres, is next considered, and this 
leads to a general solution of the problem of reflection at a plane 
surface. For the case in which the motions of the aerial parti- 
cles are not supposed small, the velocity of propagation along 
a line of air is shown to be the same as when they are small. 
This result is an inference drawn from the arbitrary disconti- 
nuity of the motion, on which it does not seem to depend. In 
a paper before alluded to J, the same result is obtained without 
reference to the principle of discontinuity. M. Poisson treats 
also of propagation in a mass of air of variable density, such as 
the earth's atmosphere. His analysis is competent to prove, in 
accordance with experience, that the velocity of sound is the 
same as in a mass of uniform density, and that its intensity at 
any place depends, in addition to the distance from the point 
of agitation, only on the density of the air where the disturbance 
is made. So that a bell rung in the upper regions of the air 
will not sound so loud as when rung by the same effort below, 
but will sound equally loud at all equal distances from the place 
where it is rung. 

In seeking for the general equations of the motion of fluids, 
(first obtained by Euler,) a quantity § is met with which, if it be 

* Memoires de V Academic, An 1779. , 

t " Memoire sur la Theorie du Son," Journal de I'Ecole Polytechnique, 
torn. vii. cah. xiv. 

J Transactions of the Philosophical Society of Cambridge, vol. iii. Part III. 
§ In M. Poisson's writings this quantity is udx + v dy + ir dz. 

140 THIRD REPORT 1833. 

an exact differential of a function of three variables, i-enders the 
subsequent analytical reasoning much simpler than it would be 
in the contrary case. This simplification has been proved by 
Lagrange to obtain in most of the problems of interest that are 
proposed for our solution *. Euler showed that the differential 
is inexact when the mass of fluid revolves round an axis so that 
the velocity is some function of the distance from the axisj-. 
But no general method exists of distinguishing the ipstances in 
which the quantity in question is a complete differential, and 
when it is not. Nor is it known to what physical circumstance 
this peculiarity of the analysis refers. To clear up this point 
is a desideratum in the theory of hydrodynamics. M. Poisson 
has left nothing to be desired in the generality with which he 
has solved the problem of propagation of motion in elastic fluids; 
for in the Memoirs of the Academy of Paris \ he has given 
a solution of the question, without supposing the initial disturb- 
ance to be such as to make the above-mentioned quantity an 
exact differential. His conclusions are, that the velocity of 
propagation is the same as when this supposition is made ; that 
the part of the motion which depends on the initial condensa- 
tions or dilations follows the same laws as in that case, but 
the part depending on the initial velocity does not return com- 
pletely to a state of repose after a determinate interval of time ; 
that at great distances from the place of agitation there is no 
essential difference between the motion in the two cases. 

HI. We turn now to the theory of musical vibrations of the 
air in cylindrical tubes of finite length. Little has been effected 
by analysis in regard to this interesting subject. The principal 
difficulty consists in determining the manner in which the mo- 
tion is affected by the extremities of the tube, whether open or 
closed, but particularly the open end. Those who first handled 
the question reasoned on the hypotheses, that at the open end 
the air is always of the same density as the external air to 
maintain an equilibrium with it, and at the closed end always 
stationary by reason of the stop. The latter supposition will 
be true only when the stop is perfectly rigid. It does not ma- 
terially affect the truth of the reasoning ; but if the other sup- 
position were strictly true, the sound from the vibrating column 
of air in the tube would not cease so suddenly as experience 
shows it does, when the disturbing cause is removed ; neither 
on this hypothesis could the external air be acted on so as to 
receive alternate condensations and rarefactions, and transmit 

* Mecanique Analytique, Part II. § xi. art. 16. 

t Memoires de I' Academic de Berlin, 1 755, p. 292. 

% torn. X. 1831. 


sonorous waves. These objections to the old theory have been 
stated by M. Poisson, who proposes a new mode of considering 
the problem*. He reasons on an hypothesis which embraces 
both the case of an open and a closed end, viz. that the velo- 
city at each is in a constant ratio to the condensation. This 
ratio will be very large for the open end, and a very small frac- 
tion for the closed end. Its exact value in the latter case de- 
pends on the elasticity of the stop, and in the other on the mode 
of action of the vibrations on the external air, — to determine 
which is a problem of great difficulty, which M. Poisson has 
forborne to meddle with. His theory is not competent to assign 
a priori either the series of tones or the gravest that can be 
sounded by a tube of given length, but is more successful in 
determining the number of nodes and loops, and the intervals 
between them, when a given tone is sounded. To find the di- 
stances of the nodes and loops from the extremities of the tubes, 
he has recourse to the hypotheses of the old theory, which 
make the closed end the position of a node, and the open end 
the position of a loop. This, he says, will not be sensibly dif- 
ferent from the truth, if, in the one case, the stop be very un- 
yielding, and, in the other, the diameter of the tube be small. 
Recent researches on this subject, which we shall presently 
speak of, show that when the diameter is not very small the 
position of the loop is perceptibly distant from the open end. 

The latter part of M. Poisson's memoir contains an applica- 
tion of the principles of the foregoing part to the vibrations of 
air in a tube composed of two or more cylinders of different dia- 
meters, and to the motion of two different fluids superimposed 
in the same tube. In the course of this latter inquiry, the au- 
thor determines the reflection which sound experiences at the 
junction of two fluids ; and by an extension of like considerations 
to luminous undulations, obtains the same expressions for the 
relative intensities of light perpendicularly incident, and re- 
flected at a plane surface, as those given by Dr. Young in the 
Article Chromatics of the Supplement to the Encyclopcedia 
Britannica. This subject was afterwards resumed by M. Pois- 
son at greater length in a very elaborate memoir " On the Mo- 
tion of two Elastic Fluids superimposed f," which is chiefly 
remarkable for the bearing which the results have upon the 
theory of light. 

At the last meeting, in May this year, of the Philosophical So- 
ciety of Cambridge, a paper was read by Mr. Hopkins, in which, 

* Memoires de V Acad^mie des Sciences, Paris, An 1817. 
t Ibid. torn. X. p. 317. 

142 THIUD REPORT — 1833. 

by combining analysis witb a delicate set of experiments, re- 
sults are obtained which are a valuable addition to this part of 
the theory of fluid motion. His experiments were made on a 
tube open at both ends, and the column of air within it was put 
in motion by the vibrations of a plate of glass applied close to 
one end. The following are the principal results. The nodes 
are not points of quiescence, but of minimum vibration ; — the 
extremity of the tube most remote from the disturbance is not 
a place of maximum vibration, but the whole system of places 
of maximum and minimum vibration is shifted in a very sensible 
degree towards it ; — the distances of the places of maximum 
and minimum vibration from each other, and from that extre- 
mity, remain the same for the same disturbance, whatever be the 
length of the tube. This last fact Mr. Hopkins proves by his 
analysis must obtain. The shifting of the places of maximum 
and minimum vibration is not accounted for by the theory : nor 
is it probable that it can be, unless the consideration of the 
mode of action of the vibrations on the external air be entered 
upon, — an important inquiry, but, as I said before, one of great 
difficulty. I think also that the effect of the vibrations of the 
tube itself on the contained air ought to be taken into account. 

IV. The problem of waves at the surface of water is princi- 
pally interesting as furnishing an exercise of analysis. The 
general difterential equations of fluid motion assume a very sim- 
ple form for the case of oscillations of small velocity and extent, 
and seem to oiFer a favourable opportunity for the application 
of analytical reasoning. Yet mathematicians have not succeed- 
ed in giving a solution of the problem in any degree satisfactory, 
which does not involve calculations of a complex nature. We 
need not stay to inquire in what way Newton found the velo- 
city of the propagation of waves to vary as the square root of 
their breadths : he was himself aware of the imperfection of his 
theory. The question cannot be well entered upon without 
partial differential equations. Laplace was the first to apply to 
it a regular analysis. His essay is inserted at the end of a 
memoir on the oscillations of the sea and the atmosphere, in 
the volume of the Paris Academy of Sciences for the year 1776. 
The differential equations of the motion are there formed on 
the supposition that the velocities and oscillations are always so 
small that their products, and the powers superior to the first, 
may be neglected. The problem without this limitation be- 
comes so complicated that no one has dared to attempt it. La- 
place's reasoning conducts to a linear partial differential equa- 
tion of the second order, consisting of two terms, which is 
readily integrated; but on account of the difficulty of obtaining a 


general solution from this integi'al, he makes a particular sup- 
position, which is equivalent to considering the fluid to be de- 
ranged from its state of equilibrium by causing the surface in 
its whole extent to take the form of a trochoid, i. e. a serpentine 
curve, of which the vertical ordinate varies as the cosine of the 
horizontal abscissa. The solution in question is of so limited 
a nature, that we may dispense with stating the results arrived 

In the volume of the Memoirs of the Academy of Berlin for 
the year 1786, Lagrange has given* a very simple way of 
proving, in the Newtonian method of reasoning, that the ve- 
locity of propagation of waves along a canal of small and con- 
stant depth and uniform width, is that acquired by a heavy 
body falling through half the depth. In the Mecanique Ana' 
li/tiquef the same result is obtained analytically. The princi- 
pal feature of the analysis in this solution is, that the linear 
partial differential equation of the second order and of four va- 
riables, to which the reasoning conducts, is integrated approxi- 
mately in a series. Lagrange is of opinion, that on account of 
the tenacity and mutual adherence of the parts of the fluid, the 
motion extends only to a small distance vertically below the 
surface agitated by the waves, of whatever depth the fluid may 
be ; and that his solution will consequently apply to a mass of 
fluid of any depth, and will serve to determine, from the ob- 
served velocity of propagation, the distance to which the motion 
extends downwards. 

The problem of waves was proposed by the French Institute 
for the prize subject of 1816. M. Poisson, whose labours are 
preeminent in every important question of Hydrodynamics, had 
already given this his attention. His essay, which was the first 
deposited in the bureau of the Institute, was read Oct. 2, 1815, 
just at the expiration of the period allowed for competition. It 
forms the first part of the memoir " On the Theory of Waves," 
published in the volume of the Academy for the year 1816, and 
contains the general formulas required for the complete solution 
of the problem, and the theory, derived from these formulae, of 
waves propagated with a nniformly accelerated motion. In the 
month of December following, an additional paper was read by 
M. Poisson on the same subject, which forms the second part 
of the memoir just mentioned, and contains the theory of waves 
propagated with a constant velocity. These are much more 
sensible than the waves propagated with an accelerated motion, 
and are in fact those which are commonly seen to spread in 

• p. 192. \ Part II. sect. xi. ait. 3G. 

144 THIRD REPORT — 1833. 

circles round any disturbance made at the surface of water. 
No theory of waves which does not embrace these can be con- 
sidered complete. In the essay of M. Cauchy, which obtained 
the prize, and is printed in the Mdmoires des Savans*, the 
theory of only the first kind of waves is given. This essay, 
however, claims to be more complete than the first part of 
M. Poisson's memoir, because it leaves the function relative to 
the initial form of the fluid surface entirely arbitrary, and conse- 
quently allows of applying the analysis to any form of the body 
immersed to produce the initial disturbance. M. Poisson re- 
stricts his reasoning to a body, of the form of an elliptic para- 
boloid, immersed a little in the fluid, with its vertex downward 
and axis vertical ; and as this form may have a contact of the 
second order, with any continuous surface, the reasoning may 
be legitimately extended to any bodies of a continuous form, 
but not to such as have summits or edges, like the cone, cy- 
linder and prism. This restriction having been objected to as a 
defect in the theoryf , M. Poisson answers J that his analysis 
is not at fault, but that one of the differential equations of the 
problem, which expresses the condition that the same particles 
of water remain at the surface during the whole time of motion, 
very much restricts the form which the immersed body may be 
supposed to have. When the initial motion is produced by the 
immersion of a body whose surface presents summits or edges, 
it is not possible, he thinks, to represent the velocities of the 
fluid particles by analytical formulae, especially at the first in- 
stants of the agitation, when the motion must be very complicated, 
and the same points will not remain constantly at the surface. 

With the exception of the particular we have been mention- 
ing, the two essays do not present mathematical processes es- 
sentially different in principle. Attached to that of M. Cauchy, 
which was published subsequently to M. Poisson's memoir, are 
Valuable and copious additions, serving to clear up several 
points of analysis that occur in the course of the work, and re- 
ferring chiefly to integration by series and definite integrals, 
and to the treatment of arbitrary functions. Among these is a 
lengthened discussion of the theory of the waves uniformly 
propagated, the existence of which, as indicated by the analysis, 
had escaped the notice of both mathematicians in their first re- 
searches. In this discussion the velocities of propagation are 
determined of the two foremost wa,ves produced by the immer- 

♦ vol. lii. 

f Bulletin de la Societe Philomatiqiie, Septembre 1818, p. 129. 
X "Note sur le Probleme des Ondes," torn. viii. of Mimoires de I'Acadhnie 
des Sciences, p. 371. 


sion and sudden elevation of bodies of the forms of a parabo- 
loid, a cylinder, a cone, and a solid, generated by the revolution 
of a parabola about a tangent at its vertex. To bodies of the 
last three forms, M. Poisson objects to extending the reasoning; 
and in the " Note" above referred to, attempts to show that such 
an extension leads to results inconsistent with the principle of 
the coexistence of small vibrations. If we are not permitted to 
receive the analysis of M. Cauchy in all the generality it lays 
claim to, we must at least assent to the reasonableness of the 
following conclusion it pretends to arrive at, viz. that "the 
heights and velocities of the different waves produced by the 
immersion of a cylindrical or prismatic body depend not only 
on the width and height of the part immersed, but also on the 
form of the surface which bounds this part." There is also 
much appearance of probability in a remark made by the 
same mathematician, that the number of the waves produced 
may depend on the form of the immersed body and the depth 
of immersion. 

We proceed to say a few words on the contents of M. Pois- 
son's memoir. He commences by showing, as well by a priori 
reasoning as by an appeal to facts, that Lagrange's solution 
cannot be extended to fluid of any depth. In his own solution 
he supposes the fluid to be of any uniform depth, but princi- 
pally has regard to the case which most commonly occurs of a 
very great depth : he neglects the square of the velocity of the 
oscillating particles, as all have done who have attempted this 
problem, and assumes, that a fluid particle which at any instant 
is at the surface, remains there during the whole time of the 
motion. This latter supposition seems necessary for the con- 
dition of the continuity of the fluid. With regard to the neg- 
lect of the square of the velocity, it does not seem that we can 
tell to what extent it may affect the calculations so well as in 
the case of the vibrations of elastic fluids, where the velocity of 
the vibrating particle is neglected in comparison of a known and 
constant velocity, that of propagation. M. Poisson treats first 
the case in which the motion takes place in a canal of uniform 
width, and, consequently, abstraction is made of one horizontal 
dimension of the fluid ; and afterwards the case in which the 
fluid is considered in its three dimensions. The former requires 
for its solution the integration of the same differential equation 
of two terms * as that occurring in Laplace's theory. No use 
is made of the common integral of this equation, as, on account 
of the impossible quantities it involves, it would be difficult 

* In M. Poissoii's works this equation is '-1? -f — = 0, 

1833. T. 

146 THIRD REPORT — 1833. 

to make it serve to determine the laws of propagation. It is 
remarkable that this integral is not necessary for solving the 
problem, although, as M. Poisson has shown in his first me- 
moir, " On the Distribution of Heat in Solid Bodies," and M. 
Cauchy in the Notes added to his " Theory of Waves," a solu- 
tion may be derived from it equivalent to that which they have 
given without its aid. We may be permitted to doubt whether 
its meaning is yet fully understood, and to hope that, by over- 
coming some difficulty in the interpretation of this integral, the 
problem of waves may receive a simpler solution than has hi- 
therto been given. Be this as it may, the process of integration 
adopted by M. Poisson leaves nothing to be wished for in regard 
to generality. It is easy to obtain an unlimited number of pai-- 
ticular equations not containing arbitrary functions, which will 
satisfy the differential equation in question, and to combine 
them all in an expression for the principal variable (ip), deve- 
loped in series of real or imaginary exponentials. This will be 
the most general integral the equation admits of, and (to use 
the words of M. Poisson,) " there exist theorems, by means of 
which we may introduce into expressions of this nature, arbi- 
trary functions, which represent the initial state of the fluid : 
the difficulty of the question consists then in discussing the re- 
sulting formulas, and discovering from them all the laws of the 
phasnomenon. The theory of waves furnishes at present the 
most complete example of a discussion of this sort." 

In a Report like the present, it is not possible to give any 
very precise idea of the analysis which has been employed for 
solving the problem of waves. I have thought it proper to call 
attention to a process of reasoning which has been very exten- 
sively employed by the French mathematicians of the present 
day, and indeed may be considered to be the principal feature 
of their calculations in the more recent applications of mathe- 
matics to physical and mechanical questions. To understand 
fully the nature and power of the method, the works of Fourier, 
particularly The Analytical Theory of Heat ^ the Notes, before 
spoken of, to M. Cauchy's " Theory of Waves," and the two 
memoirs of M. Poisson " On the Distribution of Heat in Solid 
Bodies," must be studied. I will just refer to some parts of the 
writings of the last-mentioned geometer, where he has been 
careful to state in a concise manner the principle of the method 
in question. There are some remarks on the generality of a 
main step in the process in the Bulletin de la Societe Philoi7ia- 
tique*. The note before spoken of in the eighth volume of the 

* An 1817, p. ISO. 


Memoirs of the Academy concludes with a brief account of the 
history and principle of this way of expressing the complete 
integral by a series of particular integrals, and introducing the 
arbitrary function. But 1 would chiefly recommend the peru- 
sal of the remarks at the end of a memoir by this author " On 
the Integration of some linear partial Differential Equations ; 
and particularly the general Equation of the Motion of Elastic 
Fluids." To the memoir itself I beg to refer, by the way, as 
presenting a demonstration of the constancy of the velocity of 
propagation from an irregular disturbance in an elastic fluid, 
more simple and direct than that in the Journal de VEcole Po- 
lytechniqiie. It contains also a general integral of the linear 
partial differential equation of three terms, which occurs in the 
problem of waves for the case in which the three dimensions of 
the fluid are taken account of; but the author does not consider 
this integral of much utility, because of the impossible quantities 
involved in it, and rather recommends the method of express- 
ing the principal variable by infinite series of exponentials. In 
fact, in the " Theory of AVaves " this case is treated in a manner 
exactly analogous to that in which abstraction is made of one 
dimension of the fluid. 

It may be useful to state some of the principal results ob- 
tained by theory respecting the nature of waves, to give an idea 
of what the independent power of analysis has been able to ef- 

With respect, first, to the canal of uniform width, the law of 
the velocity of propagation found by Lagrange is confirmed by 
M. Poisson's theory when the depth is small, but not other- 

When the canal is of unlimited depth, the following are the 
chief results : 

(1.) An impulse given to any point of the surface affects in- 
stantaneously the whole extent of the fluid mass. The theory 
determines the magnitude and direction of the initial velocity of 
each particle resulting from a given impulse. 

(2.) " The summit of each wave moves with a uniformly acce- 
lerated motion." 

This must be understood to refer to a series of very small 
waves, called by M. Poisson dents, which perform their move- 
ments as it were on the surface of the larger waves, which he 
calls " les ondes denteUes." Each wave of the series is found 
to have its proper velocity, independent of the primitive im- 
pulse. Waves of this kind have been actually observed : they 
are small from the first, and quickly disappear. 

(3.) At considerable distances from the place of disturbance, 


148 THIRD REPORT — 1833. 

there are waves of much more sensible magnitude than the pre- 
ceding. Their summits are propagated with a uniform velocity, 
which varies as the square root of the breadth a fleur deau of 
the fluid originally disturbed. Yet the different waves which 
are formed in succession are propagated with different veloci- 
ties : the foremost travels swiftest. The amplitude of oscilla- 
tions of equal duration are reciprocally proportional to the 
square root of the distances from the point of disturbance. 

(4.) The vertical excursions of the particles situated directly 
below the primitive impulse, vary according to the inverse ratio 
of the depth below the surface. This law of decrease is not so 
I'apid but that the motion will be very sensible at very consider- 
able depths : it will not be the true law, as the theory proves, 
when the original disturbance extends over the whole surface 
of the water, for the decrease of motion in this case will be 
much more rapid. 

The results of the theory, when the three dimensions of the 
fluid are considered, are analogous to the preceding, (1), (2), (3), 
(4), and may be stated in the same terms, excepting that the am- 
plitudes of the oscillations are inversely as the distances from 
the origin of disturbance, and the vertical excursions of the par- 
ticles situated directly below the disturbance vary inversely as 
the square of the depth. 

There is a good analysis of M. Poisson's theory, and a com- 
parison of many of the results with experiments, in a Treatise 
by M. Weber, entitled Wellenlehre aiif Experimente gegrim- 
det*. The experiments of M. Weber were made in a manner 
not sufficiently agreeing with the conditions supposed in the 
theoi'y to be a correct test of it. They, however, manifest a 
general accordance with it, and confii'm the existence of the 
small accelerated waves near the place of distui'bance, and of a 
sensible motion of the fluid particles at considerable depths 
below the surface. In one particular, in which the theory ad- 
mits of easy comparison with experiment, it is not found to 
agree. When the body employed to cause the initial agitation 
of the water is an elliptic paraboloid, with its vertex downwards 
and axis vertical, and consequently the section in the plane of 
the surface of the water an ellipse, theory determines the velo- 
city of propagation to be greater in the direction of the major 
axis than in that of the minor in the proportion of the square 
root of the one to the square root of the other. This result, 
which it must be confessed has not an appearance of probabi' 
lity, is not borne out by experience. 

* Leipzig, 182,5. 


The theory has been also put to the test of expermient by 
M. Bidone, who succeeded in overcoming in great measure an 
obstacle in the way of making the experiments according to the 
conditions supposed in the theory, arising from the adhesion of 
the water to the immersed body*. His observations confirm 
the existence and laws of motion of the accelerated waves. 

V. Scarcely anything worth mentioning has been effected 
by theory in regard to the resistance of fluids to bodies moving 
in them. The defect of every attempt hitherto made has 
arisen from its proceeding upon some hypothesis respecting 
the law of the resistance ; for instance, that it varies as the ve- 
locity, or as the square of the velocity: whereas the law, which 
cannot be known a priori, ought to be a result of the calcula- 
tion, which should embrace not only the motion of the body, 
but that of every particle of the fluid which moves simulta- 
neously with it. The only problem that has been attempted 
to be solved on this principle, is one of very considerable in- 
terest, relating to the correction to be applied to the pendulum 
to effect the reduction to a vacuum. The memoir of M. Pois- 
son, " On the Simultaneous Motions of a Pendulum and of the 
surrounding Air," was read before the Royal Academy of Paris 
in August 1831, and is inserted in vol. xi. of iheiv 3femoires. 
He takes the case of a spherical ball suspended by a very slen- 
der thread, the effect of which is neglected in the calculations ; 
the ball is supposed to perform oscillations of very small ampli- 
tude, so that the air in contact with its surface is sensibly the 
same during the motion. A simpler problem of resistance can- 
not be conceived. M. Poisson considers the effect which the 
friction of the particles of air against the surface of the ball 
may have on its motion, and comes to the conclusion that the 
time of the oscillations is not affected by it, but only their ex- 
tent. The most important result of the theoretical calculation 
is, that the correction which has been usually applied for the 
reduction to a vacuum, and calculated without considering the 
motion of the air, must be increased by one half. This he finds 
to agree sufficiently with some experiments of Captain Sabine. 
He also adduces forty-four experiments of Dubuat, made fifty 
years ago, upon oscillations in water, and three upon oscilla- 
tions in air. These give nearly the same numerical result, and 
agreeing nearly with the value 1^. The experiments, however, 
of M. Bessel give results which coincide with Dubuat's for os- 
cillations in water, but determine the correction in air for re- 
duction to a vacuum to be very nearly double that hitherto 

• See vol. XXV. of the Memoirs of the Royal Academy of Turin. 

150 THIRD REPORT — 1833. 

applied, instead of once and a half. M. Poisson thinks that 
the calculations of M. Bessel leave some room for doubt, and 
objects to the discordance of the values obtained for air and 
watei', which, according- to his own theory, ought to agree. 
More recent experiments of Mr. Baily *, which, from their num- 
ber and variety, and the care taken in performing them, are 
entitled to the utmost confidence, give the value 1*864 for 
spheres of different materials one inch and a half in diameter, 
and 1 '748 for spheres two inches in diameter, the latter being 
nearly the size of those for which M. Bessel obtained 1*946. 
The theory of M. Poisson does not recognise any difference in 
the value of the coefficient for spheres of different diameters. 
The discrepancies that thus appear between theory and expe- 
riment, and between the experiments themselves, show that 
there is much that requires clearing up in this important sub- 
ject. As far as theory is concerned, it is easily conceivable that 
much must depend upon the way in which the law of trans- 
mission of the motion from the parts of the fluid immediately 
acted on by the sphere to the parts more remote is to be deter- 
mined : and, as it is the province of this Report to point out 
any possible source of error in theory, I will venture again 
to express my doubts of the correctness of the principle em- 
ployed in the solution of this problem, of making the deter- 
mination of the law of transmission depend on the arbitrary 
discontinuity of the functions introduced by integration, the law 
itself not being arbitrary f. 

A singular fact, relating to the resistance to the motion of 
bodies partly immersed in water, has been recently established 
by experiments on canal navigation, by which it appears that a 
boat, drawn with a velocity of more than four or five miles an 
hour, rises perceptibly out of the water, so that the water-line 
is not so distant from the keel as in a state of rest, and the re- 
sistance is less than it would be if no such effect took place. 
Theory, although it has never predicted anything of this na- 
ture, now that the fact is proposed for explanation, will proba- 
bly soon be able to account for it on known mechanical prin- 

The foregoing review of the theory of fluid motion, incom- 

* PhHosophlcal Transactions for 1832, p. 399. 

t In an attempt at this problem made by myself, and published subsequently 
to the Meeting of the Association, the value of the coefficient is found to be 2, 
without accounting for any difference for spheres of different diameters. See 
the London and Edinburgh Philosophical Magazine and Journal for Septem- 
ber 1833. 


plete as it is, may suffice to show that this department of science 
is in an extremely imperfect state. Possibly it may on that ac- 
count be the more likely to receive improvements ; and I am 
disposed to think that such will be the case. But these im- 
provements, I expect, will be available not so much in practical 
applications, as in reference to the great physical questions of 
light, heat and electricity, which have been so long the subjects 
of experiment, and the theories of which require to be perfected. 
For this purpose a more complete knowledge of the analytical 
calculation proper for the treatment of fluids in motion may be 
of great utility. 


[ 153 ] 

Report on the Progress and Present Stale of our Knowledge 
of Hydraulics as a Branch of Engifieering. By George 
Rennie, Esq., F.R.S., §-c. ^-c. 

Part I. 

The paper now communicated to the British Association for 
the Advancement of Science comprises a Report on the pro- 
gress and present state of our knowledge of Hydrauhcs as a 
branch of Engineering, with reference to the principles already 
established on that subject. 

Technically speaking, the term hydraulics signifies that 
branch of the science of hydrodynamics which treats of the 
motion of fluids issuing from orifices and tubes in reservoirs, 
or moving in pipes, canals or rivers, oscillating in waves, or 
opposing a resistance to the progress of solid bodies at rest. 

We can readily imagine that if a hole of given dimensions be 
pierced in the sides or bottom of a vessel kept constantly full, 
the expenditure ought to be measured by the amplitude of the 
opening, and the height of the liquid column. 

If we isolate the column above the orifice by a tube, it ap- 
pears evident that the fluid will fall freely, and follow the laws 
of gravity. But experiment proves that this is not exactly the 
case, on account of the resistances and forces which act in a 
contrary direction, and destroy part of, or the whole, effect. 
The development of these forces is so extremely complicated 
that it becomes necessary to adopt some auxiliary hypothesis 
or abbreviation in order to obtain approximate results. Hence 
the science of hydrodynamics is entirely indebted to experi- 
ment. The fundamental problem of it is to determine the efflux 
of a vein of water or any other fluid issuing from an aperture 
made in the sides or bottom of a vessel kept constantly full, or 
allowed to empty itself. Torricelli had demonstrated that, 
abstracting the resistances, the velocities of fluids issuing from 
very small orifices followed the subduplicate ratio of the pres- 
sures. This law had been, in a measure, confused by sub- 
sequent writers, in consequence of the discrepancies which 
appeared to exist between the theory and experiment ; until 
Varignon remarked, that when water escaped from a small 
opening made in the bottom of a cylindrical vessel, there ap- 
peared to be very little, or scarcely any, sensible motion in the 

154 THIRD REPORT — 1833. 

particles of the water ; from which he concluded that the law 
of acceleration existed, and that the particles which escaped at 
every instant of time received their motion simply from the 
pressure produced by the weight of the fluid column above the 
orifice, and that the weight of this column of fluid ought to 
represent the pressure on the particles which continually escape 
from the orifice ; and that the quantity of motion or expenditure 
is in the ratio of the breadth of the orifice, multiplied by the 
square of the velocity, or, in other words, that the height of 
the water in the vessel is proportional to the square of the ve- 
locity with which it escapes ; which is precisely the theorem of 
Torricelli. This mode of reasoning is in some degree vague, 
because it supposes that the small mass which escapes from 
the vessel at each instant of time acquires its velocity from the 
pressure of the column immediately above the orifice. But 
supposing, as is natural, that the weight of the cohunn acts on 
the particle during the time it takes to issue from the vessel, it 
is clear that this particle will receive an accelerated motion, 
whose quantity in a given time will be proportional to the 
pressure multiplied by the time : hence the product of the 
weight of the column by the time of its issuing from the orifice, 
will be equal to the product of the mass of this particle by the 
velocity it will have acquired ; and as the mass is the product 
of the opening of the orifice, by the small space which the 
particle describes in issuing from the orifice, it follows that the 
height of the column will be as the square of the velocity ac- 
quired. This theory is the more correct the more the fluid 
approaches to a perfect state of repose, and the more the 
dimensions of the vessel exceed the dimensions of the orifice. 
By a contrary mode of reasoning this theory became insufficient 
to determine the motions of fluids through pipes of small dia- 
meters. It is necessary, therefore, to consider all the motions 
of the particles of fluids, and examine how they are changed 
and altered by the figure of the conduit. But experiment teaches 
us that when a pipe has a different direction from the vertical 
one, the different horizontal sections of the fluid preserve their 
parallelism, the sections following taking the place of the pre- 
ceding ones, and so on ; from which it follows (on account of 
the incompressibility of the fluid) that the velocity of each 
horizontal section or plate, taken vertically, ought to be in 
the inverse ratio of the diameter of the section. It suffices, 
therefore, to determine the motion of a single section, and the 
problem then becomes analogous to the vibration of a com- 
pound pendulum, by which, according to the theory of James 
Bernoulli, the motions acquired and lost at each instant of time 


form an equilibrium, as may be supposed to take place with 
the different sections of a fluid in a pipe, each section being 
animated with velocities acquired and lost at every instant of 

The theory of Bernoulli had not been proposed by him until 
long after the discovery of the indirect principle of vis viva by 
Huygens. The same was the case with the problem of the mo- 
tions of fluids issuing from vessels, and it is surprising that no 
advantage had been taken of it earlier. Michelotti, in his experi- 
mental researches de Separatione Fluidorum in Corpore Ani- 
mali, in rejecting the theory of the Newtonian cataract, (which 
had been advanced in Newton's Mathematical Principles, in 
the year 1687, but afterwards corrected in the year 1714,) sup- 
poses the water to escape from an orifice in the bottom of a 
vessel kept constantly full, with a velocity produced by the 
height of the superior surface ; and that if, immediately above 
the lowest plate of water escaping from the orifice, the column 
of water be frozen, the weight of the column will have no effect 
on the velocity of the water issuing from the orifice ; and that 
if this solid column be at once changed to its liquid state, the 
effect will remain the same. The Marquis Poleni, in his work 
De Castellis j)er quce derivantur Fliiviornm Aquce, published at 
Padua in the year 1718, shows, from many experiments, that 
if A be the orifice, and H the height of the column above it, 
the quantity of water which issues in a given time is represented 

by 2 A H X fTTTjT^j whereas if it spouted out from the orifice 

with a velocity acquired by falling from the height H, it ought 
to be exactly 2 A H, so that experiment only gives a little more 
than half the quantity promised by the theory ; hence, if we 
were to calculate from these experiments the velocity that the 
water ought to have to furnish the necessary quantity, we 
should find that it would hardly make it reascend ^rd of its 
height. These experiments would have been quite contrary 
to expectation, had not Sir Isaac Newton observed that water 
issuing from an orifice f ths of an inch in diameter, was contracted 
Ijths of the diameter of the orifice, so that the cylinder of water 
which actually issued was less than it ought to have been, 
according to the theory, in the ratio of 441 to Q9,^ ; and aug- 
menting it in this proportion, the opening should have been 

~ A H jTqqq, or l^ths of the quantity which ought to have issued 

on the supposition that the velocity was in the ratio of the 
square root of the height ; from which it was inferred that the 
theory was correct, but that the discrepancy was owing to cer- 

156 THIRD REPORT — 1833. 

tain resistances, which experiment could alone determine. The 
accuracy of the general conclusion was affected by several 
assumptions, namely, the perfect fluidity and sensibility of the 
mass, which was neither affected by friction nor cohesion, and 
an infinitely small thickness in the edge of the aperture. 

Daniel Bernoulli, in his great work, Hydrody namica , seu de 
Viribns et Motibus Fluidorum Commentaria, published at Stras- 
burgh in the year 1738, in considering the efflux of water from 
an orifice in the bottom of a vessel, conceives the fluid to be 
divided into an infinite number of horizontal strata, on the fol- 
lowing suppositions, namely, that the upper surface of the fluid 
always preserves its horizontality ; that the fluid forms a con- 
tinuous mass ; that the velocities vary by insensible gradations, 
like those of heavy bodies ; and that every point of the same 
stratum descends vertically with the same velocity, which is 
inversely proportional to the area of the base of the stratum ; 
that all sections thus retaining their parallelism are contiguous, 
and change their velocities imperceptibly ; and that there is 
always an equality between the vertical descent and ascent, or 
vis viva : hence he arrives, by a very simple and elegant pro- 
cess, to the equations of the problem, and applies its general 
formulae to several cases of practical utility. When the figure 
of the vessel is not subject to the law of continuity, or when 
sudden and finite changes take place in the velocities of the 
sections, there is a loss of vis viva, and the equations require 
to be modified. John Bernoulli and Maclaurin arrived at the 
same conclusions by different steps, somewhat analogous to the 
cataract of Newton. The investigations of D'Alembert had 
been directed principally to the dynamics of solid bodies, until 
it occurred to him to apply them to fluids ; but in following the 
steps of Bernoulli he discovered a formula applicable to the 
motions of fluid, and reducible to the ordinary laws of hydro- 
statics. The application of his theory to elastic and non-elastic 
bodies, and the determination of the motions of fluids in flexible 
pipes, together with his investigations relative to the resistance 
of pipes, place him high in the ranks of those who have contri- 
buted to the perfection of the science. 

The celebrated Euler, to whom every branch of science owes 
such deep obligations, seems to have paid particular attention 
to the subject of hydrodynamics ; and in attempting to reduce 
the whole of it to uniform and general formulae, he exhibited a 
beautiful example of the application of analytical investigation 
to the solution of a great variety of problems for which he was 
so famous. The Memoirs of the Academy of Berlin, from the 
year 1768 to 1771, contain numerous papers relative to fluids 


flowing from orifices in vessels, and through pipes of constant 
or variable diameters. " But it is greatly to be regretted," 
says M. Prony, " that Euler had not treated of friction and 
cohesion, as his theory of the linear motion of air would have 
applied to the motions of fluids through pipes and conduits, 
had he not always reasoned on the hypotheses of mathematical 
fluidity, independently of the resistances which modify it." 

In the year 1765 a very complete work was published at 
Milan by PauJ Lecchi, a celebrated Milanese engineer, entitled 
Idrostatica esaminata ne suoi Principi e Stabilite nelle suoi 
Regole delta Mensura delta Aeque correnti, containing a com- 
plete examination of all the different theories which had been 
proposed to explain the phgenomena of effluent water, and the 
doctrine of the resistance of fluids. The author treats of the 
velocity and quantity of water, whether absolutely or relatively, 
which issues from orifices in vessels and reservoirs, according 
to their different altitudes, and inquires how far the law applies 
to masses of water flowing in canals and rivers, the velocities 
and quantities of which he gives the methods of measuring. 
The extensive and successful practice of Lecchi as an engineer 
added much to the reputation of his work*. 

In the year 1764 Professor Michelotti of Turin undertook, 
at the expense of the King of Sardinia, a very extensive series 
of experiments on running water issuing through orifices and 
additional tubes placed at different heights in a tower of the 
finest masonry, twenty feet in height and three feet square 
inside. The water was supplied by a channel two feet in width, 
and under pressures of from five to twenty-two feet. The 
effluent waters were conveyed into a reservoir of ample area, 
by canals of brick-work lined with stucco, and having various 
forms and declivities ; and the experiments, particularly on the 
efilux of water through differently shaped orifices, and addi- 
tional tubes of different lengths, were most numerous and 
accurate, and Michelotti was the first who gave representations 
of the changes which take place in the figure of the fluid vein, 
after it has issued from the orifice. His expei'iments on the 
velocities of rivers, by means of the bent tube of Pitot, and by 
an instrument resembling a water-wheel, called the stadera 
idraulica, are numerous and interesting ; but, unfortunately, 
their reduction is complicated with such various circumstances 
that it is difficult to deinve from them any satisfactoi'y conclu- 
sions. But Michelotti is justly entitled to the merit of having 
made the greatest revolution in the science by experimental 

* Sec also Memorie Idrostatico-storiclie, 1773. 

158 THIRD REPORT 1833. 

investigation *. The example of Michelotti gave a fresh sti- 
mulus to the exertions of the French philosophers, to whom, 
after the ItaUans, the science owes the greatest obligations. 
Accordingly, the Abb^ Bossut, a most zealous and enhght- 
ened cultivator of hydrodynamics, undertook, at the expense 
of the French Government, a most extensive and accurate se- 
ries of experiments, which he published in the year 1771, 
and a more enlarged edition, in two volumes, in the year 
1786, entitled Traits Theorique et Experimental (THijdro- 
namique. The first volume treats of the general principles of 
hydrostatics and hydraulics, including the pressure and equili- 
brium of non-elastic and elastic fluids against inflexible and 
flexible vessels ; the thickness of pipes to resist the pressure 
of stagnant fluids ; the rise of water in barometers and pumps, 
and the pressure and equilibrivun of floating bodies ; the ge- 
neral principles of the motions of fluids through orifices of dif- 
ferent shapes, and their friction and resistance against the 
orifices ; the oscillations of water in siphons ; the percussion 
and resistance of fluids against solids ; and machines moved by 
the action and reaction of water. The second volume gives a 
great variety of experiments on the motions of water through 
orifices and pipes and fountains ; their resistances in rectan- 
gular or curvilinear channels, and against solids moving through 
them ; and lastly, of the fire- or steam-engine. In the course 
of these experiments he found that when the water flowed 
through an orifice in a thin plate, the contraction of the fluid 
vein diminished the discharge in the ratio of 16 to 10; and when 
the fluid was discharged through an additional tube, two or 
three inches in length, the theoretical discharge was diminished 
only in the ratio of 16 to 13. In examining the effects of fric- 
tion, Bossut found that small orifices discharged less water in 
proportion than large ones, on account of friction, and that, as 
the height of the reservoir augmented, the fluid vein contracted 
likewise ; and by combining these two circumstances together, 
he has furnished the means of measuring with precision the 
quantity of water discharged either from simple orifices or 
additional tubes, whether the vessels be constantly full, or be 
allowed to empty themselves. He endeavoured to point out 
the law by which the diminution of expenditure takes place, 
according to the increase in the length of the pipe or the num- 
ber of its bends ; he examined the effect of friction in dimi- 
nishing the velocity of a stream in rectangular and curvilinear 
channels ; and showed that in an open canal, with the same 

* Sperimenti Idraulici, 1767 and 1771. 


height of reservoh", the same quantity of water is always dis- 
charged, whatever be the decUvity and length ; that the ve- 
locities of the waters in the canal are not as the square roots 
of the declivities, and that in equal declivities and depth of the 
canal the velocities are not exactly as the quantities of water 
discharged ; and he considers the variations which take place 
in the velocity and level of the waters when two rivers unite, 
and the manner in which they establish their beds. 

His experiments, in conjunction with D'Alembert and Con- 
dorcet, on the resistance of fluids, in the year 1777, and his 
subsequent application of them to all kinds of surfaces, in- 
cluding the shock and resistance of water-wheels, have justly 
entitled him to the gratitude of posterity. The Abbe Bossut 
had opened out a new career of experiments ; but the most dif- 
ficult and important problem remaining to be solved related to 
I'ivers. It was easy to perform experiments with water running 
through pipes and conduits on a small scale, under given and 
determined circumstances : but when the mass of fluid rolled 
in channels of unequal capacities, and which were composed of 
every kind of material, from the rocks amongst which it accu- 
mulated to the gravel and sand through which it forced a pass- 
age, — at first a rapid and impetuous torrent, but latterly hold- 
ing a calm and majestic course, — sometimes forming sand-banks 
and islands, at other times destroying them, at all times capri- 
cious, and subject to variation in its force and direction by 
the slightest obstacles, — it appeared impossible to submit them 
to any general law. 

Unappalled, however, by these difficulties, the Chevalier 
Buat, after perusing attentively M. Bossut's work, undertook 
to solve them by means of a theorem which appeared to him 
to be the key of the whole science of hydraulics. He consi- 
dered that if water was in a perfect state of fluidity, and ran in 
a bed from which it experienced no resistance whatever, its 
motion would be constantly accelerated, like the motion of a 
heavy body descending an inclined plane ; but as the velocity 
of a river is not accelerated ad infinitum, but arrives at a state 
of uniformity, it follows that there exists some obstacle which 
destroys the accelerating force, and prevents it from impressing 
upon the water a new degree of velocity. This obstacle must 
therefore be owing either to the viscidity of the water, or to 
the resistance it experiences against the bed of the river ; from 
which Dubuat derives the following principle : — That when 
water runs uniformly in any channel, the accelerating force 
which obliges it to run is equal to the sum of all the resistances 
which it experiences, whether arising from the viscidity of the 
water or the friction of its bed. Encouraged by this discovery, 

160 THIRD REPORT — 1833. 

and by the application of its principles to the solution of a gi'eat 
many cases in practice, Dubuat' was convinced that the motion 
of water in a conduit pipe was analogous to the uniform motion 
of a river, since in both cases gravity was the cause of motion, 
and the resistance of the channel or perimeter of the pipes the 
modifiers. He then availed himself of the experiments of Bossut 
on conduit pipes and artificial channels to explain his theory : 
the results of which investigations were published in the year 
1779. M. Dubuat was, however, sensible that a theory of so 
much novelty, and at variance with the then received theory, 
required to be supported by experiments more numerous and 
direct than those formerly undertaken, as he was constrained 
to suppose that the friction of the water did not depend upon 
the pressure, but on the surface and square of the velocity. 
Accordingly, he devoted three years to making fresh experi- 
ments, and, with ample funds and assistance provided by the 
French Government, was enabled to publish his great work, 
entitled Principes d' Hydraidique verifies par un grand nombre 
d' Experiences, faittis par Ordre du Gouvernement, 2 vols. 1786, 
(a third volume, entitled Principes dHydrauUque et Hydro^ 
namique, appeared in 1816); — in the first instance, by repeating 
and enlarging the scale of Bossut's experiments on pipes (with 
water running in them) of difFei'ent inclinations or angles, of 
from 90° to iqMj*^ P^'^* ^^ ^ right angle, and in channels of from 
1 \ line in diameter to 7 and 8 square toises of surface, and sub- 
sequently to water running in open channels, in which he ex- 
perienced great difficulties in rendering the motion uniform : 
but he was amply recompensed by the results he obtained on 
the diminution of the velocity of the different parts of a uniform 
current, and of the relation of the velocities at the surface and 
bottom, by which the water works its own channel, and by the 
knowledge of the resistances which different kinds of beds pro- 
duce, such as clay, sand and gravel; and varying the experiments 
on the effect of sluices, and the piers of bridges, &c., he was ena- 
bled to obtain a formula applicable to most cases in practice*. 
Thus, let V = mean velocity per second, in inches. 

d = hydraulic mean depth, or quotient which arises 
from dividing the area or section of the canal, 
in square inches, by the perimeter of the part 
in contact with the water, in linear inches. 
s = the slope or declivity of the pipe, or the sur- 
face of the water. 
g = 16'087, the velocity in inches which a body 
acquires in fidling one second of time. 

* Edinburgh Encyclopedia, Art. Hydrodynamics, by Brewster. 


n = an abstract number, which was found by ex- 
periment to be equal to 243*7. 

then . = _^Il{^^I^:}L_ _ 0-3 (^^ _ o-i). 
ys — log. V s + I'o 

Such are some of the objects of M. Dubuat's work. But his 
hypotheses are unfortunately founded upon assumptions which 
render the appUcations of his theory of little use. It is evident 
that the supposition of a constant and uniform velocity in rivers 
cannot hold : nevertheless he has rendered great services to 
the science by the solution of many important questions relating 
to it ; and although he has left on some points a vast field open 
to research, he is justly entitled to the merit of originality and 

Contemporary with Dubuat was M. Chezy, one of the most 
skilful engineers of his time : he was director of the Ecole des 
Fonts et Chauss^es, and reported, conjointly with M. Perronet, 
on the Canal Yvette. He endeavoured to assign, by experiment, 
the relation existing between the inclination, length, trans- 
versal section, and velocity of a canal. In the course of this 
investigation he obtained a very simple expression of the velo- 
city, involving three different variable quantities, and capable, 
by means of a single experiment, of being applied to all cur- 
rents whatever. He assimilates the resistance of the sides and 
bottom of the canal to known resistances, which follow the law 
of the square of the velocity, and he obtains the following sim- 
ple formula: 

V = S^-, where g" is = 16"087 feet, the velocity acquired 

z s 

by a heavy body after falling one second. 

d = hydraulic mean depth, equal to the area of the section 
divided by the perimeter of the part of the canal in 
contact with the water. 

s = the slope or declivity of the pipe. 

« = an abstract number, to be determined by experiment. 

In the year 1784, M. Lespinasse published in the Memoirs 
of the Academy of Sciences at Toulouse two papers, contain- 
ing some interesting observations on the expenditure of water 
through large orifices, and on the junction and separation of 
rivers. The author had performed the experiments contained 
in his last paper on the rivers Fresquel and Aude, and on that 
part of the canal of Languedoc below the Fresquel lock, towards 
its junction with that river. 

As we before stated, M. Dubuat had classified with much 
1833. M 

162 THIRD REPORT — 1833. 

sagacity his observations on the different kinds of resistance 
experienced in the motion of fluids, and which might have led 
him to express the sum of the resistances by a rational function 
of the velocity composed of two or three terms only. Yet the 
merit of this determination was reserved to M. Coulomb, who, 
in a beautiful paper, entitled " Experiences destinees a deter- 
miner la Coherence des Fluides et les Lois de leurs Resistances 
dans les Mouvemens tres lents," proves, by reasoning and facts, 

1st, That in extremely slow motions the part of the resist- 
ance is proportional to the square of the velocity. 

2ndly, That the resistance is not sensibly increased by in- 
creasing the height of the fluid above the resisting body. 

3rdly, That the resistance arises solely from the mutual co- 
hesion of the fluid particles, and not from their adhesion to the 
body upon which they act. 

4thly, That the resistance in clarified oil, at the temperature 
of 69° Fahrenheit, is to that of water as 17*5 : 1 ; a proportion 
which expresses the ratio of the mutual cohesion of the par- 
ticles of oil to the mutual cohesion of the particles of water. 

M. Coulomb concludes his experiments by ascertaining the 
resistance experienced by cylinders that move very slowly and 
perpendicularly to their axes, &c. 

This eminent philosopher, who had applied the doctrine of 
torsion with such distinguished success in investigating the 
phasnomena of electricity and magnetism, entertained the idea 
of examining in a similar manner the resistance of fluids, con- 
trary to the doctrines of resistance previously laid down. 
M. Coulomb proved, that in the resistance of fluids against 
solids, there was no constant quantity of sufficient magnitude 
to be detected ; and that the pressure sustained by a moving 
body is represented by two terms, one which varies as the 
simple velocity, and the other with its square. 

The apparatus with which these results were obtained con- 
sisted of discs of various sizes, which were fixed to the lower 
extremity of a brass wire, and were made to oscillate under a 
fluid by the force of torsion of the wire. By observing the 
successive diminution of the oscillations, the law of resistance 
was easily found. The oscillations which were best suited to 
these experiments continued for twenty or thirty seconds, and 
the amplitude of the oscillation (that gave the most regular re- 
sults) was between 480 the entire division of the disc, and 8 or 
10 divisions from zero. 

The first who had the happy idea of applying the law of 
Coulomb to the case of the velocities of water running in na- 
tural or artificial channels was M. Girard, Ingenieur en chef 


des Fonts et Chaussees, and Director of the Works of the 
Canal I'Ourcq at Paris *. 

He is the author of several papers on the theory of running 
waters, and of a valuable series of experiments on the motions 
of fluids in capillary tubes. 

M. Coulomb had given a common coefficient to the two terms 
of his formula representing the resistance of a fluid, — one pro- 
portional to the simple velocity, the other to the square of the 
velocity. M. Girard found that this identity of the coefficients 
was applicable only to particular fluids under certain circum- 
stances ; and his conclusions were confirmed by the researches 
of M. Prony, derived from a great many experiments, which 
make the coefficients not only different, but very inferior to the 
value of the motion of the filaments of the water contiguous to 
the side of the pipe. 

The object of M. Girard's experiments was to determine 
this velocity ; and this he has effected in a very satisfactory 
manner, by means of twelve hundred experiments, performed 
with a series of copper tubes, from 1*88 to 2'96 millimetres in 
diameter, and from 20 to 222 centimetres in length ; from which 
it appeared, that when the velocity was expressed by 10, and 
the temperature was 0, centigrade, the velocity was increased 
four times when the temperature amounted to 85°. When the 
length of the capillary tube was below that limit, a variation of 
temperature exercised very little influence upon the velocity 
of the issuing fluid, &c. 

It was in this state of the science that M. Prony (then having 
under his direction different projects for canals,) undertook to 
reduce the solutions of many important problems on running 
water to the most strict and rigorous principles, at the same 
time capable of being applied with facility to practice. 

For this purpose he selected fifty-one experiments which 
corresponded best on conduit pipes, and thirty-one on open 
conduits. Proceeding, therefore, on M. Giraird's theory of the 
analogy between fluids and a system of corpuscular solids or 
material bodies, gravitating in a curvilinear channel of indefinite 
length, and occupying and abandoning successively the dif- 
ferent parts of the length of channel, he was enabled to express 
the velocity of the water, whether it flows in pipes or in open 
conduits, by a simple formula, free of logarithms, and requiring 
merely the extraction of the square root-f. 

• Essai stir le Mouvement des Eaux courantes : Paris 1804. Recherches 
sitr les Eaux publiqiies, ^-c. Devis general du Canal I'Ourcq, ^-c. 
+ Memoires dfs Savans Ef rangers, 8fc. 1815. 

M 2 

164 THIRD REPORT — 1833. 

Thus v= - 0-0469734 + a/ 0-00^2065 + 3041-47 x G, 
which gives the velocity in metres : or, in English feet, 

?;= - 0-1541131 + \/~0W3751 + 32806-6 x G. 
When this formula is apphed to pipes, we must take G = :^DK, 

rj I 7 rr 

which is deduced from the equation K = j . When 

it is applied to canals^, we must take G = R I, which is deduced 

from the equation I = :p, R being equal to the mean radius of 

Dubuat on the hydraulic mean depth, and I equal to the sine 
of inclination in the pipe or canal. M. Prony has drawn up ex- 
tensive Tables, in which he has compared the observed velo- 
cities with those which are calculated from the preceding for- 
mulae, and from those of Dubuat and Girard. In both cases 
the coincidence of the observed results with the formulae are 
very i-emarkable, but particularly with the formulae of M. Prony. 
But the great work of M. Prony is his Noiwelle Architecture 
Hydraulique, published in the year 1790. This able produc- 
tion is divided into five sections, viz. Statics, Dynamics, Hydro- 
statics, Hydrodynamics, and on the physical circumstances 
that influence the motions of Machines. The chapter on hydro- 
dynamics is particularly copious and explanatory of the motions 
of compressible and incompressible fluids in pipes and vessels, 
on the principle of the parallelism of the fluid filaments, and 
the efflux of water through different kinds of orifices made in 
vessels kept constantly full, or permitted to empty themselves ; 
he details the theory of the clepsydra, and the curves described 
by spouting fluids ; and having noticed the different phaenomena 
of the contraction of the fluid vein, and given an account of the ex- 
periments of Bossut, M. Prony deduces formulae by which the re- 
sults may be expressed with all the accuracy required in practice. 
In treating of the impulse and resistance of fluids, M. Prony 
explains the theory of Don George Juan, which he finds con- 
formable to the experiments of Smeaton, but to differ very ma- 
terially from the previously received law of the product of the 
surfaces by the squares of the velocities, as established by the 
joint experiments of D'Alembert, Condorcet and Bossut, in the 
year 1775. The concluding part of the fourth section is de- 
voted to an examination of the theory of the equilibrium and 
motion of fluids according to Euler and D'Alembert ; and by a 
rigorous investigation of the nature of the questions to be de- 
termined, the whole theory is reduced to two equations only, in 
narrow pipes, according to the theory of Euler, showing its 
approximation to the hypothesis of the parallelism of filaments. 



The fifth and last section investigates the different circum- 
stances (such as friction, adhesion and rigidity,) which influence 
the motions of machines. 

A second volume, published in the year 1796, is devoted to 
the theory and practice of the steam-engine. Previously to the 
memoir of M. Prony, Sur le Jaugeage des Eaux courantes, in 
the year 1802, no attempt had been made to establish vi^ith cer- 
tainty the correction to be applied to the theoretical expendi- 
tures of fluids through orifices and additional tubes. The phag- 
nomenon had been long noticed by Sir Isaac Newton, and illus- 
trated by Michelotti by a magnificent series of experiments, 
which, although involving some intricacies, have certainly 
formed the groundwork of all the subsequent experiments upon 
this particular subject. 

By the method of interpolation, M. Prony has succeeded in 
discovering a series of formulas applicable to the expenditures 
of currents out of vertical and horizontal orifices, and to the con- 
traction of the fluid vein ; and in a subsequent work, entitled 
Recherches sur le Mouvemens des Eaux courantes, he establishes 
the following formulae for the mean velocities of rivers. 

When V = velocity at the surface, 
and U = mean velocity, 

U = 0-816458 V, 
which is about f V. 

These velocities are determined by two methods. 1st, By a 
small water-wheel for the velocity at the surface, and the im- 
proved tube of Pitot for the velocities at different depths below 
the surface. 

If h = the height of the water in the vertical tube above the 
level of the current, the velocity due to this height will be deter- 

/ metres 

mined by the formula V= V2gh=\/ 19-606 h = 4-429 -//*. 

When water runs in channels, the inclination usually given 
amounts to between ^^^q*^ ^^^ ^o o*^ P^'^' ^^ *^^ length, which 
will give a velocity of nearly 1^ mile per hour, sufficient to 
allow the water to run freely in earth. We have seen the incli- 
nation very conveniently applied in cases of drainage at xzW*-^ 
and y^o*^' ^^^ some rivers are said to have ^^%o*^ only. 

M. Prony gives the following formulas, from a great number 
of observations : 

If U = mean velocity of the water in the canal, 
I = the inclination of the canal per metre, 
R = the relation of the area to the profile of its perimeter, 
we shall have 

U = - 0-07 + V0'005 + 3233. R71; 

166 THIRD REPORT 1833. 

and for conduit pipes, 

calling U = the mean velocity, 

Z = the head of water in the inferior orifice of the pipe, 
L = the length of the pipe in metres, 
D = the diameter of the pipe, 

we shall have 

U = - 0-0248829 + \/0-000619159 + 717-857 DZ 

or, where the velocity is small, 

U = 26-79 ^DZ; 
that is, the mean velocities approximate to a direct ratio com- 
pounded of the squares of the diameters and heads of water, 
and inversely as the square root of the length of the pipes : 
and by experiments made with great care, M. Prony has found 
that the formula 

\J = - 0-0248829 + -/ 0-000619159 + 717-857 DZ 

scarcely differs more or less from experiments than ^^^ or gL. 
The preceding formulae suppose that the horizontal sections, 
both of the reservoir and the recipient, are great in relation 
to the transverse section of the pipe, and that the pipe is kept 
constantly full *. 

In comparing the formulae given for open and close canals, 
M. Prony has remarked that these formulae are not only similar, 
but the constants which enter into their composition are nearly 
the same ; so that either of them may represent the two series 
of phaenomena with sufficient exactness. 

The following formula applies equally to open or close canals : 

U = - 0-0469734 + ^Z (^0-0022065 + 3041-47 ^V 

But the most useful of the numerous formulae given by M. Prony 
for open canals is the following : 

* According to Mr. Jardine's experiments on the quantity of water delivered 
by the Coniston Main from Coniston to Edinburgh, the following is a compa- 
rison : Scots Pints. 

Actual delivery of Coniston Main 189-4 

Ditto by Eytelwein's formula 189'77 

Ditto by Girard's formula 188-26 

Ditto by Dubuat's formula 188-13 

Ditto by Prony "s simple formula 192-32 

Ditto by Prony 's tables 180-7 


Let g = the velocity of a body falling in one second, 
w = the area of the transverse section, 
p — the perimeter of that section, 
I = the inclination of the canal, 

Q = the constant volume of water through the section, 
U = the mean velocity of the water, 
R = the relation of the area to the perimeter of the section; 

then 1st, 0-000436 U + 0-003034 V^- = glR = gl-; 
2ndly, U = ^; 
3rdly, R w' - 0-0000444499 .wj- 0-000309314 y = 0. 

This last equation, containing the quantities 

QIw and R = — , 

shows how to determine one of them, and, knowing the three 
others, we shall have the following equations : 

4thly, p = 0.000436 Qw + 0-003034 Q^' 

rfui T P (0-0000444499 Qw + 0-000 309314 Q^) 

6thly, . = 0-000436 ± ^[(0-000^ + 4 |-003034) g RI] Q^ 

These formulae are, however, modified in rivers by circum- 
stances, such as weeds, vessels and other obstacles in the 
rivers ; in which case M. Girard has conceived it necessary to 
introduce into the formulas the coefficient of correction = 1-7 
as a multiplier of the perimeter, by which the equations will be, 

• p-1'7 (0-000436 U + 0-003034 W) = gl w. 

The preceding are among the principal researches of this 
distinguished philosopher *. 

In the year 1798, Professor Venturi of Modena published a 
very interesting memoir, entitled Sur la Communication lat^- 
rale du Mouvement des Fluides. Sir Isaac Newton was well 
acquainted with this communication, having deduced from it 
the propagation of rotary motion from the interior to the exte- 
rior of a whirlpool ; and had affirmed that when motion is pro- 
pagated in a fluid, and has passed beyond the aperture, the 

• Recherchcs Physico-mathematiques sur la Thcorie des Eaux courantes, 
par M. Prony. 

168 THIRD REPORT — 1833. 

motion diverges from that opening, as from a centre, and is 
propagated in right Unes towards the lateral parts. The sim- 
ple and immediate application of this theorem cannot be made 
to a jet or aperture at the surface of still water. Circumstances 
enter into this case which transform the results of the principle 
into particular motions. It is nevertheless true that the jet 
communicates its motion to the lateral parts without the orifice, 
but does not repel it in a radial divergency. M. Venturi illus- 
trates his theory by experiments on the form and expenditure 
of fluid veins issuing from orifices, and shows how the velocity 
and expenditure are increased by the application of additional 
tubes; and that in descending cylindrical tubes, the upper ends 
of which possess the form of the contracted vein, the expense 
is such as corresponds with the height of the fluid above the 
inferior extremity of the tube. The ancients remarked that a 
descending tube applied to a reservoir increased the expendi- 
ture*. D'Alembert, Euler and Bernoulli attributed it to the 
pressure of the atmosphere. Gravesend, Guglielmini and others 
sought for the cause of this augmentation in the weight of the 
atmosphere, and determined the velocity at the bottom of the 
tube to be the same as would ai'ise from the whole height of 
the column, including the height of the reservoir. Guglielmini 
supposed that the pressure at the orifice below is the same for 
a state of motion as for that of rest, which is not true. In the 
experiments he made for that purpose, he paid no regard either 
to the diminution of expenditure produced by the irregularity 
of the inner surface of the tubes, or the augmentation occa- 
sioned by the form of the tubes themselves. But Venturi esta- 
blished the proposition upon the principle of vertical ascension 
combined with the pressure of the atmosphere, as follows : 

1st, That in additional conical tubes the pressure of the at- 
mosphere increases the expenditure in the proportion of the 
exterior section of the tube to the section of the contracted 
vein, whatever be the position of the tube. 

Sndly, That in cylindrical pipes the expenditure is less than 
through conical pipes, which diverge from the contracted vein, 
and have the same exterior diameter. This is illustrated by 
experiments with differently formed tubes, as compared with a 
plate orifice and a cylindrical tube, by which the ratios in point 
of time were found to be 41", 31" and 27", showing the advan- 
tage of the conical tube. 

3rdly, That the expenditure may be still fiirther increased, 

* " Calix devaxus amplius rapit." — Frontinus de Aqueductibus. See also 
Pneumatics of Hero. 


in the ratio of 24 to 10, by a certain form of tube, — a circum- 
stance of which he supposes the Romans were well aware, as 
appears from their restricting the length of the pipes of con- 
veyance from the pubHc reservoirs to fifty feet ; but it was not 
perceived that the law might be equally evaded by applying a 
conical frustrum to the extremity of the tube. 

M. Venturi then examines the causes of eddies in rivers; 
whence he deduces from his experiments on tubes with en- 
larged parts, that every eddy destroys part of the moving force 
of the current of the river, of which the course is permanent 
and the sections of the bed unequal, the water continues more 
elevated than it would have done if the whole river had been 
equally contracted to the dimensions of its smallest section, — a 
consequence extremely important in the theory of rivers, as the 
retardation experienced by the water in rivers is not only due 
to the friction over the beds, but to eddies produced from the 
irregularities in the bed, and the flexures or windings of its 
course : a part of the current is thus employed to restore an 
equilibrium of motion, which the current itself continually de- 
ranges. As respects the contracted vein, it had been pretended 
by the Marquis de Lorgna* that the contracted vein was no- 
thing else but a continuation of the Newtonian cataract ; and 
that the celerity of the fluid issuing from an orifice in a thin 
plate is much less than that of a body which falls from the 
height of the charge. But Venturi proved that the contraction 
of the vein is incomparably greater than can be produced by 
the acceleration of gravity, even in descending streams, the 
contraction of the stream being 0*64, and the velocity nearly 
the same as that of a heavy body which may have fallen through 
the height of the charge. These experimental principles, which 
are in accordance with the results of Bossut, Michelotti and 
Poleni, are strictly true in all cases where the orifice is small in 
proportion to the section of the reservoir, and when that orifice 
is made in a thin plate, and the internal afflux of the filaments 
is made in an uniform manner round the orifice itself. Venturi 
then shows the form and contraction of the fluid vein by in- 
creased charges. His experiments with the cone are curious ; 
and it would have been greatly to be regretted that he had 
stopped short in his investigations, but for the more extensive 
researches of Bidone and Lesbros. M. Hachette, in opposition 
to the theory of Venturi, assigns, as a cause of the increase by 
additional tubes, the adhesion of the fluid to the sides of the . 
tubes arising from capillary attraction. 

* Memorie della Societa Ifaliana, vol. iv. 

170 THIRD REPORT — 183.3. 

In the year 1801, M. Eytelwein, a gentleman well known to 

the pubUc by his ti'anslation of M. Dubuat's work into German, 

(with important additions of his own,) published a valuable 

compendium of hydraulics, entitled Handbuch der Mechanik 

und der Hydraidik, in which he lays down the following rules. 

1 . That when water flows from a notch made in the side of 

a dam, its velocity is as the square of the height of 

the head of the water ; that is, that the pressure and 

consequent height are as the square of the velocity, the 

proportional velocities being nearly the same as those 

of Bossut. 

3. That the contraction of the fluid vein from a simple orifice 

in a thin plate is reduced to 0*64. 
S. For additional pipes the coefficient is 0*65. 

4. For a conical tube similar to the curve of contraction 0*98. 

5. For the whole velocity due to the height, the coefficient 

by its square must be multiplied by 8*0458. 

6. For an orifice the coefficient must be multiplied by 7*8. 

7. For wide openings in bridges, sluices, &c., by 6'9. 

8. For short pipes 6"6. 

9. For openings in sluices without side walls 5"1. 

Of the twenty-four chapters into which M. Eytelwein's * work 
is divided, the seventh is the most important. The late Dr. 
Thomas Young, in commenting upon this chapter, says : 

. " The simple theorem by which the velocity of a river is de- 
termined, appears to be the most valuable of M. Eytelwein's 
improvements, although the reasoning from which it is deduced 
is somewhat exceptionable. The friction is nearly as the square 
of the velocity, not because a number of particles proportional 
to the velocity is torn asunder in a time proportionally short, — 
for, according to the analogy of soUd bodies, no more force is 
destroyed by friction when the motion is rapid than when slow, 
— but because when a body is moving in lines of a given curva- 
ture, the deflecting forces are as the squares of the velocities ; 
and the particles of water in contact with the sides and bottom 
must be deflected, in consequence of the minute irregularities 
of the surfaces on which they slide, nearly in the same curvi- 
linear path, whatever their velocity may be. At any rate (he 
continues) we may safely set out with this hypothesis, that the 
principal part of the friction is as the square of the velocity, 
and the friction is nearly the same at all depths f; for Professor 
. Robison found that the time of oscillation of the fluid in a bent 

* See Nicholson's translation of Eytelwein's work. 

t See my "Experiments on the Friction and Resistance of Fluids," Philo- 
sopkical Transactions for 1831. 


tube was not increased by increasing the pressure against the 
sides, being nearly the same when the principal part was si- 
tuated horizontally, as when vertically. The friction will, how- 
ever, vary, according to the surface of the fluid which is in 
contact with the solid, in proportion to the whole quantity of 
fluid ; that is, the friction for any given quantity of water will 
be as the surface of the bottom and sides of a river directly, 
and as the whole quantity in the river inversely ; or, supposing 
the whole quantity of water to be spread on a horizontal sur- 
face equal to the bottom and sides, the friction is inversely as 
the height at which the river would then stand, which is called 
the hydraulic mean depth*." It is, therefore, calculated that 
the velocities will be a mean proportional between the hydraulic 
mean depth and the fall, or -j-^tlis of the velocity per second. 

Professor Robison informs us, that by the experiments of 
Mr. Watt on a canal eighteen feet wide at the top, seven feet 
at the bottom, and four feet deep, having a fall of four inches 
per mile, the velocities were seventeen inches per second at the 
surface, fourteen inches per second in the middle, and ten inches 
per second at the bottom, making a mean velocity of fourteen 
inches per second ; then finding the hydraulic mean depth, and 

dividing the area of the section by the perimeter, we have „„ „ , 

or 29*13 inches ; and the fall in two miles being eight inches, 
we have v'(8 x 29*13) = 15*26 for the mean proportional of 
■j-^ths, or 13*9 inches, which agrees very nearly with Mr. Watt's 

The Professor has, however, deduced from Dubuat's elabo- 
rate theories 12*568 inches. But this simple theorem applies 
only to the straight and equable channels of a river. In a 
curved channel the theorem becomes more complicated ; and, 
from observations made in the Po, Arno, Rhine, and other 
rivers, there appears to be no general rule for the decrease of 
velocity going downwards. M. Eytelwein directs us to deduct 
from the superficial velocity ^-^^ for every foot of the whole 
depth. Dr. Young thinks y^g*^^ ^^ *^^ superficial velocity suf- 
ficient. According to Major Rennell, the windings of the river 
Ganges in a length of sixty miles are so numerous as to reduce 
the declivity of the bed to four inches per mile, the medium 
rate of motion being about three miles per hour, so that a mean 
hydraulic depth of thirty feet, as stated to be f rds of the 
velocity per second, will be 4*47 feet, or three miles per hour. 
Again, the river when full has thrice the volume of water in it, 
and its motion is also accelerated in the proportion of 5 to 3 ; 

* See Nicholsons Journal for 1802, vol. iii. p. 31. 

172 THIRD REPORT — 1833. ' 

and, assuming the hydraulicmean depth to be doubled at the 
time of the inundation, the velocity will be increased in the 
ratio of 7 to 5 ; but the inclination of the surface is probably 
increased also, and consequently produces a further velocity of 
from 1*4 to 1*7. M. Eytelwein agrees with Gennete*, that a 
river may absorb the whole of the water of another river equal 
in magnitude to itself, without producing any sensible elevation 
in its surface. This apparent paradox Gennete pretends to 
prove by experiments, from observing that the Danube absorbs 
the Inn, and the Rhine the Mayne I'ivers ; but the author evi- 
dently has not attended to the fact, as may be witnessed in the 
junction of rivers in marshes and fenny countries, — the various 
rivers which run through the Pontine and other marshes in 
Italy, and in Cambridgeshire and Lincolnshire in this country : 
hence the familiar expression of the waters being overridden is 
founded in facts continually observed in these districts. We 
have also the experiments of Brunings in the Architecture Hy- 
draulique Generate de Wiebeking, Wattmann's M^moires sur 
I'Art de construire les Canaux, and Funk Sur F Architecture 
Hydraidique generate, which are sufficient to determine the 
coefficients under different circumstances, from velocities of 
fths to 7| feet, and of transverse sections from J- to 19135 
square feet. The experiments of Dubuat were made on the 
canal of Jard and the river Hayne ; those of Brunings in the 
Rhine, the Waal and Ifrel; and those of Wattmann in the 
drains near Cuxhaven. 

M. Eytelwein's paper contains formulae for the contraction 
of fluid veins through orifices f, and the resistances of fluids 
passing through pipes and beds of canals and rivers, according 
to the experiments of Couplet, Michelotti, Bossut, Venturi, 
Dubuat, Wattmann, Brunings, Funk and Bidone. 

In the ninth chapter of the Handbuch, the author has en- 
deavoured to simplify, nearly in the same manner as the motion 
of rivers, the theory of the motion of water in pipes, observing 
that the head of water may be divided into two parts, one to 
produce velocity, the other to overcome the friction ; and that 
the height must be as the length and circumference of the sec- 
tion of the pipe directly, or as the diameter, — and inversely as 
the area of the section, or as the square of the diameter. 

* Experiences sur le Cours des Fleuves, ou Lettre a un Magistral Hollandais, 
par M. Gennete. Paris 1760. 

+ " Recherches sur le Mouvement de I'Eau, en ayant egard a la Contraction 
qui a lieu au Passage par divers Orifices, et a la Resistance qui retard le Mouve- 
ment, le long des Parois des Vases ; par M. Eytelwein," — Memoires de I'Aca- 
dimie de Berlin, 1814 and 1815. 


In the allowance for flexure, the product of its square, multi- 
plied by the sum of the sines of the several angles of inflection, 
and then by "0038, will give the degree of pressure employed 
in overcoming the resistance occasioned by the angles ; and de- 
ducting this height from the height corresponding to the velo- 
city, will give the corrected velocity*. 

M. Eytelwein investigates, both theoretically and experi- 
mentally, the discharge of water by compound pipes, — the mo- 
tions of jets, and their impulses against plane and oblique sur- 
faces, as in watei'-wheels, in which it is shown that the hydraulic 
pressure must be twice the weight of the generating column, as 
deduced from the experiments of Bossut and Langsdorft ; and 
in the case of oblique surfaces, the effect is stated to vary as the 
square of the sine of the angle of incidence ; but for motions 
in open water about fths of the difference of the sine from the 
radius must be added to this square. 

The author is evidently wrong in calculating upon impulse 
as forming part of the motion of overshot wheels; but his 
theory, that the perimeter of a water-wheel should move with 
half the velocity of a given stream to produce a maximum effect, 
agrees perfectly with the experiments of Smeaton and others if. 

The author concludes his highly interesting work by exa- 
mining the effects of air as far as they relate to hydraulic ma- 
chines, including its impulse against plane surfaces on siphons 

• Hence, if / denote the height due to the friction, 
d = the diameter of the pipe, 
a = a constant quantity, 

we shall have, / = V^ ^^ and V^ = •^—,. 

•' d al 

But the height employed in overcoming the friction corresponds to the differ- 

ence between the actual velocity and the actual height, that is, /= A 5-, 

where b is the coefficient for finding the velocity from the height. 

Hence we have, V^ = =^-= and V = V — — -. 

ab' I ab^l -\- d 

Now Dubuat found h to be 6-6, and aV^ was found to be 0-0211, particularly 
when the velocity is between six and twenty-four inches per second. Hence 

or more accurately, V =: 50 v ( -, ^, |. 

•' \/+ 50 d/ 

+ The author of this paper has made a great many experiments on the max- 
imum effect of water-wheels ; but the recent experiments of the Franklin Insti- 
tution, made on a more magnificent scale, and now in the course of trial, eclipse 
everything that has yet been effected on this subject. See also Poncelet, Me- 
moire sur les Roues Hydrauliques, and Aubes Courbes par dessovs, ^fc. 1 827. 

174 THIRD REPORT — 1833. 

and pumps of different descriptions, horizontal and inclined 
helices, bucket-wheels, throwing-wheels, and lastly, on instru- 
ments for measuring the velocity of streams of water. A very 
detailed account of the work was given in the Journal of the 
Royal Institutioti, by the late Dr. Young. But it is due to 
MM. Dubuat and Prony to state, that M. Eytelwein has 
exactly followed the steps of these gentlemen in his Theory of 
the Motion of Water in open Channels. 

In the year 1809 a valuable series of experiments upon the 
motions of waters through pipes, was made by MM. Mallet 
and Vici at Rome, and afterwards by M. Prony*. 

It had been proved, by experiments made with great care, 
that the diminution of velocity, and consequent expenditure in 
pipes, was not in the ratio of the capacity of the pipes, as Fron- 
tinus had supposed in his valuation of the product of the an- 
cient module or calice ; and as it was desirable to ascertain the 
actual product of the three fountains now used at Rome, a se- 
ries of experiments was undertaken by these gentlemen ; the 
principal result of which was, that a pipe, of which the gauge 
was five oncesf, furnished fth more water than five pipes of 
one once, on account of the diminution of the velocity by 
friction in the ratio of the perimeters of the orifices as com- 
pared with their areas. 

M. Mallet also made a great many researches relative to the 
distribution of water in the different cities and towns of En- 
gland and France, with a view to their application at Paris ; of 
all of which he has published an account. 

The researches that had been made hitherto on the expendi- 
ture of water through orifices, had for their object the deter- 
mination of the velocity and magnitude of the section, by which 
it is necessary to multiply the velocity to obtain the expense. 
But although these be the first elements for consideration, they 
are not sufficient ; for the fluid vein presents other phaenomena 
equally important, both in the theory and its application, 
namely, the form and direction of the vein after it has issued 
from the orifice. The former phaenomena, as we before stated, 
had been long noticed by Michelotti and others, but nothing 
precise had been established on the forms and remarkable phag- 
nomena of the fluid vein itself. Venturi had given three ex- 

M. Hachette, in two memoirs presented to the Academic 
Royale des Sciences in 1815 and 1816, also considered the 

* Notices Historiques, par M. Mallet. Paris 1830. 
t French measure, or 0-03059 French kilolitres. 


form of veins ; and in his Traits des Machines, he states that 
he had ah'eady given a description of veins issuing from circu- 
lar, elliptical, triangular and square orifices, without having 
entered into any detail respecting them, so that that part of 
the subject was in a great measure involved in doubt. In 1829 
a paper, entitled "Experiences sur la Forme et sur la Direction 
des Veines et des Courans d'Eau, lances par diverses Ouvertures," 
was read to the Academy of Sciences at Turin by M. Bidone, 
giving an account of a series of experiments made in the years 
1826 and 1827, in the Hydraulic Establishment of the Royal 
University. The results of these experiments are divided into 
five articles. The first gives a description of the apparatus 
and mode of proceeding, and the figures obtained from veins 
expended from rectilinear and curvilinear orifices, with salient 
angles pierced in vertical plates, and whose perimeters are 
formed by straight and curved lines, varying upwards of fifty 
different ways, with vai-iable and invariable changes, from zero 
to twenty-two French feet : the area of water was equal to one 
square inch. The sections of the veins were taken at different 
distances from the aperture. The results are extremely curi- 
ous, as illustrating the influence of pressure and divergence on 
part of a fluid mass not i?i equilibrio, and may be assimilated to 
the phsenomena presented by the undulation of streams of 
light. The author contents himself with stating the results, 
which are further illustrated by diagrams. 

In a second paper, read to the Accademia delle Scienze in 
April following of the same year (1829), M. Bidone enters into 
a theoretical consideration of his experiments, in which he re- 
presents the greatest contraction of the fluid vein to take place 
at a distance not exceeding the greatest diameter of the orifice, 
whatever be the shape ; from which it results that the expres- 
sion for the expense of the orifice is equal to the sum of the 
product of each superficial element multiplied by the velocity 
of the fluid vein ; and as it was determined by experiment 
that the area of the vena contracta was from 0"60 to 0*62 of 
the area of the orifice, it follows that this coefficient of con- 
traction, multiplied by the velocity due to the charge, repre- 
sents the expenditure. 

M. Bidone considers the case of a fluid vein reduced to a 
state of permanence, and expended from a very small orifice, 
as compared with the sections of the containing vessel, accord- 
ing to the theory of Euler; and finds that the magnitude of the 
section of the contracted vein does not depend upon the velocity 
of the component filaments, but solely on their direction, a re- 
sult conformable to experiment. 

176 THIRD REPORT — 1833. 

He then determines, from the results of M. Venturoli*, the 
absolute magnitude of the contracted section of the vein (issuing 
from a circular orifice) to be exactly f rds of the orifice, the 
correction due to the contraction depending upon the ad- 
hesion and friction of the fluid against the perimeter of the ori- 
fice, and the ratio of the area of the vein to the area of the 
orifice : the same for all orifices. Hitherto the magnitude of 
fluid veins, as determined by direct measurements, had given 
greater coefficients than the effective expenditure allowed. 

Michelotti, with a pressure of twenty feet, with orifices of one 
and two inches in diameter, found the coefficient 0*649 

Bossut 0-660 

Borda 0-646 

Venturi 0-640 

Eytelwein 0-640 

Hachette 0-690 

Newton 0-707 

Helsham 0-705 

Brindley and Smeaton 0-631 

Banks 0-750 

Rennief 0-621 

In several experiments the ratio rarely exceeded 0-620 ; so 
that the discrepancy must have arisen from inaccuracies in the 
measurement of the fluid vein and orifice. 

In the year 1827, it having been considered desirable to re- 
peat the experiments of Bossut and Dubuat, application was 
made to the French Government by General Sabatier, Com- 
mander-in-chief of the Mihtary School at Metz, for permission 
to undertake a series of experiments on a scale of magnitude 
sufficient to estabhsh the principles laid down by those authors, 
and serve as valuable practical rules for future calculations. 
The apparatus consisted, 1st, of an immense basin, having 
an area of 25,000 square metres ; 2nd, of a smaller reservoir, 
having a superficial area of 1 500 square metres, and a depth 
of 3-70 metres, so contrived, by means of sluices, as to have a 
complete command of the level of the water during the experi- 
ment ; 3rd, of a basin directly communicating with the second 
basin, 3-68 metres in length, and 3 metres in width, to receive 
the product of the orifices ; 4th, a basin or gauge capable of 
containing 24,000 litres. 

* Elementi di Meccanica e d'Idraulka: Milano 1818. Recherche Geome- 
triche fatte nella Scuola degli Ingegneri pontifici d'Acque e Strode, Fanno 
1821. Milano. 

t "On the Friction and Resistance of Fluids," Philosophical Transactions of 



The time was constantly noticed by an excellent stop-watch, 
made by Breguet; and the opening of the orifices, the charges 
of the fluid in the reservoir, as well as the level of the water m 
the gauge basin relative to each expense of fluid, were always 
measured to the tenth of a millimetre, so that, even under the 
most unfavourable circumstances, the approximation was at least 
to g^odth part of the total result. The total disposable fall or 
height, counting from the ordinary surface of the Moselle river, 
was four metres, from which two metres were deducted for the 
gauge basin, leaving only a fall of two metres under the most 
favourable circumstances ; and in the subsequent experiments 
of 1828 the height never exceeded 1-60 metre, sufficiently 
high for all practical purposes. An apparatus was provided 
for regulating the height of the orifice and the surface of the 
water in the reservoirs, and for tracing with the greatest accu- 
racy the forms and sections of the fluid veins before and after 
issuing from the orifices, and the depressions experienced by 
the surface of the water previously to its issuing from an open- 
ing of twenty centimetres square, the upper side of which was 
on a level with the surface of the water in the reservoir. These 
depressions are recorded in the Tables, 

1st, On the expenditure of water through rectangular verti- 
cal orifices, twenty centimetres square, and varying in height 
from one to twenty centimetres, under charges of from -0174 
of a metre to 1"6901 metre: 

2ndly, On the expenditures of water from the similar-sized 
orifices, open at the top, but under charges of from two to 
twenty-two centimetres. 

The whole is comprised in eleven Tables of 241 experiments, 
to which is added a twelfth Table, showing the value of the co- 
efficients of contraction for complete orifices, from twenty cen- 
timetres square to one centimetre, calculated according to the 
following formula: 

D for the height of the orifices, where* 

D = lo\/¥gli = l{Ji-¥) ^/2g ^^—^ being the theo- 
retical expense relative to the velocity ; 

or the theoretical expense, having regard to the influence of 
the orifice. 

• That is, where/ = 020 metre, being thehorizontal breadth of all the orifices; 
h = the charge of the fluid on the lower part of the orifice ; 
A'= the charge in the upper or variable side of the orifice ; 
oz= k — h' the thickness of the vein of water. 

18.^3. N 

178 THIRD REPORT 1833. 

-' The conclusions to be derived from these Tables are, 

1st, That for complete orifices of twenty centimetres square 
and high charges, the coefficient is 0*000 ; with the charge 
equal to four or five times the opening of the orifice, the co- 
efficient augments to 0"605 ; but beyond that charge the co- 
efficient diminishes to 0"593. 

2ndly, That the same law maintains for orifices of ten and 
five centimetres in height, the coefficients being for ten centi- 
metres 0'61], 0-618, 0"611 respectively, and for five centi- 
metres in height 0-618, 0-631, 0-623. 

Lastly, That with orifices of three, two and one centimetres 
in height, the law changes very rapidly, and the coefficients 
increase as the opening of the orifice becomes less, being for 
one centimetre 0-698, the smallest height of the orifice, to 0-640 
for three centimetres. 

These remarkable discrepancies from the results of Bidone 
and others are attributed by MM. Lesbroa and Poncelet to 
diflferences in the construction of the apparatus or in the mode 
of measurement adopted by the latter gentlemen ; but in gene- 
ral the coincidences are sufficiently satisfactory, and they are 
the more accurately confirmed by the subsequent investigations 
of MM. D'Aubuisson and Castel at Toulouse *. As respects 
water issuing from the openings or notches made in the sides 
of dams, or what we should term incomplete orifices, it appears 
that the coefficient obtained by the ordinary formula of Dubuat, 
or I h V 2gh, augments from the total charge of twenty-two cen- 
timetres when it is from 0-389 to two centimetres when it be- 
comes 0-415 ; hence we may safely adopt M. Bidone's coefficient 
of 0-405, or, according to MM. Poncelet and Lesbros' theory 
0-400, for calculating expenditures through notches in dams. 
From these and other experiments the authors are led to con- 
clude, that the law of continuity maintains for indefinite heights 
both with complete and incomplete orifices, and that the same 
coefficient can be obtained by adopting in both cases the same 
formula. The avithors observe that the area of the section of 
the greatest contraction of the vein, considered as a true 
square, is exactly two thirds of the area of the orifice ; a fact 
which goes to prove that there is no certain comparison be- 
tween the mean theoretical or calculated velocities, by means 
of the formula now used, and the mean effective velocities de- 
rived from the expenditure. 

The authors conclude their memoir by recommending their 
experiments for adoption in all cases of plate orifices situated 

* 4nnales de Chimie et de Physique for 1830, torn. xliv. p. 225. 


at a distance from the sides and bottom of the reservoir, pro- 
mising to investigate with similar accuracy in a future memoir 
the cases which may occur to the contrary. 

A note is appended to the memoir by M. Lesbros, contain- 
ing formulae for calculating the effective expenditure of com- 
plete orifices ; and also a Table of constants, which gives the 
effective expenditure of each orifice as compared with experi- 
ment. We have been thus particular in detaihng the results of 
MM. Lesbros and Poncelet's work, because they have com- 
prehended all the cases upon which there remained any doubts, 
and with very few exceptions are in accordance with the expe- 
riments of Brunacci, Navier, Christian, Gueymard, D'Aubuis- 
son, and by the author of this paper *. So that in point of 
accuracy and laborious investigation, the authors of these va- 
luable accessions to our knowledge, not only merit our grati- 
tude, but have very amply i-ephed to the liberality of the French 

Having thus endeavoured to elucidate the labours of the 
foreign philosophers who have contributed so greatly to the 
progress of hydraulics, it only remains for us to notice the 
scanty contributions of our countrymen to the science. While 
France and Germany were rapidly advancing upon the traces 
of Italy, England remained an inactive spectator of their pro- 
gress, contented with the splendour of her own Newton, to 
receive from foreigners whatever was original or valuable in 
the science. The Philosophical Transactions, rich as they 
are in other respects, scarcely contain a single paper on this 
subject founded on any experimental investigations. Some 
erroneous and inconclusive inferences from Newton, by Dr. 
Jurin ; a paper on the Measure of Force, by Mr. Eames ; a 
paper on Wiers, by Mr. Roberts ; another on the Motion and 
Resistance of Fluids, by Dr. Vince ; and a summary of Bossut 
and Dubuat's Experiments on the Motions of Fluids through 
Tubes, by Dr. Thomas Young, comprise nearly the whole of 
the papers on hydraulics in the Philosophical Transactions. 
The various treatises on the subject published by Maclaurin, 
Emerson, Dr. Matthew Young, Desaguliers, Clare and Switzer, 
with the exception of the theoretical investigations, are compiled 
principally from the works of foreigners ; and it was not until the 
subject was taken up by Brindley, Smeaton, Robison, Banks 
and Dr. Thomas Young, that we were at all aware of our defi- 
ciency. Practical men were either necessitated to follow the un- 
certain rules derived from their predecessors, or their own expe- 
rience and sagacity, for the little knowledge they possessed. 
* Philosophical Transactions for 18SI, 
N 2 

180 THIRD RKPORT — 1833. 

On the suliject of lij'drometry ^ve were equally ignorant ; and 
although the Italian collection had been published several 
years previously, and was well known on the Continent, it was 
not until Mr. Mann published an abstract of that collection 
that we were at all aware of the state of the science abroad. 

Under these circumstances the author of this paper was in- 
duced, in the year 1830, to undertake a series of experiments to 
ascertain, 1st, The friction of water against the surface of a 
cylinder, and discs revolving in it, at different depths and ve- 
locities : from which it appeared, that with slow velocities the 
friction approximated the ratio of the surfaces, but that an in- 
crease of surface did not materially affect it with increased velo- 
cities ; and that with equal surfaces the resistances approxi- 
mated to the squares of the velocities. 

2ndly, To ascertain the direct resistances against globes 
and discs revolving in air and water alternately : from which it 
resulted, that the resistances in both cases were as the squares 
of the velocities; and that the mean resistances of circular discs, 
square plates, and globes of equal area, in atmospherical air, 
were as under : 

Circular discs . . 25-180 MS 

Square plates . . 22'010 in air, . . 1 '36 in water. 

Round globes . . 10-627 0-75 

3rdly, That with circular orifices made in brass plates of 
gJg-th of an inch in thickness, and having apertures of i, ^, f , y 
of an inch respectively, under pressures varying from one to 
four feet, the average coefficients of contraction were, 

for altitudes of 1 foot 0-619 

4 feet 0-621 

For additional tubes of glass the coefficient was,. 

for 1 foot 0-817 

4 feet 0-806 

4thly, That the expenditures through orifices, additional 
tubes, and pipes of different lengths, of equal areas and under 
the same altitude as compared with the expenditui-e through a 
pipe of 30 feet in length, are as 
1 : 3 for orifices, 
1 : 4 for additional tubes, 
1 : 3-7 for a pipe 1 foot in length, 

1 : 2-6 8 feet , 

1 : 2-0 4 , 

1 : 1-4 2 . 

5thly, That with bent rectangular pipes ^ an inch in diameter, 
and 15 feet in length, the expenditures were diminished with 
fourteen bends two thirds, as compared with a straight pipe. 


and with twenty-four right angles, one third ; but did not seem 
to observe any decided law. 

In several experiments tried on a great scale, the results 
gave from one fifth to one sixth of the altitude for the fric- 
tion. In the case of the Coniston main, which conducts the 
water from the reservoir at Coniston to the castle of Edin- 
burgh, the diameter of which is 41 inches, the length 14,930 
feet, and the altitude 51 feet, it was proved by Mr. Jardine 
that the formulae of Dubuat and Eytelwein approximated to 
the real results very nearly ; and in some experiments made on 
a. great scale by the author of this paper, these formulae were 
found equally applicable. In several experiments made in the 
year 1828, on the water-works at Grenoble, by M. Gueymard, 
it was found that pipes of six and eight French inches in dia- 
meter furnished only two thirds of the water indicated by the 
formulee of M. Prony ; but when of nine inches diameter, the 
formula approximated very nearly. In M. Gueymard's expe- 
riment the altitude of the reservoir above the point of delivery 
was 8*453 metres, or 27'73 English feet. The height to which 
the water was required to be elevated was 5'514 metres, or 
18 feet ; the volume of water required was 954 litres, or 33*6 
cubic feet; the length of the pipe was 3200 metres, or 10498 
feet. There were eight gentle curves in the system, but en- 
larged beyond the average diameter of the parts of the pipe ; 
from which it resulted that the height to which the water was 
delivered was only two thirds of the height of the reservoir*. 

In the preceding short but imperfect history of the science 
of hydraulics we have confined our attention to the experi- 
mental researches that have been made on spouting fluids only. 
In a future communication I hope to examine the state of 
our knowledge of the natural phaenomena of rivers, and the 
causes by which they are influenced ; at present it is extremely 
limited, and although we have many works upon the subject, 
very little seems to be known either of their properties or of 
the laws by which they are governed. 

• According to M. Prony 's theory, the height raised would only have been 
■5-514 metres instead of 5-671 metres. The difficulty, however, of making ex- 
periments on a great scale will always prove an obstacle to the right solution 
of the question, in as much as it exacts that the pipe be of the same dia- 
meter throughout, that is, perfectly straight, and free from bends, and the 
charge of water invariable. For this purpose M. Prony has calculated Tables 
showing the relation subsisting between the expenditure, diameter, length, the 
total inclination of the pipes, and the difference of pressure at its extremities. 

182 THIRD REPORT — lS'6o. 


Since the foregoing Report was read to the British Associa- 
tion a paper, entitled " Memoire sur la Constitution des Veines 
Liquides lancees par des Orifices Circulaires en mince paroi," 
has been communicated to the Academy of Sciences at Paris, 
by M. FeUx Savart, 26 Aout 1833. The author, after detailing 
very minutely the different phsenomena presented by liquid 
veins issuing from circular orifices perforated in thin plates, 
attached to the bottom and sides of vessels, illustrates his po- 
sitions by a series of curious experiments on the vibrations and 
sounds of the drops which issue from the annular rings or pipes 
formed by the troubled part of the liquid. The results of these 
experiments are best given in his own words. 

" 1°. Toute veine liquide lancee verticalement de haut en bas 
par un orifice circulaire pratique dans une parol plane et hori- 
zontale est toujours composee de deux parties bien distinctes 
par I'aspect et la constitution. La partie qui touche a I'orifice est 
un solide de revolution dont toutes les sections horizontales 
vont en decroissant graduellement de diametre. Cette premiere 
partie de la veine est calme et transparente, et ressemble a un 
tige de cristal. La seconde partie, au contraire, est toujours 
agitde, et paralt denuee de transparence, quoiqu'elle soit ce- 
pendant d'une forme assez reguliere pour qu'on puisse facile- 
ment voir quelle est divisee en un certahi nombre de ren- 
flemens allonges dont le diametre maximum est toujours plus 
grand que celui de I'orifice. 

" 2°. Cette seconde partie de la veine est composee de gouttes 
bien distinctes les unes des autres, qui subissent pendant leur 
chute, des changemens periodiqvies de forme, auxquels sont dues 
les apparences de ventres ou renflemens reguherement espaces 
que I'inspection directe fait reconnaitre dans cette partie de la 
veine, dont la continuite apparente depend de ce que les gouttes 
se succedent a des intervalles de temps qvii sont moindres que 
la duree de la sensation produite sur la retine par chaque goutte 
en particulier. 

" 3°. Les gouttes qui forment la partie trouble de la veine 
sont produites par des renflemens annulaires qui prennent 
naissance tres pres de I'orifice, et qui se propagent a des inter- 
valles de temps ^gaux, le long de la partie limpide de la veine, 
en augmentant de volume a mesure qu'ils descendent, et qui 
enfin se separent de I'extremite inferieure de la partie limpide 
et continue a des intervalles de temps egaux a ceux de leur 
production et de leur propagation. 


"4°. Ces renflemens annulaires sent engendr^s par une suc- 
cession periodique de pulsations qui ont lieu a rorifice meme ; 
de sorte que la vitesse de lecouleuient, au lieu d'etre uniforme, 
est periodiquement variable. 

"5". Le nombre de ces pulsations, m^me pour des charges 
foibles, est toujours assez grand, dans un temps donne, pour 
qu'elles soient de I'ordre de celles qui, par la frequence de leur 
retour, peuvent donner lieu a des sons perceptibles et compa- 
rables. Ce nombre ne depend que de la vitesse de I'ecoule- 
ment, a laquelle il est directement proportionnel, et du diametre 
des orifices, auquel il est inversement proportionnel. II ne pa- 
rait alter e ni par la nature du liquide, ni par la temperature. 

"6°. L'amplitude de ces pulsations pent etre considerable- 
ment augmentee par des vibrations de m^me periode commu- 
niquees a la masse entiere du liquide et aux parois du reservoir 
qui le contient. Sous cette influence etrangere, les dimensions 
et I'etat de la veine peuvent subir des changemens remarqua- 
bles : la longueur de la partie limpide et continue pent se 
reduire presqu'a rien, tandis que les ventres de la partie trouble 
acquierent vine regularite de forme et une transparence qu'ils 
ne possedent pas ordinairement. Lorsque le nombre des vibra- 
tions communiquees est diiferent de celui des pulsations qui 
ont lieu a I'orifice, leur influence peut meme aller jusqu'a 
changer le nombre de ces pulsations, mais seulement entre de 
certaines limites. 

" 7°. La depense ne paralt pas alteree par l'amplitude des 
pulsations, ni meme par leur nombre. 

" 8°. La resistance de I'air n'influe pas sensiblement sur la 
forme et les dimensions des veines, non plus que sur le nombre 
des pulsations. 

" 9°. La constitution des veines lancees horizontalement ou 
m&me obliquement de bas en haut ne difFere pas essentiellement 
de celle des veines lancees verticalement de haut en bas ; seule- 
ment le nombre des pulsations a I'orifice parait devenir d'autant 
moindre que le jet approche plus d'etre lance verticalement de 
bas en haut. 

" 10°. Quelle que soit la direction de la veine, son diametre 
decroit toujours tres rapidement jusqu'a une petite distance de 
I'orifice ; mais quand la veine tombe verticalement, le decroisse- 
ment continue jusqu'a ce que la partie limpide se perde dans 
la partie trouble : il en est encore de m^me quand la veine est 
lancee horizontalement, quoiqu'alors le decroissement suive une 
loi moins rapide. Lorsque le jet est lance obliquement de 
bas en haut, et qu'il forme avec I'horizon un angle de 25° a 45°, 
toutes les sections normales a la courbe qu'il decrit deviennent 

184 THIRD REPORT — 1833. 

sensiblement %ales entre elles, a partir de la partie contractee 
que touche a rorifice. Enfin, pour des angles plus grands que 
45°, le diametre de la veine va en augmentant depuis la partie 
contractee jusqu'a la naissance de la portion trouble ; de sorte 
que c'est seulement alors qu'il existe une section qu'on peut 
k juste titre appeler section contractee." 


Report on the Recent Progress and Present State of certain 
Branches of Analysis. By George Peacock, M.A., F.R.S., 
F.G.S., F.Z.S., F.R.A.S., F.C.P.S., Fellow and Tutor of 
Trinity College, Cambridge. 

The present Report was intended in the first instance to have 
comprehended some notice of the recent progress and present 
state of analytical science in general, including algebra, the 
application of algebra to geometry, the differential and integral 
calculus, and the theory of series : a very little progress, how- 
ever, in the inquiries which were required for the execution of 
this tmdertaking convinced me of the necessity of confining 
them within much narrower limits, unless I should have ven- 
tured to occupy a much larger space in the annual publication 
of the Proceedings and Reports of the British Association than 
could be properly or conveniently allotted to one department 
of science, when so many others were required to be noticed. 
It is for these reasons that I shall restrict my observations, 
in the following Report, to Algebra, Trigonometry, and the 
Arithmetic of Sines ; at the same time I venture to indulge a 
hope of being allowed, upon some future occasion, to bring 
before the Members of the Association some notice of those 
higher branches of analysis which at present I feel myself 
compelled, though reluctantly, to omit. 

Algebra. — The science of algebra may be considered under 
two points of view, the one having reference to its principles, 
and the other to its applications : the first regards its complete- 
ness as an independent science ; the second its usefulness and 
power as an instrument of investigation and discovery, whether 
as respects the merely symbolical results which are deducible 
from the systematic developement of its principles, or the ap- 
plication of those results, by interpretation, to the physical 

Algebra, considered with reference to its principles, has re- 
ceived very little attention, and consequently very little im- 
provement, during the last century ; whilst its applications, 
using that term in its largest sense, have been in a state of 
continued advancement. Many causes have contributed to this 
comparative neglect of the accurat^^ and logical examination of 
the first principles of algebra : in the first place, the proper 

186 THIRD REPORT — 18o3. 

assumption and establishment of those principles involve meta- 
physical difficulties of a very serious kind, which present them- 
selves to a learner at a period of his studies when his mind has 
not been subjected to such a system of mathematical discipline 
as may enable it to cope with them : in tlie second place, we are 
commonly taught to approach those difficulties under the cover 
of a much more simple and much less general science, by steps 
which are studiously smoothed down, in order to render the 
transition from one science to the other as gentle and as little 
startling as possible ; and lastly, from the peculiar relation 
which the first principles of algebra, in common with those of 
other sciences of strict demonstration, bear to the great mass 
of facts and reasonings of which those sciences are composed. 

It is this last circumstance which constitutes a marked distinc- 
tion between those sciences which, like algebra and geometry, 
are founded upon assumed principles and definitions, and the 
physical sciences : in one case we consider those principles and 
definitions as ultimate facts, from which our investigations pro- 
ceed in one direction only, giving rise to a series of conclusions 
which have reference to those facts alone, and whose correct- 
ness or truth involves no other condition than the existence of 
a necessary connexion between them, in whatever manner the 
evidence of that existence may be made manifest ; whilst in the 
physical sciences there are no such ultimate facts which can be 
considered as the natural or the assumable limits of our inves- 
tigations. It is true, indeed, that in the application of algebra 
or geometry to such sciences, we assume certain facts or prin- 
ciples as possessing a necessary existence or truth, investing 
them, as it were, with a strictly mathematical character, and 
making them the foundation of a system of propositions, whose 
connexion involves the same species of evidence with that of the 
succession of propositions in the abstract sciences ; but in as- 
signing to such propositions their proper interpretation in the 
physical world, our conclusions are only true to an extent which 
is commensurate with the truth and universality of application 
of our fundamental assumptions, and of the various conditions 
by which the investigation of those propositions has been sup- 
posed to be limited ; in other words, such conclusions can be 
considered as approximations only to physical truth ; for such 
assumed first principles, however vast may be the superstruc- 
ture which is raised upon them, form only one or more links in 
the great chain of propositions, the tennination and foundation 
of which must be for ever veiled in the mystery of the first 

It is not my intention to enter upon the examination of the 


general relations which exist between the speculative and physi- 
cal sciences, but merely to point out the distinction between the 
ultimate objects of our reasonings in the one class and in the 
other : in the first, we merely regard the results of the science 
itself, and the logical accuracy of the reasoning by which they 
are deduced from assumed first principles; and all our conclu- 
sions possess a necessary existence, without seeking either for 
their strict or for their approximate interpretation in the nature 
of things : in the second, we found our reasonings equally upon 
assumed first principles, and we equally seek for logical accu- 
racy in the deduction of our conclusions from them ; but both 
in the principles themselves and in the conclusions from them, 
we look to the external world as furnishing by interpretation 
corresponding principles and corresponding conclusions ; and 
the physical sciences become more or less adapted to the ap- 
plication of mathematics, in proportion to the extent to which 
our assumed first principles can be made to approach to the 
most simple and general facts or principles which are discover- 
able in those sciences by observation or experiment, when di- 
vested of all incidental and foreign causes of variation ; and 
still more so, when the causes of such variation can be di- 
stinctly pointed out, and when their extent and influence are 
reducible to approximate at least, if not to accurate estimation. 

The first principles, therefore, which form the foundation 
of our mathematical reasonings in the physical sciences being 
neither arbitrary assumptions nor necessary truths, but really 
forming part of the series of propositions of Avhich those sci- 
ences are composed, can never cease to be more or less the 
subject of examination and inquiry at any point of our re- 
searches : they form the basis of those interpretations which 
are perpetually required to connect our mathematical with the 
corresponding physical conclusions ; and even supposing the 
immediate appeal to them to be superseded, as will frequently 
be the case, by other propositions which are deducible from 
them, they still continue to claim our attention as the proposi- 
tions which terminate those physical and logical inquiries at 
which our mathematical reasonings begin. But in the abstract 
sciences of geometry and algebra, those principles which are 
the foundation of those sciences are also the proper limits of 
our inquiries ; for if they are in any way connected with the phy- 
sical sciences, the connexion is arbitrary, and in no respect af- 
fects the truth of our conclusions, which respects the evidence 
of their connexion with the first principles only, and does not 
require, though it may allow, the aid of physical interpretation. 

It is true that there exists a connexion between physical and 

188 THIRD REPORT — 1833. 

speculative geometry, as well as between physical and specula- 
tive mechanics ; and if in speculative geometry we regarded 
the actual construction and mensuration of the figures and solids 
in physical geometry alone, the transition from one science to 
the otlier being made by interpretation, then speculative geo- 
metry and speculative mechanics must be regarded as sciences 
which were similar in their character, though different in 
their objects : but we cultivate speculative geometry without 
any such exclusive reference to physical geometry, as an in- 
strument of investigation more or less applicable, by means 
of interpretation, to all sciences which are reducible to mea- 
sure, and whose abstract conclusions, in whatever manner 
suggested or derived, possess a great practical value altogether 
apart from their applications to practical geometry ; whilst the 
conclusions in speculative mechanics are valuable from their 
applications to physical mechanics only, and are not other- 
wise separable from the conclusions of those abstract sciences 
which are employed as instruments in their investigation. 

This separation of speculative and physical geometry was 
perfectly understood by the ancients, though their views of its 
application to the physical sciences were extremely limited; 
and it is to the complete abstraction of the principles of specu- 
lative geometry that we must in a great measure attribute the 
vast discoveries which were made by its aid in the hands of 
Newton and his predecessors, when a more enlarged and phi- 
losophical knowledge of the laws of nature supplied those phy- 
sical axioms or truths which were required as the medium of 
its applications ; and though it was destined to be superseded, 
at least in a great degree, by another abstract science of much 
greater extent and applicability, yet it was enabled to maintain 
its ground for a considerable time against its more powerful 
rival, in consequence of the superior precision of its prin- 
ciples and the superior evidence of its conclusions, when con- 
sidered with reference to the form under which the principles 
and conclusions of algebra were known or exhibited at that 

Algebra was denominated in the time of Newton specious or 
universal arithmetic, and the view of its principles which gave 
rise to this synonym (if such a term may be used) has more or 
less prevailed in almost every treatise upon this subject which 
has appeared since his time. In a similar sense, algebra has 
been said to be a science which arises from that generalization 
of the processes of arithmetic which results from the use of 
symbolical language : but though in the exposition of the prin- 
ciples of algebra, arithmetic has always been taken for its foun- 


dation, and the names of the fundamental operations in one 
science have been transferred to the other without any imme- 
diate change of their meaning, yet it has generally been found 
necessary subsequently to enlarge this very narrow basis of so 
very general a science, though the reason of the necessity of 
doing so, and the precise point at which, or the extent to which, 
it was done, has usually been passed over without notice. The 
science which was thus formed was perfectly abstract, in what- 
ever manner we arrived at its fundamental conclusions ; and 
those conclusions were the same whatever view was taken of 
their origin, or in whatever manner they were deduced ; but a 
serious error was committed in considering it as a science which 
admitted of strict and rigorous demonstration, when it certainly 
possessed no adequate principles of its own, whether assumed 
or demonstrated, which could properly justify the character 
which was thus given to it. 

There are, in fact, two distinct sciences, arithmetical and 
symbolical algebra, which are closely connected with each 
other, though the existence of one does not necessarily deter- 
mine the existence of the other. The first of these sciences 
would be, properly speaking, universal arithmetic : its general 
symbols would represent numbers only ; its fundamental ope- 
rations, and the signs used to denote them, would have the same 
meaning as in common arithmetic ; it would reject the inde- 
pendent use of the signs + and — , though it would recognise the 
common rules for their incorporation, when they were preceded 
by other quantities or symbols : the operation of subtraction 
would be impossible when the subtrahend was greater than 
the quantity from which it was required to be taken, and there- 
fore the proper imjjossible quantities of such a scienee-would 
be the negative quantities of symbolical algebra ; it would re- 
ject also the consideration of the multiple values of simple 
roots, as well as of the negative and impossible roots of equa- 
tions of the second and higher degree : it is this species of al- 
gebra which alone can be legitimately founded upon arithmetic 
as its basis. 

Mr. Frend *, Baron Maseres, and others, about the latter 
end of the last century, attempted to introduce arithmetical 

* The Principles of Algebra, by William Frend, 1796; and The true The- 
ory of Equations, established on Mathematical Demonstration, 1799. The fol- 
lowing extracts from his prefaces to these works will explain the nature of his 
views : 

" The ideas of number are the clearest and most distinct of the human mind : 
the acts of the mind upon them are equally simple and clear. There cannot 
be confusion in them, unless numbers too great for the comprehension of the 

190 THIRD REPORT— 1833. 

to the exclusion of sijmboUcal algebra, as the only form of it 
which was capable of strict demonstration, and which alone, 
therefore, was entitled to be considered as a science of strict and 
logical reasoning. The arguments which they made use of 
were unanswerable, M^hen advanced against the form under 
which the principles of algebra were exhibited in the elemen- 
tary and all other works of that period, and which they have 
continued to retain ever since, with very trifling and unimpor- 
tant alterations ; and the system of algebra which was formed 
by the first of these authors was perfectly logical and complete, 
the connexion of all its parts being capable of strict demon- 
stration; but there were a great multitude of algebraical re- 
sults and propositions, of unquestionable value and of unques- 
tionable consistency with each other, which were irreconcila- 
ble with such a system, or, at all events, not deducible from it ; 
and amongst them, the theory of the composition of equations, 
which Harriot had left in so complete a form, and which made 
it necessary to consider negative and even impossible quan- 

learner are employed, or some arts are used which are not justifiable. The 
first error in teaching the first principles of algebra is obvious on per\ising a few 
pages only of the first part of Maclaurin's Algebra. Numbers are there divided 
into two sorts, positive and negative : and an attempt is made to explain the 
nature of negative numbers, by allusions to book debts and other arts. Now 
when a person cannot explain the pi-inciples of a science, without reference to 
a metaphor, the probability is, that he has never thought accurately upon the 
subject. A number may be greater or less than another number : it may be 
added to, taken from, multiplied into, or divided by, another number ; but in 
other respects it is very intractable; though the whole world should be destroyed, 
one will be one, and three will be three, and no art whatever can change their 
nature. You may put a mark before one, wiiich it will obey ; it submits to be 
taken away from another number greater than itself, but to attempt to take it 
away from a number less than itself is ridiculous. Yet this is attempted by 
algebraists, who talk of a number less than nothing, of multiplying a negative 
number into a negative number, and thus producing a positive number, of a 
number being imaginary. Hence they talk of tvyro roots to every equation of 
the second order, and the learner is to try which will succeed in a given equa- 
tion : they talk of solving an equation which requii-es two impossible roots to 
make it soluble : they can find out some impossible numbers, which being 
multiplied together produce unity. This is all jargon, at which common sense 
recoils ; but from its having been once adopted, like many other figments, it 
finds the most strenuous supporters among those who love to take things upon 
trust and hate the colour of a serious thought." 

" From the age of Vieta, the father, to this of Maseres, the restorer of alge- 
bra, many men of the greatest abilities have employed themselves in the pursuit 
of an idle hypothesis, and have laid down rules not founded in truth, nor of any 
sort of use in a science admitting in every step of the plainest principles of 
reasoning. If the name of Sir Isaac Newton appears in this list, the number 
of advocates for errour must be considerable. It is, however, to be recollected, 
that for a much longer period, men scarcely inferiour to Newton in genius, and 
his equals, probably, in industry, maintained a variety of positions in philoso- 


titles as having a real existence in algebra, however vain might 
be the attempt to interpret their meaning. 

Both Mr. Frend and Baron Maseres were sensible of the con- 
sequences of admitting the truth of this theory of the compo- 
sition of equations as far as their system was concerned, and it 
must be allowed that they have struggled against it with con- 
siderable ingenuity: they admitted the possibility of multiple 
real, that is, positive roots, and which are all equally congruous 
to the problem whose solution was required through the medium 
of the equation, indicating an indetermination in the problem 
proposed : but it would be easy to propose problems leading to 
equations whose roots were real and positive, and yet not con- 
gnious to the problem proposed, whose existence must be ad- 
mitted upon their own principles ; and if so, why not admit the 
existence of other roots, whether negative or impossible, to 
which the algebraical solution of the problem might lead, though 
they might admit of no very direct interpretation, in conformity 
with the expressed conditions of the problem*? 

phy, which were overthrown by a more accurate investigation of nature ; and 
if the name Ptolemy can no longer support liis epicycles, nor that of Des 
Cartes his vortices, Newton's dereliction of the principles of reasoning cannot 
establish the fallacious notion, that every equation has as many roots as it has 

" This notion of Newton and others is founded on precipitation. Instead of 
a patient examination of the subject, an hypothesis which accounts for many 
appearances is formed; where it fails, unintelligible terms are used; in thos6 
terms indolence acquiesces ; much time is wasted on a jargon whicli has the 
appearance of science, and real knowledge is retarded. Thus volumes upon 
volumes have been written on the stupid dreams of Athanasius, and on the im- 
possible roots of an equation of w dimensions." 

This work of Mr. Frend, though containing many assertions which show 
great distrust of the results of algebraical science which were in existence at 
the time it was written, presents a very clear and logical view of the principles 
of arithmetical algebra. 

The voluminous labours of Baron Maseres are contained in his Scriptores 
Logarithmici, and in a thick volume of Tracts on the Resolution of Cubic and 
Biquadratic Equations. He seems generally to have forgotten that an}' change 
had taken place in the science of algebra between the age of Ferrari, Cardan, 
Des Cartes, and Harriot, and the end of the 1 8th century ; and by considering 
all algebraical formulae as essentially arithmetical, he is speedily overwhelmed 
by the same multiplicity of cases (which are all included in the same really al- 
gebraical formula) which embarrassed and confounded the first authors of the 

* Thus, in the solution of the following problem : " Sold a horse for 24?., 
and by so doing lost as much per cent, as the horse cost me : required the 
prime cost of the horse ?" we arrive at the equation 

100 X —x" = 2400 ; 
if we subtract both sides of this equation from 2500, we get 
2500 — 100 a; + x- = 100, 
or X- — 100 X -\- 2500 = 100, 
inastnuch as the quantities upon each side of the sign = are in both cases 

192 THIRD REPORT — 1833. 

If the authors of this attempt at algebraical reform had been 
better acquainted with the more modern results of the science, 
they would have felt the total inadequacy of the very limited 
science of arithmetical algebra to replace it ; and they would 
probably have directed their attention to discover whether any 
principles were necessary to be assumed, which were not neces- 
sarily deducible as propositions from arithmetic or arithmetical 
algebra, though they might be suggested by them. As it was, 
however, these speculations did not receive the consideration 
which they really merited ; and it is very possible that the 
attempt which was made by one of their authors to connect the 
errors in reasoning, which he attacked, as forming part only of 
a much more extensive class to which the human mind is liable 
from the influence of prejudice or fashion, had a tendency to 
divert men of an enlarged acquaintance with the results of 
algebra from such a cautious and sustained examination of them 
as was required for their refutation, or rather for such a correc- 
tion of them as was really necessary to establish the science of 
algebra upon its proper basis. 

I know that it is the opinion of many persons, even amongst 
the masters * of algebraical science, that arithmetic does supply 

identical with each other : if we extract the square root on both sides, re- 
jecting the negative value of the square root, we get in the first case 

50 — a; = 10, 
and in the second, 

a? — 50 = 10. 
The first of these simple equations gives us « = 40, and the second x = 60, both 
of which satisfy the conditions of the problem proposed : the two roots which 
are thus obtained, strictly by means of arithmetical algebra, show that the pro- 
blem proposed is to a certain extent indeterminate. Mr. Frend and Baron 
Maseres contended that multiple real roots, which are always the indication 
of a similar indetermination in the problems which lead to such equations, 
might be obtained by arithmetical algebra alone, and that all other roots were 
useless fictions, which could lead to no practical conclusions. But it is very easy 
to show, that incongruous and real, as well as negative and impossible roots, 
may equally indicate the impossibility of the problem proposed : thus, if it 
was proposed " to find a number the double of whose square exceeds three 
times the number itself by 5," we shall find 4 and — 1 for the roots of the 
resulting equation, both of which equally indicate the impossibility of the pro- 
blem proposed, if by number be meant a lohole positive number. 

* Cauchy, who has enriched analysis with many important discoveries, 
and who is justly celebrated for his almost unequalled command over its lan- 
guage, has made it the principal object of his admirable work, entitled Cours 
d' Analyse de I'Ecole Royale Polytechnique, to meet the difficulties which pre- 
sent themselves in the transition from arithmetical to symbolical algebra : and 
though he admits to the fullest extent the essential distinction between them, 
in the ultimate form which the latter science assumes, yet he considers the 
principles of one as deducible from those of the other, and presents the rules 
for the concurrence and incorporation of signs ; for the inverse relation of the 
operations called addition and subtraction, multiplication and division ; for 


a sufficient basis for symbolical algebra considered under its 
most general form ; that symbols, considered as representing 
numbers, may represent every kind of concrete magnitude ; 

the indifference of the order of succession of different algebraical operations, 
as so many theorems founded upon the ordinary principles and reasonings of 
arithmetic. In order to show, however, the extraordinary vagueness of the 
reasoning which is employed to establish these theorems, we will notice some 
of them in detail : On repvesente, says he, les grandeurs qui doivent servir d'ac- 
croissements, par des nomhres precedes du signs +. <?' l^s grandeurs qui doivent 
servir de diminutions par des nomhres precedes du signe — . Cela pose, les signes 
-\- et — places devant les nombres peuvent itre compares, suivani la remarque 
qui en a He faite^, a. des adjectifs place's aupres de leurs suhstantifs. It is unques- 
tionable, however, that in the most common cases of the interpretation of 
specific magnitudes affected with the signs + and — , there is no direct refer- 
ence either to increase or diminution, to addition or to subtraction. He sub- 
sequently gives those signs a conventional interpretation, as denoting quan- 
tities which are opposed to each other ; and assuming the existence of quan- 
tities affected by independent signs, and denoting + A by a, and — A by 6, 
he savs that 

4-«=-|-A + h=z — K 

— a = — A — b = + A; 

and therefore, 

-f(+A) = + A +(_A) = -A 

- (-h A) = - A - (- A) = -h A ; 

which he considers as a sufficient proof of the rule of the concurrence of 
signs in whatever operations they may occur ; though it requires a very slight 
examination of this process of reasoning to show that it involves several ar- 
bitrary assumptions and interpretations which may or may not be consistent 
with each other. In the proofs which he has given of the other fundamental 
theorems which we have mentioned above, we shall find many other instances 
of similar confusion both in language and in reasoning : thus, " subtraction 
is the inverse of addition in arithmetic ; then therefore, also, subtraction is 
the inverse of addition in algebra, even when applied to quantities affected 
with the signs -\- and — , and whatever those quantities may be." But is 
this a conclusion or an assumption? or in what manner can we explain in 
words the process which the mind follows in effecting such a deduction ? 
" If a and h be whole numbers, it may be proved that a 6 is identical with 
h a : therefore, a b is identical with b a, whatever a and b may denote, and 
whatever may be the interpretation of the operation which connects them." 
But any attempt to establish this conclusion, without a previous definition 
of the meaning of the operation of multiplication when applied to such quanti- 
ties, will show it to be altogether impracticable. The system which he has fol- 
lowed, not merely in the establishment of the fundamental operations, but 
likewise in the interpretation of what he terms symbolical expressions and 
symbolical equations, requires the introduction of new conventions, which are not 
the less arbitrary because they are rendered necessary for the purpose of 
making the results of the science consistent with each other : some of those 
conventions I believe to be necessary, and others not ; but in almost every in- 
stance I should consider them introduced at the wrong place, and more or 
less inconsistently with the professed grounds upon which the science is 

' By Buee in the Philosophical Transactions, 1806. 

1833. o 

194 THIRD KEPOUT — 1833. 

that the operations of addition, subtraction, multipUcation and 
division are used in one science and in the other in no sense 
which the mind may not comprehend by a practicable, though 
it may not be by a very simple, process of generalization ; that 
we may be enabled by similar means to conceive both the use 
and the meaning of the signs + and — , when used independ- 
ently ; and that though we may be startled and somewhat em- 
barrassed by the occurrence of impossible quantities, yet that 
investigations in which they present themselves may generally 
be conducted by other means, and those difficulties may be 
evaded which it may not be very easy or very prudent to en- 
counter directly and openly. 

In reply, however, to such opinions, it ought to be remarked 
that arithmetic and algebra, under no view of their relation to 
each other, can be considered as one science, whatever may be 
the nature of their connexion with each other ; that there is 
nothing in the nature of the symbols of algebra which can es- 
sentially confine or limit their signification or value ; that it is 
an abuse of the term generalization* to apply it to designate 
the process of mind by which we pass from the meaning of a — b, 
when a is greater than b, to its meaning when a is less than b, 
or from that of the product a b, when a and b are abstract num- 
bers, to its meaning when a and b are concrete numbers of the 
same or of a different kind ; and similarly in every case where 
a result is either to be obtained or explained, where no pre- 
vious definition or explanation can be given of the operation 
upon which it depends : and even if we should grant the legiti- 
macy of such generalizations, we do necessarily arrive at a new 
science much more general than arithmetic, whose principles, 
however derived, may be considered as the immediate, though 
not the ultimate foundation of that system of combinations of 
symbols which constitutes the science of algebra. It is more 
natural and philosophical, therefore, to assume such principles 
as independent and ultimate, as far as the science itself is con- 
cerned, in whatever manner they may have been suggested, so 
that it may thus become essentially a science of sypibols and 
their combinations, constructed upon its own rules, which may 

* The operations in arrithmetical algebra can be previously defined, whilst 
those in symbolical algebra, though bearing the same name, cannot : their 
meaning, however, when the nature of the symbols is known, can be generall_y, 
but by no means necessarily, interpreted. The process, therefore, by which we 
pass from one science to the other is not an ascent from particulars to generals, 
which is properly called generalization, but one which is essentially arbitrary, 
though restricted with a specific view to its operations and their results admit- 
ting of such interpretations as may make its applications most generally useful. 


be applied to arithmetic and to all other sciences by intei'preta- 
tion : by this means, interpretation -w'tM follow, and woiprecede, 
the operations of algebra and their results ; an order of suc- 
cession which a very slight examination of their necessary 
changes of meaning, corresponding to the changes in the spe- 
cific values and applications of the symbols involved, will very 
speedily make manifest. 

But though the science of arithmetic, or of arithmetical al- 
gebra, does not furnish an adequate foundation for the science 
of symbolical algebra, it necessarily suggests its principles, or 
rather its laws of combination ; for in as much as symbolical al- 
gebra, though arbitrary in the authority of its principles, is not 
arbitrary in their application, being required to include arith- 
metical algebra as well as other sciences, it is evident that their 
rules must be identical with each other, as far as those sciences 
proceed together in common : the real distinction between them 
will arise from the supj^osition or assumption that the symbols 
in symbolical algebra are perfectly general and unlimited both 
in value and rejjresentation, and that the operations to which 
they are subject are equally general likewise. Let us now 
consider some of the consequences of such an assumption. 

A system of symbolical algebra will require the assumption 
of the independent use of the signs + and — . 

For the general rule in arithmetical algebra* informs us, 
that the result of the subtraction of 6 + c from a is denoted 
hy a — b — c, or that a — {b + c) = a — b — c, its application 
being limited by the necessity of supposing that 6 + e is less 
than a. The general hypothesis made in symbolical algebra, 
namely, that symbols are unlimited in value, and that operations 
are equally applicable in all cases, would necessarily lead us 
to the conclusion that a — {b + c) = a — b — c for all values 
of the symbols, and therefore, also, when 6 = a, in which case 
we have 

a — {a + c) = a — a — c=:— c. 

In a similar manner, also, we find 

a — {a — c) = a — a + c=^ -|-e = cf. 
We are thus necessarily led to the assumption of the exist- 
ence of such quantities as — c and + c, or of symbols preceded 

• Whatever general symbolical conclusions are true in arithmetical algebra 
must be true likewise in symbolical algebra, otherwise one science could not 
include the other. This is a most important principle, and will be the subject 
of particular notice hereafter. 

t For it appears from arithmetical algebra that a — a^O, and that a — a 

196 THIRD REPORT 1833. 

by the independent signs * + and — , which no longer denote 
operations, though they may denote affections of quantity. It 
appears likewise that + c is identical with c, but that — c is a 
quantity of a different nature from c : the interpretation of its 
meaning must depend upon the joint consideration of the spe- 
cific nature of the magnitude denoted by a, and of the symbolical 
conditions which the sign — , thus used, is required to satisfyj-. 
In a similar manner, the result of the operation, or rather 
the operation itself, of extracting the square root of such a 
quantity as « — 6 is impossible, unless a is greater than b. To 
remove the limitation in such cases, (an essential condition in 
symbolical algebra,) we assume the existence of a sign such 
as -v/ — I ; so that if we should suppose b =■ a -\- c, we should 

get V {a — b) = '/{« — (« + (?)} = V [a — a—c) = V [ — c) 

= V' — \ cX- In a similar manner, in order to make the ope- 
ration universally applicable, when the ?«"' root of a — 6 is 
required, we assume the existence of a sign v^ — 1, for which, 
as will afterwards appear, equivalent symbolical forms can al- 
ways be found, involving a/ — 1 and numerical quantities. 

By assuming, therefore, the independent existence of the 
signs +, — , X/\, and v^ — 1, (1)", and ( — 1)''§, we shall obtain 
a symbolical result in all those operations, which we call addi- 
tion, subtraction, multiplication, division, extraction of roots, 
and raising of powers, though their meaning may or may ?iot be 
identical with that which they possess in arithmetic. Let us 
now inquire a little further into the assumptions which deter- 
mine the symbolical character and relation of these funda- 
mental operations. 

The operations called addition and subtraction are denoted 
by the signs + and — . 

They are the inverse of each other. 

* That is, not preceded by other symbols as in the expressions a — c and 
a + c. 

\ Amongst these conditions, the principal is, that if — c be subjected to 
the operation denoted by the sign — , it will become identical with + c: thus, 
a — ( — c) ■:= a-\- c. It does not follow, however, that the sign — thus used, 
must necessarily admit of interpretation. 

X The symbolical form, however, of this and of similar signs is not arbi- 
trary, but dependent upon the general laws of symbolical combination. 

§ I do not assert the necessity of considering such signs as V — 1. (1)", 
( — 1)», as forming essentially a part of the earliest and most fundamental as- 
sumptions of algebra : the necessity for their introduction will arise when 
those operations with which they are connected are first required to be con- 
sidered, and will in all cases be governed by the general principle above men- 


In the concurrence of the signs + and — , in whatever man- 
ner used, if two hke signs come together, whether + and + , or 

— and — , they are replaced by the single sign + ; and when 
two unlike signs come together, whether + and — , or — and + , 
they are replaced by the single sign — . 

When different operations are performed or indicated, it is 
indifferent in what order they succeed each other. 

The operations called multiplication and division are de- 
noted by the signs x and h-, or more frequently by a conven- 
tional position of the quantities or symbols with respect to 
each other : thus, the product of a and b is denoted by a x b, 
a . b, or a b ; the quotient of a divided by b is denoted by 

« -r- A, or by -V. 
•' b 

The operations of multiplication and division are the inverse 
of each other. 

In the concurrence of the signs + and — in multiplication or 
division, if two like signs come together, whether + and + , or 

— and — , they are replaced by the single sign + ; and if two un- 
like signs come together, whether + and — , or — and +, they 
are replaced by the single sign — . 

When different operations succeed each other, it is not indif- 
rent in what order they are taken. 

We arrive at all these rules, when the operations are defined 
and when the symbols are numbers, by deductions, not from 
each other, but from the definitions themselves : in other words, 
these conclusions are not dependent upon each other, but upon 
the definitions only. In the absence, therefore, of such defini- 
tions of the meaning of the operations which these signs or 
forms of notation indicate, they become assumptions, which are 
independent of each other, and which serve to define, or rather 
to interpret* the operations, when the specific nature of the 
symbols is known ; and which also identify the results of those 
operations \mth the corresponding restdts in arithmetical alge- 
bra, lohen the symbols are numbers and when the operations are 
arithmetical operations. 

The rules of symbolical combination which are thus assumed 

* To define, is to assign beforehand the meaning or conditions of a term or 
operation ; to interpret, is to determine the meaning of a term or operation 
conformably to definitions or to conditions previously given or assigned. It is 
for this reason, that we define operations in arithmetic and arithmetical alge- 
bra conformably to their popular meaning, and we interpret them in symboli- 
cal algebra conformably to the svmbolical conditions to which they are sub- 

198 THIRD REPORT — 1833. 

have been suggested only by the corresponding rules in arith- 
metical algebra. They cannot be said to he founded xx^on them, 
for they are not deducible from them ; for though the opera- 
tions of addition and subtraction, in their arithmetical sense, 
are applicable to all quantities of the same kind, yet they ne- 
cessarily require a different meaning when applied to quanti- 
ties which are different in their nature, whether that difference 
consists in the kind of quantity expressed by the unaffected 
symbols, or in the different signs of affection of symbols de- 
noting the same quantity ; neither does it necessarily follow 
that in such cases there exists any interpretation which can be 
given of the operations, which is competent to satisfy the re- 
quired symbolical conditions. It is for such reasons that the 
investigation of such interpretations, when they are discover- 
able, becomes one of the most important and most essential of 
the deductive processes which are required in algebra and its 

Supposing that all the operations which are required to be 
performed in algebra are capable of being symbolically de- 
noted, the results of those operations will constitute what are 
called equivalent forms, the discovery and determination of 
which form the principal business of algebra. The greatest 
part of such equivalent forms result from the direct applica- 
tion of the rules for the fundamental operations of algebra, 
when these rules regard symbolical combinations only : but 
in other cases, the operations which produce them being nei- 
ther previously defined nor reduced to symbolical rules, unless 
for some specific values of the symbols, we are compelled to 
resort, as we have already done in the discovery and assump- 
tion of the fundamental rules of algebra themselves, to the re- 
sults obtained for such specific values, for the purpose of dis- 
covering the rules which determine the symbolical natiu-e of 
the operation for all values of the symbols. As this principle, 
which may be termed the principle of the permanence of equi- 
valent forms, constitutes the real foundation of all the rules of 
symbolical algebra, when viewed in connexion with arithmeti- 
cal algebra considered as a science of suggestion, it may be 
proper to express it in its most general form, so that its autho- 
rity may be distinctly appealed to, and some of the most im- 
portant of its consequences may be pointed out. 

Direct proposition : 

Whatever form is algebraically/ equivalent to another when 
expressed in general symbols, must contifiue to be equivalent, 
whatever those sijmbols denote. 

Converse proposition : 


Whatever equivalent form is discoverable in arithmetical 
algebra considered as the science of suggestion, when the sym- 
bols are general in their form, though specif c in their value, 
will continue to be an equivalent form when the symbols are 
general in their nature as well as in their form *. 

The direct proposition must be true, since the laws of com- 
bination of symbols by which such equivalent forms are de- 
duced, have no reference to the specific values of the symbols. 

The converse proposition must be true, for the following 
reasons : 

If there be an equivalent form when the symbols are general 
in their nature as well as in their form, it must coincide with 
the form discovered and proved in arithmetical algebra, where 
the symbols are general in their form but specific in their na- 
ture ; for in passing from the first to the second, no change in 
its form can take place by the first proposition. 

Secondly, we may assume the existence of such an equivalent 
form in symbols which are general both in their form and in 
their nature, since it will satisfy the only condition to which 
all such forms are subject, which is, that of perfect coincidence 
with the results of arithmetical algebra, as far as such results 
are common to both sciences. 

Equivalent forms may be said to have a necessary existence 
when the operation which produces them admits of being de- 
fined, or the rules for performing it of being expressly laid 
down : in all other cases their existence may be said to be 
conventional or assumed. Such conventional results, however, 
are as much real results as those which have a necessary ex- 
istence, in as much as they satisfy the only condition of their 
existence, which the principle of the permanence of equivalent 
forms imposes upon them : thus, the series for (1 + ;r)" has a 
necessary existence whenever the nature of the operation upon 
\ + X which it indicates can be defined ; that is, when « is a 
whole or a fractional, a positive or negative, number f ; but for 
all other values of n, where no previous definition or interpre- 
tation of the nature of the operation which connects it with its 
equivalent series can be given, then its existence is conventional 
only, though, symbolically speaking, it is equally entitled to be 
considered as an equivalent form in one case as in the other. 

It is evident that a system of symbolical algebra might be 

• Peacock's Algebra, Art. 132. 

+ The meaning of (1 + «)" cannot properly be said to be defined when n 
is a fractional number, whether positive or negative, or a negative whole num- 
ber, but to be ascertained by interpretation conformably to the principle of 
the permanence of equivalent forms. 

200 THIRD REPORT 1833. 

formed, in which the symbols and the conventional operations 
to which they were required to be subjected would be perfectly 
general both in value and application. If, however, in the con- 
sti-uction of such a system, we looked to the assumption of such 
rules of operation or of combination only, as would be sufficient, 
and not more than sufficient, for deducing equivalent forms, 
without any reference to any subordinate science, we shovdd be 
altogether without any means of interpreting either our opera- 
tions or their results, and the science thus formed would be 
one of symbols only, admitting of no applications whatever. It 
is for this reason that we adopt a subordinate science as a sci- 
ence of suggestion, and we frame our assumptions so that our 
results shall be the same as those of that science, when the 
symbols and the operations upon them become identical like- 
wise : and in as much as arithmetic is the science of calculation, 
comprehending all sciences which are reducible to measure and 
to number ; and in as much as arithmetical algebra is the imme- 
diate form which arithmetic takes when its digits are replaced 
by symbols and when the fundamental operations of arithmetic 
are applied to them, those symbols being general in form, 
though specific in value, it is most convenient to assume it as 
the subordinate science, which our system of symbolical algebra 
must be required to comprehend in all its parts. The principle 
of the permanence of equivalent forms is the most general ex- 
pression of this law, in as much as its truth is absolutely neces- 
sary to the identity of the results of the two sciences, when the 
symbols in both denote the same things and are subject to the 
same conditions. It was with reference to this principle that 
the fundamental assumptions respecting the operations of ad- 
dition, subtraction, multiplication and division were said to be 
suggested by the ascertained rules of the operations bearing 
the same names in arithmetical algebra. The independent use 
of the signs -}- and — , and of other signs of affection, was an as- 
sumption requisite to satisfy the still more general principle of 
symbolical algebra, that its symbols should be unlimited in value 
and representation, and the operations to which they are sub- 
ject unlimited in their application. 

In arithmetical algebra, the definitions of the operations de- 
termine the rules ; in symbolical algebra^ the rules determine 
the meaning of the operations, or more properly speaking, they 
furnish the means of interpreting them : but the rules of the 
former science are invariably the same as those of the latter, 
in as much as the rules of the latter are assumed with this view, 
and merely differ from the former in the universality of their 
applications : and in order to secure this universality of their 


applications, such additional signs* are assumed, and of such a 
symbolical foi-m, as those applications may render necessary. 
We call those rules, or their equivalent symbolical consequences, 
assumptions, in as much as they are not deducible as conclusions 
from any previous knowledge of those operations which have 
corresponding names: and we might call them arhitrary as- 
sumptions, in as much as they are arhitrar'ily imposed upon a 
science of symbols and their combinations, which might be 
adapted to any other assumed system of consistent rules. In 
the assumption, therefore, of a system of rules such as will make 
its symbolical conclusions necessarily coincident with those of 
arithmetical algebra, as far as they can exist in common, we in 
no respect derogate from the authority or completeness of sym- 
bolical algebra, considered with reference to its own conclu- 
sions and to their connexion with each other, at the same time 
that we give to them a meaning and an application which they 
would not otherwise possess. 

It follows from this view of the relation of arithmetical and 
symbolical algebra, that all the results of arithmetical algebra 
which are general in form are true likewise in symbolical 
algebra, whatever the symbols may denote. This conclusion 
may be said to be true in virtue of the principle of the perma- 
nence of equivalent forms, or rather it may be said to be the 
proper expression of that principle. Its consequences are most 
important, as far as the investigation of the fundamental pro- 
positions of the science are concerned, in as much as it enables 
us to investigate them in the most simple cases, when the 
operations which produce them are perfectly defined and un- 
derstood, and when the general symbols denote positive whole 
numbers. If the conclusions thus obtained do not involve in 
their expression any conditiofi which is essentially connected 
v-it/i the specific values of the symbols, they may be at once 
transferred to symbolical algebra, and considered as true for 
all values of the symbols whatsoever^. 

Thus, coefficients in arithmetical algebra, such as m in m a, 
which are general in form, lead to the interpretation of such 

* There is no necessary limit to the multiplication of such signs : the signs 
+ . — I (1)" and (—1)" and their equivalents (for the symbolical form of such 
signs is not arbitrary), comprehend all those signs of affection which aie re- 
quired by those operations with which we are at present acquainted. 

+ Some formula; are essentially arithmetical : of this kind is 1 . 2 . 3 . . . r, 

in which r must be a whole number. The formula "K"' — ^) • • • (w — r+ 1) 

1.2 . . . r 
IS symbolical with respect to m, but arithmetical with respect to r. Such cases, 
and their extension to general values of/-, will be more particularly considered 

202 THIRD REPORT 1833. 

expressions as m a in symbolical algebra, when m is a number 
whole or fractional, and a any symbol whatsoever. When m, n 
and a are whole numbers, it very readily appears that ma + na 
= (/« + n) a, and that ma — n a ^= {m — n) a : the same con- 
clusions are true likewise for all values of ?n, n and a. In 
arithmetical algebra we assume a^, aP, «"*, &c., to represent a a, 
a a a, aaaa, &c., and we readily arrive at the conclusion that 
«"* X «" = «'"■'" ", when m and n are whole numbers : the same 
conclusion must be true also when m and n are any quantities 
whatsoever. In a similar manner we pass from the result 
{cf) " = «"", when « is a whole number, to the same conclusion 
for all values of the symbols *. 

The preceding conclusions are extremely simple and element- 
ary, but they are not obtainable for all values of the symbols 
by the aid of any other principle than that of the permanence 
of equivalent forms : they are assumptions which are made in 
conformity with that principle, or rather for the purpose of 
rendering that principle universal ; and it will of course follow 
that all interpretations of those expressions where m and n are 
not whole numbers must be subordinate to such assumptions. 

Thus, Tr~''"0~('o+'2") a, = a, and therefore -^ must 

mean one half of a, whatever a may be ; a- x a = a^ 

= a = a, and therefore a must mean the square root of a, 
whatever a may be, whenever such an operation admits of 

interpretation. In a similar manner -^ must mean one third 

part, and a^ the cube root of a, whatever a may be, and simi- 
larly in other cases : it follows, therefore, that the interpreta- 
tion of the meaning of a , a^, &c., is determined by the general 

* The general theorems ma -\- na = (m -\- n) a and ma — na= {m — n) a, 

m —VI 

ar'Xa!' = 0'" + " and \ = a*"" ", (a"')" = o™" and (o*") " =a»' which 


are deduced by the principle of the permanence of equivalent forms, and which 
are supplementary to the fundamental rules of algebra, are of the most essen- 
tial importance in the simplification and abridgement of the results of those 
operations, though not necessary for the formation of the equivalent results 
themselves. It also appears from the four last of the above-mentioned theorems 
that the operations of multiplication and division, involution and evolution, are 
performed by the addition and subtraction, multiplication and division, of the 
indices, when adapted to the same symbol or base. If such indices or logarithms 
be calculated and registered with reference to a scale of their corresponding 
numbers, they will enable us to reduce the order of arithmetical operations by 
two unities, if their orders be regulated by the following scale ; addition (1), sub- 
traction (2), multiplication (3), division (4), involution (5), and evolution (6). 


principle of indices, and also that we ought not to say that we 

assume a to denote ^/a, and a^ to denote ^a, as is commonly 
done *, in as much as such phrases would seem to indicate that 
such assumptions are independent, and not subject to the same 
common principle in all cases. 

In all cases of indices which involve or designate the inverse 
processes of evolution, we must have regard likewise to the 
other great principle of symbolical algebra, which authorizes 
the existence of signs of affection. The square root of a may 

be either affected with the sign + or with the sign — ; for + or 

X + «*, and — «^ X — « , will equally have for their result 
+ a or a, by the general rule for the concurrence of similar 
signs and the general principle of indices : in a similar manner 

d^ may be affected with the multiple sign of affection (1)^, if 

there are any symbolical values of (1)^ different from + 1 (equi- 
valent to the sign +), which will satisfy the requisite symbo- 
lical conditions f . It is the possible existence of such signs of 
affection, which is consequent upon the universality of alge- 
braical operations, which makes it expedient to distinguish be- 
tween the resvdts which are not affected by such signs, and 
the same results when affected by them. The first class of 
results or values are such as are alone considered in arithmeti- 
cal algebra, and we shall therefore term them arithmetical va- 
lues, though the quantities themselves may not be arithmetical : 
the second class may be termed algebraical values, in as much 
as they are altogether, as far as they are different from the 
arithmetical values, the results of the generality of the opera- 
tions of symbolical algebra. 

This distinction may generally be most conveniently ex- 
pressed by considering such a sign as a factor, or a symbolical 
quantity multiplied according to the rule for that operation 
into the arithmetical value : in this sense + 1 and — I may be 
considered as factors which are equivalent to the signs + and 
— , that is, equivalent to affecting the quantities into which 
they are multiplied with the signs + and — , according to the 

• Wood's Algebra, Definitions. 

f That is, it' there is any symbolical expression different from 1, such as 

^ '-, and ^ , the cubes of which are identical with 1. 

In a similar manner we may consider the existence of multiple values of 1" or 
( — 1) , and, therefore, of multiple signs of affection corresponding to them, as 
consequent upon the general laws of combination of symbolical algebra, and as 
results to be determined from those laws, and whose existence, also, is de- 
pendent upon them. 

204 THIRD REPORT — 1833. 

general rule for the concurrence of signs. In a similar manner 

we may consider (1)* (a)* as equivalent to (r/)* ; (1)^ (o)^ as 

equivalent to («)^; (1)" a" as equivalent to «" ; (—1)" («)" as 
equivalent to ( — a)", and similarly in other cases : in all such 
cases the algebraical quantity into which the equivalent sign 
or its equivalent factor is multiplied, is supposed to possess its 
arithmetical value only*. 

The series for (1 + x)", when n is a whole number, may be 
exhibited under a general form, which is independent of the 
specific value of the index ; for such a series may be continued 
indefinitely in form, though all its terms after the (« + l)th 
must become equal to zero. Thus, the series 

(1 + £)• = (1)" (l + » 0,- + -f-^' ^ + 

+ "i'ra'\"/".T" -'' + ^°-) 

indefinitely continued, in which n is particular in value (a whole 
number) though general in form, must be true also, in virtue 
of the principle of the permanence of equivalent forms, when 
n is general in value as well as in formf . 

This theorem, which, singly considered, is, of all others, the 
most important in analysis, has been the subject of an almost 
unlimited variety of demonstrations. Like all other theorems 
whose consequences present themselves very extensively in 
algebraical results, it is more or less easy to pass from some 
one of those consequences to the theorem itself: but all the 
demonstrations which have been given of it, with the excep- 
tion of the principle of one given by Euler^, have been con- 
fined to such values of the index, namely, whole or fractional 
numbers, whether positive or negative, as made not only 
the development depend upon definable operations, but like- 
wise assumed the existence of the series itself, leaving the form 
of its coefiicients alone undetermined. It is evident, however, 
that if there existed a general form of this series, its form could 

* This separation of the symbolical sign of affection from its arithmetical 
subject, or rather the expression of the signs of affection explicitly, and not im- 
plicitly, is frequently important, and affords the only means of explaining many 
paradoxes (such as the question of the existence of real logarithms of negative 
numbers), by which the greatest analysts have been more or less embarrassed. 

f If such a series should, for any assigned value of n, have more symbolical 
values than one, one of them will be the arithmetical value, inasmuch as one 
symbolical value of l" is always 1. 

X In the Nov. Comm. Petropol. for 1771. 


be detected for any value of the index whatever, which was 
general in form, and therefore, also, when that index was a 
whole number ; a case in which the interpretation of the opera- 
tion designated by the index, as well as the performance of the 
operation itself, was the most simple and immediate. 

That such a series, likewise, would satisfy the only sym- 
bolical conditions which the general principles of indices ini- 
poses upon the binomial, might be very easily shown; for if 
m and n be whole numbers, then if the two series 

{\ \ xf -V^\\ ^■ mx ■\- -\ ^ x^ -j- &c. J 

(1 + a-)« = 1« (l + n X + '--^^^-^^^2 +- &c.) 

be multiphed together, according to the rule for that purpose, 
we must obtain 

(1 4- xy^''^ l™-*-" ( 1 + (m+ n)x + ^ \2 ^^7 

a series in which m + n has replaced m or n in its component 
factors : and in as much as we must obtain the same symbolical 
result of tliis multiplication, whatever be the specific values of 
m and n, it follows, that if the same form of these series repre- 
sents the development of (1 -|- or)" and (1 + x^, whatever m and 
n may be, then, likewise, the series for the product of (1 + a:)"' 
and (I + xY, or (1 + x)'"+", would be that which arose from 
putting wi + w in the place of m or n in each of the component 
factors. If, therefore, we assumed S {uri) and S (ra) to represent 
the series for (1 + xj" and (1 + x^, when m and n are any 
quantities whatsoever, then (1 + a-)" x (1 + .r)« = (1 + ^)'« + " 
= S (w -f- w) = S (;h) X S (w) ; or, in other words, the series 
will possess precisely the same symbolical properties with the 
binomial to which they are required to be equivalent. 

It is the equation a"" x «" = a*""*"", for all values of ra and n, 
which determines the interpretation of «"' or a", when such an 
interpretation is possible ; in other words, such quantities pos- 
sess no properties which are independent of that equation. The 
same remark of course extends to (1 + xj", for all values of w, 
and similarly, likewise, to those series which are equivalent to 
it. That all such series must possess the same form would be 
evident from considering that the symbolical properties of 
(1-1- J")" undergo no change for a change in the value of n, and 
that no series could be permanently equivalent to it whose form 

206 THIRD REPORT 1833. 

was not equally permanent likewise. In assuming, therefore, the 
existence of such a permanent series, our symbolical conclu- 
sions are necessarily consistent with each other, and it is the 
interpretation of the operations which produce them, which 
must be made in conformity with them. It is true that we can 
extract the square or the cube root of 1 + x, and we can also 
determine the corresponding series by the processes of arith- 

metical algebra ; and we likewise interpret (1 + xf and (1 + xf 
to mean the square and the cube root of 1 + x, in confoi-mity 
with the general principle of indices. The coincidence of the 

series for (1 + xf and (1 + xf, whether produced by the 
processes of arithmetical algebra, or deduced by the principle 
of the permanence of equivalent forms from the series for 
(1 + x)", would be a proof of the correctness of our interpreta- 
tion, not a condition of the truth of the general principle itself. 

In order to distinguish more accurately the precise limits of 
hypothesis and of proof in the establishment of the fundamental 
propositions of symbolical algebra, it may be expedient to re- 
state, at this point in the progress of our inquiry, the order in 
which the hypotheses and the demonstrations succeed each 

We are supposed to be in possession of a science of arith- 
metical algebra whose symbols denote numbers or arithmetical 
quantities only, and whose laws of combination are capable of 
strict demonstration, without the aid of any principle which is 
not furnished by our knowledge of common arithmetic. 

The symbols in arithmetical algebra, though general in form, 
are not general in value, being subject to limitations, which are 
necessary in many cases, in order to secure the practicability 
or possibility of the operations to be performed. In order to 
effect the transition from arithmetical to symbolical algebra, we 
now make the following hypotheses : 

(1.) The symbols are unlimited, both in value and in repre- 

(2.) The operations upon them, whatever they may be, are 
possible in all cases. 

(3.) The laws of combination of the symbols are of such a 
kind as to coincide universally with those in arithmetical algebra 
when the symbols are arithmetical quantities, and when the 
operations to which they are subject are called by the same 
names as in arithmetical algebra. 

The most general expression of this last condition, and of its 
connexion with the first hypothesis, is the law of the perma- 


nence of equivalent forms, which is our proper guide in the 
establishment of the fundamental propositions of symbolical 
algebra, in the invention of the requisite signs, and in the de- 
termination of their symbolical form : but in the absence of the 
complete enunciation of that law, we may proceed with the in- 
vestigation of the fundamental rules for addition, subtraction, 
multiplication and division, and of the theorems for the collec- 
tion of multiples, and for the multiplication and involution of 
powers of the same symbol, which will, in fact, form a series of 
assumptions which are not arbitrary, but subordinate to the 
conditions which are imposed by our hypotheses : but if we 
suppose those conditions to be incorporated into one general 
law, whose truth and universality are admitted, then those as- 
sumptions become necessary consequences of this law, and 
must be considered in the same light with other propositions 
which follow, directly or indirectly, from the first principles of 
a demonstrative science. In the same manner, if we assume the 
existence of such signs as are requisite to secure the universality 
of the operations, the symbolical form of those signs, and the 
laws which regulate their use, will be determined by the same 
principles upon which the ordinary results of symbolical al- 
gebra are founded. 

The natural and necessary dependence of these two methods 
of proceeding upon each other being once established, we may 
adopt either one or the other, as may best suit the form of the 
investigation which is under consideration : the great and im- 
portant conclusion to which we arrive in both cases being, the 
transfer of all the conclusions of arithmetical algebra which are 
general in form (that is, which do not involve in their expres- 
sion some restriction which limits the symbols to discontinuous 
values,) to symbolical algebra, accompanied by the invention or 
use of such signs (with determinate symbolical forms) as may 
be necessary to satisfy so general an hypothesis. 

There are many expressions which involve symbols which 
are necessarily discontinuous in their value, either from the 
form in which they present themselves in such expressions or 
from some very obvious conventions in their use : thus, when 
we say that 

cos x = COS {2r i: -\- x), 
and — COS x = cos { (2 r -|- 1) tt 4 x} 

propositions which are only true when r is a whole number, 
the limitation is conveyed (though imperfectly) by the con- 
ventional use of 2 r and 2 r + 1 to express even and odd num- 
bers ; for otherwise there would be no sufficient reason for not 

208 THIRD REPORT— 1833. 

using the simple symbol r both in one case and the other. In 
a similar manner, in the expression of Demoivre's theorem 

(cos 9 + \/^^l sin fl)" 

= cos (2 ?• w TT + ?i fl) + -v/ — 1 sin {2 r mr + n 6), 

we may suppose w to be any quantity whatsoever *, but r is ne- 
cessarily a whole number. 

In some cases, however, the construction of the formula it- 
self will sufficiently express the necessary restriction of the 
values of one or more of its symbols, without the necessity of 
resorting to any convention connected with their introduction : 

thvis, the formula 1 x 2 x 3 r, commencing from 1 , is 

essentially arithmetical, and limited by its form ro whole and 
positive values of r. The same is the case with the formula 
r (r — 1) . . . . 3 . 2 . 1, where some of the successive and strictly 
arithmetical values of the terms of the series r, r — 1, &c., are 
put down ; but the formula r (r — 1) ('' — 2) .... is subject to 
no such restriction, in as much as any number of such factors 
may be formed and multiplied together, whatever be the value 
of r. In a similar manner, the formula 

n {n — I) ... {n — r + I), 
r72 77. r 

which is so extensively used in analysis, is unlimited with re- 
spect to the symbol w, and essentially limited with respect to 
the symbol r : it is under such circumstances that it presents 
itself in the development of (1 + a;)". 

In the differential calculus we readily find 

'^ = n{n-l). ..{n-r + l)x"-', 

and in a similar manner also 

•^(.r« + C, x'^-' + C, x'-^ + . . C„) = w (w-1) . . (H-r+ 1>«-'-: 
dx^ 1 

in both these cases the value of n is unlimited, whilst the value 
of r is essentially a positive whole number ; in other words, 

* The investigation of this formula (like the equivalent series for (1 -|- x) 
when n is a general symbol,) requires the aid of the principle of the perma- 
nence of equivalent forms, in common with all other theorems connected with 
the general theory of indices. The formula above given involves also impli- 
citly any sign of affection which the general value of n may introduce : for 

(cos 6 + V"^ sin tf)" = (1)" (cos n6+ V^-i sin n 6) 

= (cos 2 /• M TT + -\/— 1 sin 2rn-!r) (cos n6 + -v/ — 1 sin n 6) 

= cos {2 rn TT -{■ n d) -\- V' — 1 sin (2 r n rr -\- n 6) 


the principle of equivalent forms might be extended to this for- 
mula (supposing it to be investigated for integral values of n 
and r,) as far as the symbol n is concerned only. This arithme- 
tical coefficient of differentiation (if such a term may be ap- 
pHed to it,) will present itself in the expression of the rth dif- 
ferential coefficient (r being a whole positive number,) of all al- 
gebi-aical functions *; and it is for this reason that we aveojjpa- 
rently debarred from considering fractional or general indices 
of differentiation when applied to such functions, and that we 
are consequently prevented from treating the differential and 
integral calculus as the same branch of analysis whose general 
laws of derivation are expressed by common formulae. 

But is it not possible to exhibit the coefficient of differentia- 
tion under some equivalent form which may include general 
values of the index of differentiation ? It is well known that 
the definite integral 

/ d X \ log — ) 

(adopting Fourier's notation,) is equal to 1 . 2 . . . . 7z, when n is 
a whole number ; and that consequently^ under the same cir- 
cumstances, the coefficient of differentiation or 

/ d X \ log — ) 

n (rt — 1) . . . (w — r + 1) = 

^'^.(logiy ' 

and in as much as \hQ form of this equivalent expression is not 
restricted to integral and positive values of r, we may assume 

• IT, -f 1 ^ f^' « (-l)'-.2.3...(r+l)a;' - 

* Inus if M = -; — ; — r, we have -. — 5 =: 7^ — \ — srrxi 

X (1 - ^ ('^ - ^) , r(r-l)(r-2) (r - 3) _ ^^ ] . 
\ 2.3a;2+ a;'' '/* 

I , d'u 1.2 r 

and if M = -TT. — I — 5;, we have 

V (1 + a;7 dxr — (1 — a"- ) »•+ 4 

X ( 1+ JL . '• (>• - ^) , 1 ^(,.-l)(r-2) (r-3) , & 1 
12= x" "^ 2= .4= a-4 J 

+ This definite integral, the second of that class of transcendents to which 
Legendre has given the name of Intecjrales Euleriennes, was first considered 
by Euler in the fifth volume of the Commentarii PetropoUtani, in a memoir 
on the interpolation of the terms of the series 

1 + 1x2 + 1x2x3 + 1x2X3X4 + &c., 

which is full of remarkable views upon the generalization of formula; and 
their interpretation. The same memoir contains the first solution of a pro- 
blem involving fractional indices of differentiation, 
1833. P 

210 THIRD REPORT — 1833. 

■it to be permanent, so long as we do not at the same time as- 
sume the necessary existence and interpretation of equivalent 
results. If, however, such results can be found, either gene- 
rally, or for particular values (not integral and positive) of r, 
apart from the sign of integration, the consideration of the values 
of the corresponding differential coefficients will involve no 
other theoretical difficulty than that which attends the transition 
from integral to fractional and other values of common indices. 

Euler, in his Differential Calculus *, has given the name of 
inexplicable functions to those functions which are apparently 
resti'icted by their form to integral and positive values of one or 
more of the general symbols which they involve : of this kind 
are the functions 

Ix2x3x X, 

I '^ 2 '^ 3 '^ x' 

111 1 

1 H i. — 

1" 2" ' -^" 

1 4- " ~ ^ I ^ ~ ^^ a — (x — 1) b 

'^ a + b '^ a + 3b '^ a + {x + l)b' 

and innumerable others which present themselves in the theory 
of series. The attempts which he has made to interpolate the 
series of which such functions form the general terms, are 
properly founded upon the hypothesis of the existence of per- 
manent equivalent forms, though it may not be possible to ex- 
hibit the explicit forms themselves by means of the existing 
signs and symbols of algebra. In the cases which we have 
hitherto considered, the forms which were assumed to be per- 
manent had a real previous existence, which necessarily re- 
sulted from operations which were capable of being defined. 
In the case of inexplicable functions, the corresponding perma- 
nent forms which hypothetically include them, may be consi- 
dered as having an hypothetical existence only, whose form 
degenerates into that of the inexplicable function in the case of 
integral and positive values of the independent variable or va- 
riables. It is for the expression of such cases that definite 
integrals find their most indispensable usage. 

* Instiiutiones Calculi Differentialis, Capp. xvi. et xvii. See also an admira- 
ble posthumous memoir of the same author amongst the additions to the 
Edition of that work printed at Pavia in 1787- He had been preceded in such 
researches by Stirling, an author of great genius and originality, whose la- 
bours upon the interpolation of series and other subjects have not received 
the attention to which they are justly entitled. 


It is easy to construct formulae which may exhibit the possi- 
bility of their thus degenerating into others of a much more 
simple form, when one or more of the independent variables be- 
come whole numbers : of this kind is the formula 

« + <S sin (2 r TT + ^) + y sin (g rir + flp + &c. ,. 

a + /3sin9 + ysinfl' +,&c. ^ fv)> 

which is, or is not, identical with 4) (r), according as r is a whole 
or a fractional number : such functions are termed nticlulating 
functions by Legendre *. We can conceive also the possible 
existence of many other transcendents amongst the unknown 
and undiscovered results of algebra, which may possess a simi- 
lar property. 

The transcendent 

X '^ ^ O^s D' 

mentioned above, possesses many properties which give it an 
uncommon importance in analysis, and most of all from its fur- 
nishing the connecting link in the transition from integral 
and positive to general indices of differentiation in algebraical 
functions. If we designate, as Legendre has done, 

we shall readily derive the fundamental equation 

r(l +r) = rr(r) t (1) 

which is in a form which admits of all values of r. It appears 

* Traits des Fonctions Elliptiques, torn. xi. p. 476. 
t In as much as 

t^"-" ,„ _ r (1 + n) 

-x^ = 

X'' = A a;^ 

d j;"-' r (1 + r) 


dn-r+l r (1 + W) 

(lx«-r + 1 * — r (r) '^ — D X —TAX , 

it follows that r A = B, and therefore also that 

"which is the equation (l) : and it is obvious that the transition from 

dn-r d»-r+ 1 

.x" to 

(which is equivalent to the simple differentiation of A .V, when A is a 
constant coefficient), will lead to the same relation between T (1 + r) and 
r(r), ivhateiJe)-he the value of »•, whether positive or negative, whole or frac- 
tional. Legendre has apparently limited this equation to positive values of r. 


212 THIRD REPORT — 183J}. 

also from this equation that if the values of the transcendent 
r (r) can be determined for all values of r which are included 

(Fonctions Elliptiques, torn. ii. p. 415,) a restriction which is obvious!)' unne- 
There are two cases in which the coefficient of .x"-'' in the equation 

d'- x" r (1 + n) 

dxr ~ r(l + n — r) 
requires to be particularly considered : the first is that in which this coeffi- 
cient becomes infinite ; the second, that in which it becomes equal to zero. 

The numerator F (1 -|- w) will be infinite when n is any negative whole 
number ; the denominator T (l +n — r) will become infinite when n — r is 
any negative whole number, and in no other case : if n be a negative whole 
number, and if r be a whole number, either positive or negative, such that n — r 

r(l -l-n) ^ . . 

is negative, then the coefficient fT/T^; — ;— Tn becomes finite, in as much as 

r (— i) (if t be a whole number) = j— ^ 1( — IV ' ^^^ ^ ^^^ disap- 

d~r 1 . „ . 
pears, therefore, by division : thus all the coefficients of , _^ • — are infinite, 

unless r be a negative whole number, such as — m, in which case it becomes 
1 . 2 . . TO . ( — l)*", a result which is easily verified. In a similcir manner it 

d'' 1 
would appear that the coefficients of , _y • — are infinite, when w is a posi- 
tive number, unless r be a negative whole number equal to, or greater than, n. 
The coefficient „ , , — — ^ will become equal to zero, when I + n — r 

is, and when 1 -|- n is not, equal to zero or to any negative whole number ; for, 
under such circumstances, the denominator is infinite and the numerator is 

As the most important consequences will be found to result from these 
critical values of the coefficient of differentiation, we shall proceed to examine 
them somewhat in detail. 

(1.) The simple differentials or differential coefficients of constant quantities 
are equal to zero, whilst the differentials or differential coefficients to general 
indices (positive whole numbers being excepted,) are variable. 

da d . a x' a T (l) . , a d~^a 

dx^ dx^ I'Ci) " V-^x' ^^-i 

a (r 1) , , 2 a/j: d-ifl r(l) 

= "rnr • "- * = ^v ■■ d^i = F(2j • «*"■"' = «^' 

and similarly in other cases. 

(2.) The differentials of zero to general indices (positive whole numbers 
being excepted) are not necessarily equal to zero. 

Thus, if we suppose 

C „ dro C r (1 - «) ,._„_^ . 

a = = ^prTTvi x-", we get 

r (0) ' ^ - dx^ — r (0) T(l — n — r) 
if n be a positive whole number, F (1 — ?^) = oo , and this expression is finite un- 
less r (1 — n — »■) = oo, in which case it is zero : if r be also a positive whole 

= Ca; + C 


between any two successive whole numbers, they can be der 
termined for all other values of r. Euler * first assigned the 

number, it is always zero : if r = — 1, it is finite when m = 1 : if r := — 2, 
it is finite when >j = 1 or « = 2 : if r = — 3, it is finite when « = 1, or w = 2, 
or M = 3 : and generally if r be any negative whole number, there will be 
finite values corresponding to every value of n from 1 to — r ; wo thus get 

rf-'O _ n 



3 5 = r-S + Ci a; + C2 

d x-^ 1 . 2 

d-« _ C A-"-i , Ci . ^"-2 

d^»- 1.2. .(».-]) + 1.2..(«-2) +U-2.* + U-i. 
This is the true theory of the introduction of complementary arbitrary func- 
tions in the ordinary processes of integration. 
More generally, if r be not a whole number, 

d^O _ _C_ Y{\-n ) ^_„_^ 

dx''- r (0) ■ r(i — M — ?•) 

which will be finite when n is a positive whole number and when 1 — m — r is 
not a negative whole number : thus if w be any number in the series 1, 2, 3 , . ., 
and iir=k, then 



. X T 

r (-i) 











so on for ever 

: consequently. 

tFO _ 

^ + ^ + 



-f &c. 

in hi) 




d x^ x^ x^ a; - 

In a similar manner, we shall find 

.fi r\ 3 1 . C2 . C3 .... 

? 2 = C a;^ + Ci a^ H r "I T + &c. tw tvjimtum, 

a a; - 
The knowledge of these complementary arbitrary functions will be found of 
great importance for the purpose of explaining some results of the general 
differentiation of the same function under different forms which would other- 
wise be irreconcileable with each other. 

(3.) The differential coefficient will be zero, when n is 7iot, and whenn — r 
is, a negative whole number. 


d^x „ d3«2_ d^ x^ _ dJ x~^ _^ d~^ ^~ ^-(\. 
= 0, — — 0, — 0, u, — _ u . 

d x^ dx^ 

d x^ d x^ dx 

_ 5 


and similarly in other cases. 

(4.) The differentials of 00 are not necessarily equal to », but may he finite. 
If we represent 00 by C F (0), we shall find 

Commetitarii Petrop., vol. v. 1731. 

214 THIRD REPORT — 1833. ' 

value of r ( "H" ) = V'"', by the aid of the very remarkable ex- 
pression for TT, which Wallis derived from his theory of inter- 

^'(^)^° = Cr(0) . r(l).iZ!::=(_l)r_l.l.2..(r-l).^-r, 
d X'' r (1 — r) 

whenever / is a positive whole number. 

Conversely also, 

rf-i . a;-i r (0) T, r (0) 

= — -^ . x", where — ■^—' = oo . 

dx-^ T{1) r(l) 

d-^ . x -i _ r(o) 

d a;-2 r (2) 

d-^ . a;-i _ r (0) 

d a;-3 r (3) 

d-T . a-' _ r(0) 


d x-r V {r) 

the arbitrary complementary functions being omitted. 

(5.) The occurrence of infinite values of the coefHcient of differentiation will 
generalljf be the indication of some essential change of form in the transition 
from the primitive function to its corresponding differential coefficients. 


<f-i 1 r (0) „ , , ^ 

. - = — ^-1 . aO = log X + C ; 

dx-i X r(l) o -r . 

this last result or value of -~— :■ . a:" being obtained by the ordinary process 

of integration : and generally, 

^'_^^-^ = LM xr-i + c^'-'' + Ci ^'-' + &c 
dx-r r (r) re*) r(7-— 1) ' 

the first term of which is infinite, in all cases in which r is not a negative 
whole number, in which case it becomes equal to ( — 1)"'' 1.2... ( — r) x''-^, 
the complementary arbitrary functions also disappearing. If we suppose, 

however, r to be a positive whole number, and if we replace frjvs • "^ ^Y 
its transcendental value already determined, we shall get 

- , = Hog « + C !■ + „ + &c., 

dx-r r (r) '■ " ^ -I r (;• — 1) 

which may be replaced by 

^' =m {'»«'+ (-')'rW.ra-0 + c} +-£l^,',+ . . .. 

which is in a form which is true for all values of r whatsoever, and which 
coincides, for integral values of r, with the form determined by the ordinary 
process of integration. 

More generally, 
d-r . x-» _ r (1 — r) ^~„+r ; C ■ A"-" , Ci ..r'-"- ' 
.dx-r r(l—n + r)'^ r(r — n + l)T{r — n) '^ ' ' " 

which is finite, whilst r is less than n ; and when r and n ai'e whole numbers, 

p ffi f\ 

becomes = (— 1)'' . — i ^ '' a''-", omitting complementary functions. 
1 ()i) 


polations ; and subsequently, by a much more direct process, 
which lead to the equation, 

r (r) r (1 r) = -r^ — (when r > < 1) : 
^ < ^ sin r ir ^ 

If, under the same circumstances, r be greater than n, the coefficient of dif- 
ferentiation becomes infinite, and its value, determined as above, becomes 
x'—» ,, , Ci a*"-"-' 

= roOVir-n + l) {'°S*' + ^} + r (r - u) + ^'=- 

^iWrcT^TTTT) ^^"° " + ^^ '^' "" ^'•-«+i) r («-r) + c} 

(r — n) 
which is in a form adapted to all values of r. 

The cases which we have considered above are the only ones in which the 
coefficient of differentiation will become ivfinite, in consequence of the intro- 
duction of log « in the expression of its value. We shall have occasion here- 
after to notice more particularly the meaning of infinite values of coefficients 
as indications of a change in the constitution of the function into which they 
are multiplied. 

(6.) Uti = (a X -I- i)«, then 

dru r(l-J-w)«'' . .V Ci"--! 

mrr = Y\\--\-i^-V) • («* + ^y-" + rT=7) + • • • • 

For if V = a X -j- b, then -j— = a and -7— j = ; and therefore 
d^ u T (\ ->t- 7i) /dv\- , C«'-> , „ 

Thus if M = (a; -|- \y, we get 

rf«^ ^ ^^' V (- i) x" r (- 4) a;^ 

8 .„, ,^4 , C , C, , C2 

(x+ir + ^ + ^ + ^ -I-&C. 

*^ X- x' X- 

If we replace (a; -f 1)2 by a'2 -)- 2 -|-a; 1, we shall get 

^ _ L(^) ^.^ , 2 £11) J 4. Eil) _L 
d„4 - r a) '^ + 2 J, ^^^ * + r (i) • x4 

C Ci 

r (— \) x^ r (— 4-) «* 

It thus appears that the two results may be made to coincide with each 

other, when (ar -)- 1)t jn the fi.rst of them is developed, by the aid of the proper 
arbitrary functions. 

The necessity of this introduction of arbitrary functions to restore the re- 
quired identity of the expressions deduced for the same differential coefficients, 
presents itself also in the ordinary processes of the integral calculus : thus, 
if u = (a; -I- 1)2, we find 

216 THIRD REPORT — 1833. 

Legendre, following closely in the footsteps of this illustrious 
analyst, has succeeded in the investigation of methods by which 
the values of this transcendent F (r) may be calculated to any 
required degree of accuracy for all positive values of r, and has 

a"* x^ X" a-' 1 „ „ 

If we replace (« + 1)2 by a;2 + 2 a; + 1, we find 

d-^u X* '-^^ «:- r. . r. 

I^=r2 + T + T+C^+Ci. 

d-^ u 
It is obvious that these two values of -, 5 cannot be made identical, 

without the aid of the proper arbitrary functions. 

dr u 
(7.) Let u = f" where v =.f{x) : and let it be required to find TTr' 

d'' u 
llie general expression for -j-^ , when r is a whole number, is generally 

extremely complicated, though the law of formation of its terms can always 
be assigned. If the inexplicable expressions in the resulting series be re- 
placed by their proper transcendents, the expression may be generalized for 
any value of r. 

d V , .^d^ V , dr u 

li -J— =:p and if -j—^ = c, a constant quantity, then ~r^'=- n (n — 1) 

.... {n — r -\- I) v"-'' f 

c r (r — I) £_^ _i_ r (r— 1) (r — 2) (?• — 3) c^ v^ 7 

V + 1 (ra — r + 1) 'y^l.2.Qi — r+l) {n— r + 2)"^'^ ^H 

r (1 +w) „ r r (r — 1) cc , „ ) 

+ T(^ - --' + r(-;-i) --^ + ^- 

which is in a form adapted to any value of r. 

Tf _ ' 1 ■ 



X- x^ 

Rational functions of x may be resolved into a series of fractions, whose 
denominators are of the form {x -f- a)", and whose numerators are constant 
quantities, whose rth differential coefficients may be found by the methods 
given above. Irrational functions must be treated by general methods similar 
to that followed in the example just given, which will be more or less com- 
plicated according to the greater or less number of successive simple differ- 
entials of the function beneath the radical sign, which are not equal to zero. 


- (^- /o 

3 . 5 



!) l 2* a;2 

2^* x^ 

+ ^ + ^.; 

+ &c. 


given tables of its logarithmic values to twelve places of decimals, 
with colmiins of three orders of differences for 1000 equal in- 
tervals between 1 and 2 * ; and similar tables have been given 
by Bessel and by others. We may therefore consider ourselves 
to be in possession of its numerical values under all circum- 
stances, though we should not be justified in concluding from 
thence that their explicit general symbolical forms are either 
discoverable or that they are of such a nature as to be ex- 
pressible by the existing language and signs of algebra. 
The equation 

r (/•) = (r - 1) (r - 2) .... (r - 711) F (r - m), 

or r {r — m) = -, ^r-, \^ -, ., 

^ ' (r — 1) (r — 2) . . (r — m) 

where m is a whole number, will explain the mode of passing- 
from the fundamental transcendents, when included between 
r = and I, or between r = 1 and 2, to all the other derived 
transcendents of their respective classes f. The most simple of 
such classes of transcendents, are those which correspond to 

(^) = s/-, 

which alone require for their determination the aid of no higher 
transcendents than circular arcs and logarithms. In all cases, 
also, if we consider F (r) as expressing the arithmetical yaiwe of 
the corresponding transcendent, its general form would require 
the introduction of the factor V, considered as the recipient 
of the multiple signs of aftection which are proper for each dif- 
ferential coefficient, if we use that term in its most general 

In the note, p. 211, we have noticed the principal properties 
of these fractional and general differential coefficients, partly 
for the purpose of establishing upon general principles the 
basis of a new and very interesting branch of analysis %, and 

* Fonctions Elliptiques, torn. ii. p. 490. 

tTh„s,r(-L) = V». r(i-) = i. V^,r(±) = IJ Vx. 
^/ Tc, &c. 

X The consideration of fractional and general indices of differentiation was 
first suggested by Leibnitz, in many passages of his Commercium EpistoKcum 
with John Bernouilli, and elsewhere ; but the first definite notice of their 
theory was given by Euler in the Petersburcjh Commentaries for 1731 : they 
have also been considered by Laplace and other writers, and particularly by 
Fourier, ill his great work. La Tkcorie de la Propagation de la Cliakur. The 
last of these illustrious authors has considered the general dillcrential coeffi- 

218 THIRD REPORT 1833. 

partly for the purpose of illustrating the principle of the per- 
manence of equivalent forms in one of the most remarkable 
examples of its application. The investigations which we have 
given have been confined to the case of algebraical functions, 

cients of algebraical functions, through the medium of their conversion into 
transcendental functions by means of the very remarkable formula, 
2 /*+ 00 /»+ CO 

<?•«= — ^ <p(«)d« / (p{a.)dqcosq{x — oi.), 

- 03 - CO 

which immediately gives us, 

6r (px _ 2 /*+ CO />+ CD (?»• 

~d^-lrj <^{«-)dciJ <P{cc)dq-^^r COS q(x-o,); 

- CD - 03 

which can be determined, therefore, if ^ cos q (,x — a.) can be determined, 

and the requisite definite integrations effected. If, indeed, we grant the prac- 
ticability of such a conversion of (p («) in all cases, and if we suppose the 
difficulties attending the consideration of the resulting series, which arise 
from the peculiar signs, whether of discontinuity or otherwise, which they 
may implicitly involve, to be removed, then we shall experience no embarrass- 
ment or difficulty whatever in the transition from integral to general indices 
of differentiation. 

In the thirteenth volume of the Journal de I'Ecole Polytechnique for 1832, 
there are three memoirs by M. Joseph Liouville, all relating to general in- 
dices of differentiation, and one of them expressly devoted to the discussion 
of their algebraical theory. The author defines the differential coefficient of 
the order f<, of the exponential funetion e"""^ to be mf^ e'"'^, and consequently 
the ^th differential coefficient of a series of such functions denoted by 2 A^ e*" * 
must be represented by 2 A^ rtT e""'. If it be granted that we can properly 
define a general differential coefficient, antecedently to the exposition of any 
general principles upon which its existence depends, then such a definition 
ought to coincide with the necessary conclusions deduced by those principles 
in their ordinary applications : but the question will at once present itself, 
whether such a definition is dependent or not upon the definition of the simple 
differential coefficient in this and in all other cases. In the first case it will be 
a proposition, and not a definition, merely requiring the aid of the principle 
of the permanence of equivalent forms for the purpose of giving at least an 

hypothetical existence to ^ ^^ for general, as well as for integral values 

of ft.. M. Liouville then supposes that all rational functions of x are ex- 
pressible by means of series of exponentials, and that they are consequently 
reducible to the form 2 A^ e*"^, and are thus brought under the operation of 
his definition. Thus, if x be positive, we have, 

— = / e-"*^' 
X .J 


and therefore. 

d^\ r^ 


and have been chiefly du'ected to meet the difficulties connected 
with the estimation of the values of the coefficient of differen- 
tiation in the case of fractional and general indices. If we 
should extend those investigations to certain classes of tran- 

Tvhich is easily reducible to the form, 

,.1 +M 

^1 (-irr(i + ^) 

an expression which we have analysed in the note on p. 211. This part of 
M. Liouville's theory is evidently more or less included in M. Fourier's views, 
which we have noticed above. The difficulties which attend the complete 

/ -1 \ M -r* ^1 I \ 

developement of the formula — for all values of u, which the 

principle of equivalent forms alone can reconcile, will best show how little 

progress has been made when the ^ttth differential coefficient of — is reduced 
to such a form. ^ 

M.Liouville adopts an opinion, which has been unfortunately sanctioned by 
the authority of the great names of Poisson and Cauchy, that diverging series 
should be banished altogether from analysis, as generally leading to false 
results ; and he is consequently compelled to modify his formulas with refer- 
ence to those values of the symbols involved, upon which the divergencj' or 
convergency of the series resulting from his operations depend. In one sense, 
as we shall hereafter endeavour to show, such a practice may be justified ; but 
if we adopt the principle of the permanence of equivalent forms, we may 
safely conclude that the limitations of the formulae will be sufficiently ex- 
pressed by means of those critical values which will at once suggest and re- 
quire examination. The extreme multiplication of cases, which so remark- 
ably characterizes M. Liouville's researches, and many of the errors which he 
has committed, may be principally attributed to his neglect of this important 

It is easily shown, if /3 be an indefinitely small quantity, that 

p/Sx p— /3r pm/Ji _ „— n/3i 

iy — e e ^j. e e 

2/3 (m + w) /3 

and that consequently any integral function A -j- A x -)- . . A„ xP, involving 
integral and positive powers of x only, may be expressed by 2 A^ t"*^, where 
m is indefinitely small ; and conversely, also, 2 A^j^ e"''' may, under the same 
circumstances, be always expressed by a similar integral function oix. M. Liou- 
ville, by assuming a particular form, 

^^^ 2/3 ' 

where C is arbitrary, and /3 indefinitely small, to represent zero, and differen- 
tiating, according to his definition, gets 

_ dx^ '^ 2V/3 2 ' 

but it is evident that by altering the form of this expression for zero we might 

show that was equal cither to sera or to ivjinity ; and that in the latter 

220 THIRD REPORT — 1833. 

scendental functions, such as e"**, sin m x, and cos m x, we 
shall encounter no such difficulties, in as much as the differen- 
tials of those functions corresponding to indices which are ge- 
neral in form, though denoting integral numbers, are in a form 

case the critical value infinity might be merely the indication of the existence 

of negative or fractional powers of a; in the expression for , which were 

not expressible by any rational function of e^^ under a finite form and in- 
volving indefinitely small indices only. And such, in fact, would be the re- 
sult of any attempt to differentiate this exponential expression for x or its 
powers, with respect to fractional or negative indices. It has resulted from 
this very rash generalization of M. Liouville that he has assigned as the ge- 
neral form of complementary arbitrary functions, 

C + Ci a; + C2 a:2 + Cs x^ + &c., 
which is only true when the index of differentiation is a negative whole 

Most of the rules which M. Liouville has given for the differentiation of 
algebraical functions are erroneous, partly in consequence of his fundamental 
error in the theory of complementary arbitrary functions, and partly in 
consequence of his imperfect knowledge of the constitution of the formula 

r ( 1 + >»; . |.jjyg ^fjgy deducing the formula 
T{l+n — r) 

d'' • 7 VtT^ {-ly.a'-. r (n + r) 

("'" + ^) = 1 .2...(«-l) (ax + b)n + r' 

which is only true when n is a whole number, he says that no difficulty pre- 
sents itself in its treatment, whilst n + r is > 0, but that T {n + r) be- 
comes infinite, when »i + r < 0, in which case he says that it must be 
transformed into an expression containing finite quantities only, by the aid 
of complementary functions ; whilst, in reality, T {n + r) is only infinite 
when n -\- r is zero or a negative whole number, and the forms of the com- 
plementary functions, such as he has assigned to them, are not competent to 
effect the conversion required. In consequence of this and other mistakes, 

dr 1 

in connexion with the important case ' {ax -\- &)" , nearly all his conclu- 

sions with respect to the general differentials of rational functions, by means 
of their resolution into partial fractions, are nearly or altogether erroneous. 

The general differential coefficients of sines and cosines follow immediately 
from those of exponentials, and present few difficulties upon any view of their 
theory. In looking over, however, M. Liouville's researches upon this sub- 
ject, I observe one remarkable example of the abuse of the first principles of 

„, , „ d^ cos mx ... , 

reasoning m algebra. There are two values 01 , one positive and 


the other negative, considered apart from the sign of m, whether positive or 

negative : but if we put cos m x = — cos m x + — cos m x, we get 

' ,v li 

dr cosmx 1 «■ cos mx "i. d' cos m x _ 

dx- ^ d^^ ^ dx^ 


which is adapted to the immediate apphcation of the general 
principle in question. 

Thus, ii' u = e'" *■, we get 

d x-"^^ ' d x-" "^ "" d X" ~ "* ^ ' 

when r is a whole number, and therefore, also, when r is any 
quantity whatsoever. 

it u = sm m X, -J— = m sm {-—■+mx), -z — -„ = 7n^ sin 
dx \2 / d X- 

d^ U ( T "K \ 

(w + m x), . . . . y—y = m'' sin ( —^ + m x j when r is a whole 

number, and therefore generally. In a similar manner if 

u = cos m X, or rather ti — cos m {Vf x, (introducing P as 
a factor in order to express the double sign of m x, if de- 
termined from the value of its cosine,) then we shall find 

fjh' -ft \ T TT I 

- — = (m a/ ly COS < —^ + {m \^ I) x >, whatever be the 
value of r. If u = e" * cos m x, we get, by very obvious re- 
ductions, making o = . - and fl = cos ' — , 

-, — = p'' e"'^ cos (m X + n fl). 
dx^ '^ ^ ' 

It is not necessary to mention the process to be followed in ob- 

and if we combine arbitrarily the double values of the two parts of the second 
member of this equation, we shall get four values of — ^°^ *" ^, instead of 
two ; and, in a similar manner, if we should resolve cos m x into any number 

of parts, we should get double the number of values of cos m x ^ j^ ^j^.^ 

principle of arbitrary combinations of algebraical values derived from a com- 

mon operation was admitted, we must consider -^ — as having two values, 

and its equivalent series 

x^ + x^-\-x- + &c. 
as having an infinite number. But it is quite obvious that those expressions 
which involve implicitly or explicitly a multiple sign must continue to be 
estimated with respect to the same value of this sign, however often the reci- 
pient of the multiple sign may be repeated in any derived series or expression. 
The case is difteretit in those cases where the several terms exist indepen- 
dently of any explicit or implied process of derivation. 

222 THIRD REPORT — 1833. 

taining the general differential coefficients of other expressions, 
such as (cos x)", cos tn x x cos n x, &c., which present no kind 
of difficulty. In all such cases the complementary arbitrary 
functions will be supplied precisely in the same manner as for 
the corresponding differential coefficients of algebraical func- 

The transition from the consideration of integral to that of 
fractional and general indices of differentiation is somewhat 
starthng when first presented to our view, in consequence of 
our losing sight altogether of the principles which have been 
employed in the derivation of differential coefficients whose in- 
dices are whole numbers : but a similar difficulty will attend 
the transition, in every case, from arithmetical to general values 
of symbols, through the medium of the principle of the perma- 
nence of equivalent forms, though habit and in some cases im- 
perfect views of its theory, may have made it familiar to the 
mind. We can form distinct conceptions of m . m, m . m . m, 
m .m .7)1 , . . . (r), where m is a whole number repeated twice, 
thrice, or r times, when r is also a whole number ; and we 
can readily pass from such expressions to their defined or as- 
sumed equivalents m^, m^, ... tm'': in a similar manner we can rea- 
dily pass from the factorials * 1.2, 1.2.3,... 1 . 2 ... r, to 
their assumed equivalents r(3), r(4), . . . r(l + r), as long as r 
is a whole number. The transition from m^ and r (1 + r) when 
r is a whole nvimber, to 7n'' and r (1 + r) when r is a general 
symbol, is made by the principle of the equivalent forms ; but by 
no effort of mind can we connect the first conclusion in each case 
with the last, without the aid of the intermediate formula, involv- 
ing symbols which are general in form though specific in value ; 
and in no instance can we interpret the ultimate form, for 
values of the symbols which are not included in the first, by 
the aid of the definitions or assumptions which are employed 
in the establishment of the primary form. In all such cases 
the interpretation of the ultimate form, when such an interpre- 
tation is discoverable, must be governed and determined by a 
reference to those general properties of it which are inde- 
pendent of the specific values of the symbols. 

» Legendre has named the function r(l + r) = 1 . 2 . . . r, the function 
gamma. Kramp, who has written largely upon its properties, gave it, in his 
Analyse des Refractions Astronomiques, the name oifaculte nimierique; but in his 
subsequent memoirs upon it in the earlier volumes of the Annales des Ma- 
thematiques of Gergonne he has adopted the name oi' factorial function, which 
Arbagost proposed, and which I think it expedient to retain, as recalling to 
mind the continued product which suggests this creature of algebraical lan- 


The law of derivation of the terms in Taylor's series, 
, d i( J cP u li^ cP u h^ , p 

'' = ^^ + ^ • ^' + ^- O + ^^ • TT^TS + ^^^ 

is the same as in the more general series 

and if we possess the law of derivation of —, — and of —, — ^, we 

(JL X (IX 

can find all the terms of both these series, whatever be the 
value of r. The first of these terms must be determined through 
the ordinary definitions of the differential calculus ; the second 
must be determined in form by the same principles, and gene- 
ralized through the medium of the principle of equivalent 
forms. Both these processes are indispensably necessary for 

d'' u . . 

the determination of -. — : but it is the second of them which 


altogether separates the interpretation of-r— ^ from that of -7—, 


or rather of -7—^ when r is a whole number, unless in the par- 
ct oc 

ticular cases in which the symbols in both are identical in 

There are two distinct processes in algebra, the direct and 
the inverse, presenting generally very different degrees of dif- 
ficulty. In the first case, we proceed from defined operations, 
and by various processes of demonstrative reasoning we arrive 
at results which are general in form though particular in value, 
and which are subsequently generalized in value likewise : in 
the second, we commence from the general result, and we are 
either required to discover from its form and composition some 
equivalent result, or, if defined operations have produced it, to 
discover the primitive quantity froni which those operations 
have commenced. Of all these processes we have already given 
examples, and nearly the whole business of analysis will consist 
in their discussion and developement, under the infinitely varied 
forms in which they will present themselves. 

The disappearance of undulating and of determinate func- 
tions with arbitrary constants, upon the introduction of inte- 
gral or other specific values of certain symbols involved, is one 
•of the chief sources * of error in effecting transitions to equiva- 

* The theory of discontinuous functions and of the signs of discontinuity 
will show many others. 

224 THIRD REPORT — 1833. 

lent forms, whethe, the process followed be direct or inverse. 
Many examples of the first kmd may be found in the researches 
of Poinsot respecting certain trigonometrical series, which 
will be noticed hereafter, and wliich had been hastily gene- 
ralized by Euler and Lagrange ; and a remarkable example of 
the latter has already been pointed out, in the disappearance 
of the functions with arbitrary constants in the transition from 

u to ■^—^, when r becomes a whole positive number. The gene- 

ral discussion of such cases, however, would lead me to an 
examination of the theory of the introduction of determinate 
and arbitrary functions in the most difficult processes of the 
integral calculus and of the calculus of functions, which would 
carry me far beyond the proper limits and object of this Re- 
port. I have merely thought it necessary to notice them in 
this place for the purpose of showing the extreme caution 
which must be used in the generalization of equivalent results 
by means of the application of the principle of the permanence 
of equivalent forms*. 

The preceding view of the principles of algebra would not 
only make the use and form of derivative signs, of whatever 
kind they may be, to be the necessary results of the same ge- 
neral principle, but would also show that the interpretation of 
their meaning would not precede but follow the examination of 
the circumstances attending their introduction. I consider it 
to be extremely important to attend to this order of succession 
between results and their interpretation, when those results 
belono- to symbolical and not to arithmetical algebra, in as much 
as the neglect of it has been the occasion of much of the con- 
fusion and inconsistency which prevail in the various theories 
which have been given of algebraical signs. I speak of deri- 

* Euler, in the Peiersburgh Acts for 1774, has denied the universality of 
this principle, and has adduced as an example of its failure the very remark- 
able series 

1 _ am (1 _ a»») (1 — a™-i) (1 — C") (1 — a*"-!) (1 — C'-Z) 
T^^ + 1^^2 + 1 - «' "^ ' 

which is equal to m, when m is a whole number, but which is apparently not 
equal to m, for other values of m, unless at the same time a = 1 : the occur- 
rence, however, of zero as a factor of the (»n + 1)* and following terms in 

the first case, and the reduction of every term to the fonn -^ in the second, 

would form the proper indications of a change in the constitution of the equi- 
valent function corresponding to these values of w and a, of which many ex- 
amples will be given in the text. 


vative signs as distinguished, from those p«^itive signs of ope- 
ration which are used in arithmetical algebra ; but such signs, 
though accurately defined and hmited in their use in one sci- 
ence, will cease to be so in the other, their meaning being de- 
pendent in symbolical algebra, in common with all other signs 
which are used in it, upon the symbolical conditions which they 
are required to satisfy. 

I will consider, in the first place, signs of affection, which are 
those symbolical quantities which do not affect the magnitudes, 
though they do affect the specific nature, of the quantities into 
which they are incorporated. 

Of this kind are the signs + and — , when used independ- 
ently ; or their equivalents + 1 and — 1, when considered as 
symbolical factors ; the signs (-}- 1)" and (— 1)", or their sym- 
bolical equivalents 

cos 2rmT-\- \/ — \ sin 2 r w tt and cos {2r -\- \) n w + 

-v/ — 1 sin (2 r + 1 ) w T ; 

2r?nr-v/~l 1 (2j-+ 1) n!r\/— 1 

or e and e^ 

The affections symbolized by the signs + 1 and — 1 admit 
of very general interpretation consistently with the symbolical 
conditions which they are required to satisfy,- and particularly 
so in geometry : and it has been usual, in consequence of 
the great facility of such interpretations, to consider all quan- 
tities aftected by them (which are not abstract) as possible, 
that is, as quantities possessing in all cases relations of exist- 
ence which are expressible by those signs. It should be kept 
in mind, however, that such interpretations are in no respect 
distinguished from those of other algebraical signs, except in 
the extent and clearness with which their conditions are sym- 
bolized in the nature of things. 

The other signs of affection, different from + 1 and — 1, 
which ai'e included in (1)" and ( — 1)", are expressible generally 
by cos fl + v^ — 1 sin fl, or by « + /3 V^ — 1 ,where a and |3 may have 
any values between 1 and — 1 , zero included, and where «^ -f- /3^ 
= 1. To all quantities, whether abstract or concrete, expressed 
by symbols affected by such signs, the common tei'm impossible 
has been applied, in contradistinction to those possible magni- 
tudes which are affected by the signs -|- and — only. 

If, indeed, the affections symbolized by the signs included 
under the form cos 9 -|- -/ — 1 sin fl, admitted in no case of an in- 
terpretation which was consistent with their symbolical condi- 
tions, then the term impossible would be correctly npplied to 
quantities affected by them : but in as much as the signs + and 

1 833. Q 

226 THIRD REPORT — 1833. 

— , when used indepefndently, and the sign cos 6 + V — I sm d, 
when taken in its most enlarged sense, equally/ originate in the 
generalization of the operations of algebra, and are equally in- 
dependent of any previous definitions of the meaning and extent 
of such operations, they are also equally the object of inter- 
pretation, and are in this respect no otherwise distinguished 
from each other than by the greater or less facility with which 
it can be applied to them. 

Many examples * of their consistent interpretation may be 
pointed out in geometry as well as in other sciences : thus, if 
+ a and — a denote two equal lines whose directions are op- 
posite to each other, then (cos 9 + v^ — 1 sin fl) a may denote 
an equal line, making an angle 9 with the line denoted by -^ a ; 
and consequently a ^ — \ will denote a line which is perpen- 
dicular to -1- a. This interpretation admits of very extensive 
application, and is the foimdation of many important conse- 
quences in the application of algebra to geometry. 

The signs of operation 4- and — may be immediately inter- 
preted by the terms addition and subtraction, when applied to 
unaffected symbols denoting magnitudes of the same kind : if 
they are applied to symbols affected with the sign — , these 
signs, and the terms used to interpret them, become convertible. 
Thus a + {— h) — a — b, and a — {— b) = a + b; or the al- 
gebraical sum and difference of a and — b, is equivalent to the 
algebraical difference and sum of a and b : but if they are applied 
to lines denoted by symbols affected by the signs cos 9 + V —I. 
sin 6, and cos 9' + -/ — 1 sin 9', the results will no longer de- 
note the arithmetical (or geometrical) su?n and difference of the 
lines in question, but the magnitude and position of the dia- 
gonals of the parallelogram constructed upon them, or upon 
lines which are equal and parallel to 
them. Thus, if we denote the line 
A B by a, and the h ne A C at right 
angles to it by 6 V — 1 , and if we 
complete the parallelograms AB D C 
and AB C E, then a + b \/ - i will 
denote the diagonal A D, and a — b \/ — I will denote the 
other diagonal B C, or the equal and parallel line A E. 

It is easily shown that a + b \/ ^^ = V'Ca^ + *^ (cos 9 

cos" ^ a 

+ »/ —\ sin 9), (where 9 = — r ^ , J , and a — b i/ — I 

a: v'(«' + b^) {cos 9 — a/^^I sin 9} ; it follows, therefore, that 
* Peacock's Algebra, chap. xii. Art. 437, 447, 448, 449. 


a + b V — I and a — b ^Z — I may be considered as repre- 
senting respectively a single line, equal in magnitude to 

\/{a^ + b^) *, and aiFected by the sign cos 9 + -/ — 1 sin 9 in 
one case, and by the sign cos 9 — •i/ — 1 sin Hn the other ; or 
as denoting the same lines through the medium of the opera- 
tions denoted in the one case by +, and in the other by — , 
upon the two lines at right angles to each other, which are de- 
noted by a and b V —\. 

We have spoken of the signs of operation + and — , as di- 
stinguished from the same signs when used as signs of affection, 
and we have also denominated a -\- b \Z — 1, and a — b V' — 1, 
the sum and difference of a and b V —\, though they can no 
longer be considered to be so in the arithmetical or geometrical 
sense of those terms ; but it is convenient to explain the mean- 
ing of the same sign by the same term, though they may be 
used in a sense which is not only very remote from, but even 
totally opposed to f , their primitive signification ; and such a 
licence in the use both of signs and of phrases is a necessary 
consequence of making their interpretation dependent, not upon 
previous and rigorous definitions as is the case in arithmetical 
algebra, but upon a combined consideration of their symbolical 
conditions, and the specific nature of the quantities represented 
by the symbols. It is this necessity of considering all the re- 
sults of symbolical algebra as admitting of interpretation sub- 
sequently to their formation, and not in consequence of any 
previous definitions, which places all those results in the same 
relation to the whole, as being equally the creations of the 
same general principle : and it is this circumstance which jus- 

• The arithmetical quantity ^{a- -)- b^) has been called the modulus of 
a + 6<^ — 1 by Cauchy, in his Cours d' Analyse, and elsewhere. It is the single 
unaffected magnitude which is included in the affected magnitude a + 6 V — 1 : 
conversely the affected magnitude (cos 6 + V— 1 sin d) ^Ja^ + b"- is reducible 

to the equivalent quantity a -f- 6 a/— 1, if cos &=. „ , and therefore 

sin ^ = 

t The sum of a and — b, or a -\- (— b], is identical with the difference of a 
and b, or with a — b. The term operation, also, which is applied generally to 
the fact of the transition from the component members of an expression to the 
final symbolical result, will only admit of interpretation when the nature of the 
process which it designates can be described and conceived. In all other cases 
we must regard the final result alone. Thus, if a and b denote lines, we can 
readily conceive the process by which we form the results a + 6 and a — 6, at 
least when a is greater than b. But when we interpret a + 6 \/— 1 to mean a 
determinate single line with a determinate position, we are incapable of con- 
ceiving any process or operation through the medium of which it is obtained. 


2^g THIRD REPORT— 1835. 

tifies the assertion, which we have made above, that quantities 
or their symbols affected by the signs +, — , or cos 9 + \^ — \. 
sin 6, are only distinguished from each other by the greater or 

less facility of their interpretation. 

The geometrical interpretation of the sign V — 1, when 
applied to symbols denoting lines, though more than once 
suggested by other authors, was first formally maintained by 
M. Buee in a paper in the Philosojihical Transactions for 1806*, 
which contains many original, though very imperfectly deve- 
loped views upon the meaning and application of algebraical 
signs. In the course of the same year a small pamphlet was pub- 
lished at Paris by M. Argand, entitled Essai sur une Maniere 
de repr^senter les Quanfites Imaginaires, dans les Construc- 
tions G^om^triques, written apparently without any knowledge 
of M. Buee's paper. In this memoir M. Argand arrives at this 
proposition. That the algebraical sum f of two lines |, estimated 
both according to magnitude and dii'ection, would be the dia- 
gonal of the parallelogram which might be constructed upon 
them, considei'ed both with respect to direction and magnitude, 
which is, in fact, the capital conclusion of this theory. This 
memoir of M. Argand seems, however, to have excited very 
little attention ; and his views, which were chiefly founded upon 
analogy, were too little connected with, or rather dependent 
upon, the great fundamental principles of algebra, to entitle 
his conclusions to be received at once into the great class 
of admitted or demonstrated truths. It would appear that 
M. Argand had consulted Legendre upon the subject of his me- 
moir, and that a favourable mention of its contents was made 
by that great analyst in a letter which he wrote to the brother 
of M. J. F. Fran9ais, a mathematician of no inconsiderable 
eminence. It was the inspection of this letter, upon the death 
of his brother, which induced M. Franpais to consider this 
subject, and he published, in the fourth volume of Gergonne's 
Annales des Mathematiqties for 1813, a very curious memoir 
upon it, containing views more extensive, and more completely 
developed than those of M. Argand, though generally agreeing 
with them in their character, and in the conclusions deduced 
from them. This publication led to a second memoir upon the 
same theory from M. Argand, and to several observations 
upon it, in the same Journal, from MM. Servois, Franpais, 
and Gergonne, in which some of the most prominent objections 
to it were proposed, and partly, though very imperfectly, an- 

* This paper was read in 1805. f -ta somme dirigee. 

I Lignes dirigies. 


swered. No further notice appears to have been taken of these 
researches before the year 1828, when Mr. Warren's treatise on 
the geometrical representation of the square roots of negative 
quantities * w as published. In this work Mr. Warren proposes 
to give a geometi'ical representation to every species of quan- 
tity ; and after premising definitions of addition, subtraction, 
multiplication and division, involution and evolution, which are 
conformable to the more enlarged sense which interpretation 
would assign to those operations when applied to lines repre- 
sented in position as well as in magnitude ; and after showing 
in great detail the coincidence of the symbolical results obtained 
from such definitions with the ordinary results of arithmetical 
and symbolical algebra, he proceeds to determine the meaning 
of the different symbolical roots of 1 and — I, when applied 
to symbols denoting lines, under almost every possible circum- 
stance. The course which Mr. Warren has followed leads 
almost necessarily to very embarrassing details, and perhaps, 
also, to the neglect of such comprehensive propositions as can 
only derive their authority from principles which make all the 
results of algebra which are general in form independent of the 
specific values and representation of the symbols : but at the 
same time it must be allowed that his conclusions, when viewed 
in connexion with his definitions, were demonstrably true ; a 
character which could not be given to similar conclusions when 
they were attempted to be derived by the mere aid of the arith- 
metical definitions of the fundamental operations of algebra. 

This objection to the course pursued by Mr. Warren will 
more or less apply to all attempts which are made to make the 
previous interpretations of algebra govern the symbolical con- 
clusions ; for though it is always possible to assign a meaning 
to algebraical operations, and to pursue the consequences of 
that meaning to their necessary conclusions, yet if the laws of 
combination which lead to such conclusions are expressed 
through the medium of general signs and symbols, they will 
cease, when once formed, to convey the necessary limitations 
of meaning which the definitions impose upon them. It is for 
this reason that we must in all cases consider the laws of com- 
bination of general symbols as being arbitrary and independent 
in whatever manner suggested, and that we must make our in- 
terpretations of the results obtained conformable to those laws, 
and not the laws to the interpretations : it is for the same reason, 
likewise, that our interpretations will not be necessary, though 

• A Treatise on the Geometrical Representation of the Square Moots of Ne- 
tfntive Quantities, by the Rev. John Warren, M.A., Fellow and Tutor of Jesus 
College Cambridge. J 828. 

230 THIRD REPORT — 1833. 

governed by necessary laws, except so far as those interpreta- 
tions are dependent upon each other. Thus, if a be taken to 
represent a line in ynagnitude , it is not necessary that (cos 3 
+ -v/ — 1 sin 6) « should represent a line equal in length to the one 
represented by o, and also making , an angle 5 with the line re- 
presented by a ; but if (cos 3 + v^ — 1 sin 6) a, may, consistently 
with the symbolical conditions, represent such a line, without 
any restriction in the value of 9, then, if it does represent such 
a line for one value of 9, it must represent such a line for every 
value of 9 included in the formula. It is only in such a sense 
that interpretations can be said in any case to have a necessary 
and inevitable existence. 

It is this confusion of necessary and contingent truth which 
has occasioned much of the difficulty which has attended the 
theories of the interpretation of algebraical signs. It has been 
sui:>posed that a meaning could be transmitted through a suc- 
cession of merely symbolical operations, and that there would 
exist at the conclusion an eqvially necessary connexion between 
the primitive definition and the ultimate interpretation, as be- 
tween the final symbolical result and the laws which govern it. 
So long as the definitions both of the meaning of the symbols 
and of the operations to which they are required to be subject 
are sufficient to deduce the results, those results will have a 
necessary interpretation which will be dependent upon a joint 
consideration of all those conditions ; but whenever an operation 
is required to be performed under circumstances which do not 
allow it to be strictly defined or interpreted, the chain of con- 
nexion is broken, and the interpretation of the result will be 
no longer traceable through its successive steps. This must 
take place whenever negative or other affected quantities are 
introduced, and whenever operations are to be performed, 
either with them, or upon them, even though such quantities 
and signs should altogether disappear from the final result. 

This principle of interpretation being once established, we 
must equally consider — I, \/ — \, cos 9 + -v^ — 1 sin 9, as signs 
of impossibility, in those cases in which no consistent meaning 
can be assigned to the quantities which are aflPected by them, 
and in those cases only : and it must be kept in mind that the 
impossibility which may or may not be thus indicated, has re- 
ference to the interpretation only, and not to the symbolical 
result, considered as an equivalent form : for all symbolical 
results must be considered as equally possible which the signs 
and symbols of algebra, whether admitting of interpretation or 
not, are competent to express. But there will be found to be 
many species of impossibility which will present themselves in 


considering the relations of formulae with a view to their equi- 
valence, and also under other circumstances, which will be in- 
dicated by such means as Avill destroy all traces of the equiva- 
lence which would otherwise exist. 

The capacity, therefore, possessed by the signs of affection 

involving -v/ — 1 of admitting geometrical or other interpreta- 
tions under certain circumstances, though it adds greatly to 
our power of bringing geometry and other sciences under the 
dominion of algebra, does not in any respect affect the general 
theory of their introduction or of their relation to other signs : 
for, in the first place, it is not an essential or necessary pro- 
perty of such signs ; and in the second place, it in no respect 
affects the form or equivalence of symbolical results, though it 
does affect both the extent and mode of their application. It 
would be a serious mistake, therefore, to suppose that such inci- 
dental properties of quantities affected by such signs constituted 
their real essence, though such a mistake has been generally 
made by those who have proposed this theory of interpretation, 
and has been made the foundation of a charge against them by 
others, who have criticised and disputed its correctness*. 

* This charge is made by Mr. Davies Gilbert in a very ingenious paper in 
the Philosophical Transactions for 1831, " On the Nature of Negative and Im- 
possible Quantities." He says that those mathematicians take an incorrect 
view of ideal quantities, — mistaking, in fact, incidental properties for those 
which constitute their real essence, — who suppose them to be principles of 
perpendicularity, because they may in some cases indicate extension at right 
angles to the directions indicated by the correlative signs + and — ; for with 
an equal degree of propriety might the actually existing square root of a quan- 
tity be taken as the principle of obliquity, in as much as in certain cases it 
indicates the hypothenuse of a right-angled triangle. In reply to this last 
observa tion, it may be observed, that I am not aware that in any case the 
sign /\/— I has had such an interpretation given to it. 

It is quite impossible for me to give an abridged, and at the same time a fair 
view of Mr. Davies Gilbert's theory, within a compass much smaller than the 
contents of his memoir. But I might venture to say that his proof of the rule 
of signs rests upon some properties of ratios or proportions which no arith- 
metical or geometrical view of their theory would enable us to deduce. In con- 
sidering, also, imaginary quantities as creations of an arbitrary definition, en- 
dowed with properties at the pleasure of him who defines them, he ascribes to 
them the same character as to all other symbols and operations of algebra ; 

but in saying "that quantities affected by the sign /^— 1 possess a. potential 
existence only, but that they are ready to start into energy whenever that sign 
is removed," he appears to me to assert nothing more than that symbols are 

impossible or not, according as they are affected by the sign ^ — 1 or not. 
Again, in examining the relation of the terms of the equation 

n(n— I) Q , „ , » (« — 1) (« — 2) „_« „ ,.! , - 

232 THIRD REPORT— 1833. 

Signs of transition are those signs which indicate a change 
in the nature or form of a function, when considered in the 
whole course of its passage through its different states of ex- 
istence. Such signs, if they may be so designated, are gene- 
rally sero and infinity. 

Zero and infinity are negative terms, and if applied to desig- 

he denies the correctness of the reasoning by which it is inferred that the 
second term of the first, and the even terras of the second members of this 
equation are equal to oue anotlier (when x is less than 1), because they are 
the only terms whicli are homogeneous to each other, in as much as we thus 
ascribe real properties to ideal quantities ; and he endeavours to make this 
equality depend upon an assumed arbitrary relation between x and y, though 
it is obvious that if y = cos &, we shall find x = cos n 6, and that, therefore, 
this relation is determinate, and not arbitrary. A little further examination 
of this conclusion would show that it did not depend upon any assumed 
homogeneity of the parts of the members of this equation to each other, but 
upon the double sign of the radical quantity which is involved upon both 

In arithmetical algebra, where no signs of affection are employed or recog- 
nised, both negative and imaginary quantities become the limits of operations ; 
and when this science is modified by the introduction of the independent 
signs + and — and the rule for their incorporation, the occurrence of the 
square roots of negative quantities, by presenting an apparent violation of the 
rule of the signs, becomes a new limit to the application of this new form of 
the science. The same algebraists who have acquiesced in the propriety of 
making the first transition in consequence of the facility of assigning a meaning 
to negative quantities, at the same time that they retained the definitions and 
principles of the first science, were startled and embarrassed when they came 
to the second ; for it was very clear that no attempt could be made to recon- 
cile the existence and use of such quantities, consistently with the main- 
tenance of that demonstrative character in our reasonings which exists in 
geometry and arithmetic, where the mind readily comprehends the nature of 
the quantities employed, and of the operations performed upon them. The 
proper conclusion in such a case would be that the operations performed, as 
well as the quantities employed, were symbolical, and that the results, though 
they might be suggested by the primitive definitions, were not dependent 
upon them. If no real conclusions had been obtained by the aid of such 
merely symbolical quantities, they would probably have continued to be re- 
garded as algebraical monsters, whose reduction under the laws of a regular 
system was not merely unnecessary, but altogether impracticable. But it was 
soon found that many useful theories were dependent upon them ; that any 
attempt to guard against their introduction in the course of the progress of 
our operations with symbols would not merely produce the most embarrassing 
limitations, when such limitations were discoverable, but that they would 
present themselves in the expression of real quantities, and would furnish at 
the same time the only means by which such quantities could be expressed. 
A memorable example of their occurrence under such circumstances presents 
itself in what has been called the irreducible case of cubic equations. 

In the Philosophical Transactions for 1778 there is a paper by Mr. Playfair 
on the arithmetic of impossible quantities, in which the definable nature of 
algebraical operations is asserted in the most express terms, and in which 
the truth of conclusions deduced b}^ the aid of imaginary symbols is made to 
depend upon the analogy which exists between certain geometrical properties 


nate states of quantity, are equally inconceivable. We are ac- 
customed, however, to speak of quantities as infinitely great 
and infinitely small, as distinguished from finite quantities, 
whether great or small, and to represent them by the symbols 
00 and 0. It is this practice of designating such inconceivable 
states of quantity by symbols, which brings them, in some de- 

of the circle and the rectangular hyperbola. It is -well known that the circle 
and rectangular hyperbola are included in the same equation y=^^{\ — sr), 
if we suppose x to have 
any value between + °° 
and — 00 : let a circle he 
described with centre C 
and radius C A = 1, and 
upon the production of 
this radius,; let a rectan- 
gular hyperbola be de- 
scribed whose semiaxis is 
1, in a plane at right an- 
gles to that of the circle: 
iftf denote the angle AC P, 
then the circular cosine and sine (C M and P M) are expressed by 

and — = 

2 2 -/— 1 

respectively ; whilst the hyperbolic cosine and sine (to adopt the terms pro- 
posed by Lambert) corresponding to the angle 6 -v/— 1 (in a plane at right 
angles to the former) are expressed by 

-^ and V- 1 y ^ )' o"- by -^2~~ 2 ' 

if they be considered as determined by the following conditions ; namely, 
that (hyp. cosine)^ — (hyp- sine)- = 1, and that hyp. cos 6 = hyp. cos — ^, 
and hyp. sine ^ = — hyp. sine — ^. A comparison of these processes in the 
circle and hyperbola would show, says Mr. Playfair, that investigations which 
are conducted by real symbols, and therefore by real operations, in the hy- 
perbola, would present analogovs imaginary symbols, and therefore analogous 
imaginary operations in the circle, and conversely ; and that the same species 
of analogy which connects the geometrical properties of the circle and hyper- 
bola, connects the conclusions, of the same symbolical forms, when conducted 
by real and imaginary symbols. 

This attempt to convert an extremely limited into a very general analogy, 
and to make the conclusions of symbolical algebra dependent upon an insu- 
lated case of geometrical interpretation, would certainly not justify us in 
drawing any genera! conclusions from processes involving imaginary symbols, 
unless they could be confirmed by other considerations. The late Professor 
Woodhouse, who was a very acute and able scrutinizer of the logic of ana- 
lysis, has criticised this principle of Mr. Playfair with just severity, in a paper 
in the Philosophical Transactions for 1802, "On the necessary truth of certain 
conclusions obtained by means of imaginary expressions." The view which he 
has taken of algebraical equivalence, in cases where the connexion between 
the expressions which were treated as equivalent could not be shown to be 
the result of a defined operation, makes a very near approach to the principle 

234 THIRD REPORT— 1833. 

gree, under the ordinary rules of algebra, and which compels us 
to consider different orders both ofinjinities and of zeros, though 
when they are considei*ed without reference to their symbo- 
lical connexion, they are necessarily denoted by the same sim- 
ple symbols oo and : thus there is a necessary symbolical di- 
stinction between (00)2, 00 and (oo ) , and between (0)2, and 
(0) ; though when considered absolutely as denoting infinity 
in one case and zero in the other, they are equally designated 
by the simple symbols 00 and respectively. 

Though the fundamental properties of and co , considered 
as the representatives of zero and infinity, are suggested by the 
ordinary interpretation of those terms, yet their complete in- 
terpretation, like that of other signs, must be founded upon the 

of the permanence of equivalent forms : thus, supposing, when x is a real 
quantity, we can show that 

«^ = 1 + ^ + rr2 + r:V^ + ^^•' 

but that we cannot show in a similar or any other manner that 

1 . 2 1 . 2 . 3 ^^ °^*^"' 

then the equivalence in the latter case is assumed, by considering c* 
as the abridged symbol for the series of terms 

1 + ^ V- 1 -Y72- TT2T3 + ^^- ' 

in other words, the form which is proved to be true for values of the symbols 
which are general in form, though particular in value, is assumed to be true 
in all other cases. 

It is true that such a generalization could not be considered as legitimate, 
without much preparatory theory and without considerable modifications of 
our views respecting nearly all the fundamental operations and signs of arith- 
metical algebra ; but I refer with pleasure to this incidental testimony to the 
truth and universality of this important law, from an author whose careful 
and bold examination of the first principles of analytical calculation entitle 
his opinion to the greatest consideration. 

Mr. Gompertz published, in 1817 and 1818, two tracts on the Principles and 
Application of Imaginary Quantities, containing many ingenious and novel 
views both upon the correctness of the conclusions obtained by means of ima- 
ginary quantities and also upon their geometrical interpretation. The first 
of these tracts is principally devoted to the establishment of the following 
position: "That wherever the operation by imaginary expressions can be 
used, the propriety may be explained from the capability of one arbitrary 
quantity or more being introduced into the expressions which are imaginary 
previously to the said arbitrary quantity or quantities being introduced, so 
as to render them real, without altering the truth they are meant to express ; 
and that, in consequence, the operation will proceed on real quantity, the 
introduced arbitrary quantity or quantities necessary to render the first steps 
of the reasoning arguments on real quantity, vanishing at the conclusion ; 


consideration of all the circumstances under which they pre- 
sent themselves in symbolical results. In order, therefore, to 
determine some of the principles upon which those interpreta- 
tions must be made, it will be proper to examine some of the 
more remarkable of their symbolical properties. 

and from whence it will follow that the non-introduction of such can pro- 
duce nothing wrong." Thus, x^ -I- a x -\- b, which is equal to 


is also equal to 

whatever be the value of the quantity /3 ; a conclusion which enables us to 
reason upon real quantities and to make /3 = 0, when the primitive factors 

are required. Similarly, if mstead of ^ ^ y, we suppose 

■ = V — "■ . and if mstead of ^ = x. 

2 2a/-1 

we suppose = x — R, we shall find, whatever /3 may be, 

2 V/3— 1 

g*V/3-i — j,_j^f_|_ ^^ — I (^ — R)^ a result which degenerates into 

the well known theorem e '^~ ^ = y -\- V — 1 x, if /3 = 0. Many other ex- 
amples are given of this mode of porismatizing expressions, (a term derived by 
Mr.Gompertz from the definition of porisms in geometry,) by which operations 
are performed upon real quantities which would be otherwise imaginary : 
and if it was required to satisfy a scrupulous mind respecting the correctness 
of the real conclusions which are derived by the use of imaginary expressions, 
there are few methods which appear to me better calculated for this purpose 
than the adoption of this most refined and beautiful expedient. 

The second tract of Mr. Gompertz appears to have been suggested by 
M. Buee's paper in the Philosophical Transactions, to which reference has been 
made in the text : it is devoted to the algebraical representation of lines both 
in position and in magnitude, as a part of a theory of what he terms func- 
tional projections, and embraces the most important of the conclusions obtained 
by Argand and Fran9ais, with whose researches, however, he does not appear 
to have been acquainted. I should by no means consider the process of rea- 
soning which he has followed for obtaining these results to be such as would 
naturally or necessarily follow from the fundamental assumptions of algebra : 
but it would be unjust to Mr. Gompertz not to express my admiration of the 
skill and ingenuity which he has shown in the treatment of a very novel 
subject and in the application of his principles to the solution of many curious 
and difficult geometrical problems. 

236 THIRD REPORT — \833. 

If we assume a to denote a finite quantity, then 
(1.) a + = a, and a + oo = + oo . 

Consequently does not affect a quantity with which it is 

connected by the sign + or — , whilst co , similarly connected 

with such a quantity, altogether absorbs it. 

(2.) axO = 0, axoo =oo;--- = co and — = 0. 


It is this reciprocal relation between siero and infinity which 
is the foundation of the great analogy which exists between 
their analytical properties. 

(3.) If these symbols be considered absolutely by themselves, 
without any reference to their symbolical origin, then we must 

consider ~ = \ and = 1. 


But if those symbols be considered as the representatives 
equally of all orders of zeros and infinities respectively, then 

~ and may represent either I or « or or co , its final 

form and value being determined, when capable of determina- 
tion, by an examination of the particular circumstances under 
which those symbols originated. The whole theory of vanish- 
ing fractions will depend upon such considerations. 

Having ascertained the principal symbolical conditions which 
and CO are required to satisfy, we shall be prepared to con- 
sider likewise the principle of their interpretation. The exami- 
nation of a few cases of their occurrence may serve to throw 
some light upon this inquiry. 

Let us consider, in the first place, the interpretation of the 

critical values 0, oo and ~ in the formulae which express the 

values of x and y in the simultaneous equations, 

a X + b y = c \ 
a' X + b' y ■= c' J 
In this case we find 

^ t:- and y = 


'•4^-!} -'{I'-l.}' 

„' „ „' 

If -7- = -jT, — = — 7) and therefore -z- — -n, then a: = -^ 
b a a bo 

and y = -Q- 


In this case «' = in a, U = m a, and c' = m c, and the second 
equation is deducible from the first, and does not furnish, there- 
fore, a new condition: under such circumstances, therefore, 
the values of x and y are really indeterminate, and the occur- 
rence of -r- in the values of the expressions for x and y is the 
sign, or rather the indication of that indetermination. 

If 4- be not equal to -,-t, but if — be equal to —f, then x — co 
b o, a 

and y = 00 . In this case vfe have a! — m a,h' =■ mh, but c' is 
not equal to m c; and the conditions furnished are inconsistent, 
or more properly speaking impossible. In this case, the occur- 
rence of the sign oo in the expressions for x and y is the sign 
or indication of this inconsistency or impossibility: and it should 
be observed that no infinite values of x and y, if the infinities 
thus introduced were considered as real existences and identi- 
cal in both equations, would satisfy the two equations any more 
than any two finite values of x and y which would satisfy one 
of them. We may properly interpret go in this case by the 
term impossible. 

c e' b' b 

If -^ = ^7, but if —7 be not equal to — , then x is zero and y 
h V a a 

is finite, and therefore possible. It is in this sense that we 
should include %ero amongst the possible values of x or y, a 
use or rather an abuse of language to which we are somewhat 
familiarized, from speaking of the zero of quantity as an exist- 
ing state of it in the transition from one affection of quantity to 

If we should take the equations of two ellipses, whose semi- 
axes are a and b, a' and b' respectively, which are 

f! J. ^ - ] 
„2 -f- ^,2 - i' 

^ ^ it - \ 

and consider them as simultaneous when expressing the co- 
ordinates of their points of intersection, then we should find 

X = //6^ W\ and y = /f^ ^ . 
V la^ ~ ci'^S V 16^ ~ b'^S 

If we suppose — = —7, or the ellipses to be similar, and at the 
same time b not equal to b', then or = op and y = cc , Avhich 

238 THIRD REPORT — 1833. 

would properly be interpreted to mean that under no cir- 
cumstances whatever, whether in the plane of x y or in the 
plane at right angles to it, in which the hyperbolic portions* of 
curves expressed by those equations are included, would a point 
of intersection or a simultaneous value of x and y exist : or in 
other words, the sign or symbol oo would in this case mean 

that such intersection was impossible. If we supposed — = — ;; 

and also h = b',or the ellipses to be coincident in all their parts, 

then we should find o^ = -y- and y = ^, indicating that their 

values were indeterminate, in as much as every part in the iden- 
tical curves would be also a point of intersection, and would fur- 
nish therefore simultaneous values. If we should suppose b 

greater than b', a greater than a\ and -j- not equal to -77 , then 

we should find 

X = u and y = ^ \^—U 
or x = a. \/ — 1 and y = j3^ 

according as -j- is less or greater than -77. In this case, one 

ellipse entirely includes the other, but the hyperbolic portions 
at right angles to their planes, which are in the direction of 
the major axis in one case and in that of the minor axis in the 
other, will intersect each other at points whose coordinates are 
the values of x and y above given : it wovdd appear, therefore, 
that the impossible intersection of the curves would be indi- 
cated by the sign or symbol go alone, and not by \/—l. 

The preceding example is full of instruction with respect to 
the interpretation of the signs of algebra, when viewed in con- 
nexion with the specific values and representations of the sym- 
bols ; and there are few problems in the application of algebra 
to the theory of curve lines which would not furnish the mate- 
rials for similar conclusions respecting them : but it is chiefly 
with reference to the connexion of those signs with changes in 
the nature of quantities, and in the form and constitution of ex- 
pressions, that their interpretations will require the most care- 
ful study and examination. We shall proceed to notice a few of 
such cases. 

a;2 2/2 y 

* If in the equation — + -Tg- = 1, we suppose y replaced by y v — 1, 

and the line which it represents when not afifected by V — 1 to be moved 
through 90° at right angles to the plane of x y, we shall find an hyperbola 
included in the equation of the ellipse. 


The second member of the equation 

— ~~ + T2 + «3 + • • • • 

a — b a a'- a" 

preserves the same form, whatever be the relation of the values 
of a and b, and the operation, which produces it, is equally prac- 
ticable in all cases. As long as a is greater than b, a~b is po- 
sitive, and there exists, or may be conceived to exist, a perfect 
arithmetical equality between the two members of the equa- 
tion. If, however, a = b, we have -jr upon one side and the 

sum of an infinite series of units multiplied into — upon the 

other, and both the members are correctly represented by oo ; 
but if a be less than b, we have a negative and a finite value 
upon one side of the equation, and an infinite series of perpe- 
tually increasing terms upon the other, forming one of those 
quantities to which the older algebraists would have applied 
the term plus quam infinitum, and which we shall represent by 
the sign or symbol co . It remains to interpret the occurrence 
of such a sign under such circumstances. 

The first member of this equation — — -r is said to pass 

through infinity when its sign changes from + to — , or con- 
versely : its equivalent algebraical form presents itself in a se- 
ries which is incapable of indicating the peculiar change in the 

nature of the quantity designated by — — -y , which accompa- 
nies its change of sign. The infinite values, therefore, of the 
equivalent series (for in its general algebraical form, where no 
regard is paid to the specific values of the symbols, it is still 
an equivalent form,) is the indication of the impossibility of ex- 
hibiting the value of ^ in a series of such a form under 

such circumstances. 

Let us, in the second place, consider the more general series 

for {a — by, or 

„ f, b ^ n{n- 1) b^ 

n {n -I) in - 2) 1% & 1 

The inverse ratio of the successive coefficients of this sei'ies 

S40 THIRD REPORT — 1833. 

approximates continually to — 1 as a limit, and tlie terms be- 
come all positive or all negative, according as the first negative 

coefficient is that of an odd or of an even power of — . It follows, 


therefore, that if a be greater than b, the series will be conver- 
gent and finite in all cases ; if « be equal to b, it will be 0, 1, 
or 00 , according as n is positive, 0, or negative ; and if a be 
less than b, it will be infinite. 

The occurrence of tlie last of these signs or values is an in- 
dication generally that some change has taken place in the na- 
ture of the quantity expressed by {a — i)", in the transition 
from a > 6 to a <C 6, which is of such a kind that the correspond- 
ing series is not competent to express it : thus, if w = 4» then 
{a — by is affected with the sign V — I when a is less than b, 
whilst no such sign is introduced nor introducible into the equi- 
valent series corresponding to such relative values of a and b : 
and a similar change will take place, whenever a transition 
through zero or infinity takes place. 

In this last case [a — by would appear to attain to zero or in- 
finity, but not to pass through it, and no change would appa- 
rently take place in its affection corresponding to the change of 
affection of « — b; but the corresponding series will under the 
same circumstances change from being finite to infinite, a cir- 
cumstance which we shall afterwards have occasion to notice, 
and which we shall endeavour to explain in the course of our 
observations upon the subject of diverging and converging 

In the preceding examples the sign or symbol go has not 
presented itself immediately, but has replaced an infinite series 
of terms, whose sum exceeded any finite magnitude ; and it may 
be considered as indicating the incompetence of such a series 
to express the altered state or conditions of the quantity or 
fraction to which it was required to be altogether, as well as 
algebraically, equivalent. In the examples which follow, it will 
present itself immediately and will be found to be the indica- 
tion of a change in the algebraical form of the term or terms in 
which it appears, or rather that no terms of the form assigned 
can present themselves in the required equivalent series or 

The integral I x^ d x = ^ + C is said to fail when 

n ■= — 1, in as much as it appears that under such circum- 

x"' ~ ' 
stances 7 becomes 00 , which is an indication that the va- 
ra + 1 


riable part of ^" d x is no longer expressible by a function 

under the form r' but by one which must be determined 

n + i 

by independent considerations. A knowledge, however, of the 

nature of its form in this particular case has enabled algebraists 

to bring it under a general form, by which the sign of failure 

or impossibility is replaced by the sign of indetermination 

A /r" + ' a;" "*" ' a" "*" ' 

jr-; for if we put — -— r + C = — — ^ + C, (borrowing 

from the arbitrary constant,) we shall get an expression 

-«" + ' 

n + 1 

which becomes -^ when n = — 1, and whose value, determined 

according to the rules which are founded upon the analytical 
properties of 0, will be log a; + C. 

A more general example of the same kind, including the one 
which we have just considered, is given in the note to page 
211, where it is required to determine the general form of 

— — — and of -, — . — (where n is a positive number) for all 
dx"" X dx'' x" ^ 

values of r : a formula is there constructed, from our knowledge 

of the form in the excepted case, which is capable of correctly 

expressing its value in all cases whatever. 

The cases in which the series of Taylor is said to fail are of 

a similar nature. Thus, if u = <f (x) = x + \f x — a, then 

du , , d^u W , d^u h^ ^ 
u' = <p (:c + h) = u + j-^h + j^, Y72 + J^' T7273 ^^- ' 

ft U 

and if we suppose x =i a, all the differential coefficients ^, 
-^—2, &c., become infinite, which is an indication that no terms 

of such a form exist in its developement, which becomes, under 
such circumstances, a + V'/'- The reasons of this failure in 
such cases have been very completely explained by Lagrange 
and other writers ; but it is possible, by presenting the deve- 
lopement which constitutes Taylor's series under a somewhat 
different and a somewhat more general form, that the series may 
be so constructed as to include all the excepted cases. 

There are two modes in which the developement of <p (.r + h) 
according to powers of h may be supposed to be effected. In 
the first and common mode we begin by excluding all those 
terms in the developement whose existence would be incon- 

1833. R 

242 THIRD REPORT — 1833. 

sistent with general values of the symbols : m the second we 
should assume the existence of all the terms which may cor- 
respond to values of the symbols, whether general or specific, 
and then prescribe the form which they must possess, con- 
sistently with the conditions which they are required to satisfy. 
If we adopt this second course, and assuming u ■= (^ {x) and 
2^' = (p (a; + li), if we make 

u' = « + A /i" + B /^* + C h" + &c., 

the inquiry will then be, if there be such a term as A h", where 
A is a function of x or a constant quantity, and a is any quantity 
whatsoever, what are the properties of A by which it may be 
determined ? For this purpose we shall proceed as follows. 
It is very easy to show, from general considerations, that 
if iH be considered successively as a function of x and of h, 

-7—7 = -rrr. , for all values of r, whether whole or fractional, 

positive or negative : it will follow, therefore, (adopting the 
principles of differentiation to general indices which have been 
laid down in the note, p. 211,) that 

d'^u' _ r(i + a) ^ r(i + &) B,,_„ . . 

omitting the arbitrary complementary functions, which will in- 
volve powers of h. In a similar manner we shall get 

d'u' d^u d" K j„ d^B ,, 

dx" dx" dx" dx"' 

If these results be identical with each other, we shall find 

r (1 + a) y _ d^ 

rji) ' dx"' 

and, therefore, A = -=-?! — ; — n • -i-~^, since r(l) = 1. It is easy 

r{l + a) dx"' ^ ^ •' 

to extend the same principle to the determination of the other 
coefficients, and we shall thus find 

d" u h" # u //* o ,, . 

or, in other words, it follows that the coefficient of any power 
of h whose index is r will be 

1 d' u 



The next step is to adapt the series (1) to the different cases 
which an examination of the constitution of the function ?/ will 
present to us. 

If we suppose x to possess a general value, then u' and u 
will possess the same number of values, and no fractional 
power of h can present itself in the developement. In this case 
T(l + a) = I . 2 . . . a, and it may be readily proved that 
the successive indices a, b, c, &c., are the successive numbers 
1, 2, 3, &c., and that consequently, 

, d u J cV^ u h^ cPu h^ o 

du dx^ 1.2 dx^ 1.2.3 

It will also follow that the series for m' can involve no negative 
and integral power of h ; for in that case the factorial T (1 + «), 
which appears in its denominator, would become oo, and the 
term would disappear. If it should appear, also, that for spe- 
cific values of x any differential coefficient and its successive 
values should become infinite, they must be rejected from the 
developement, in as much as in that case the equation 

would no longer exist, which is the only condition of the intro- 
duction of the corresponding terms. In other words, those 
terms in the developement of u' must be equally obliterated, 
which, under such circumstances, become either or oo. 

If the general differential coefficient of u could be assigned, 
its examination would, generally speaking, enable us to point 
out its finite values wherever they exist, for those specific va- 
lues of the symbols which make the integral differential coeffi- 
cients zero or infinity. For all such values there will be a cor- 
responding term in the developement of u under those circum- 
stances. Thus, if we suppose u ■= x t/a — x, we shall find 

1 .J-u (-')"'^(l) [ K«-.)-(|-r).r ] 
r{\ + r)r[^-rj 1^ {^--rya-xy-i J 
if we make x = a, this expression will be neither zero nor 

infinit?/ in two cases only, which are when r =i—, and when 


»' = -g- : in the first case we get, 

rf* u , — - 


R 'Z 

844 THIRD REPORT — 1833. 

and in the second we get, 

— — (t'u 2 ^ ' ■ 



\a — a) 

= >/•=!; 

dx^ Ar(0) X X (a -a) 

since r(l) =s 1 = r(0), and the symbol in the denominator 

3 3 

= — ^, is a simple zero. The corresponding developement 

of m' under such circumstances is 

^/-«2.A^ + -/-I. A^ 
a result which is very easily verified. 

If we pay a proper regard to the hypotheses which deter- 
mine the existence of terms in the series for u' for specific 
values of the independent variable, we shall be enabled without 
difficulty to select the indices of the differential coefficients 
which can present themselves amongst the coefficients of the 
different powers of h in the developement. For, in the first 

place, /*», and the differential coefficient whose index is — , will 

possess the same number of values, and the same signs of affec- 

tion. If there be a term in u which = P (ar — «)», where P 
neither becomes zero nor infinity, when x — a, and where the 
multiple values of P, if any, are independent of those contained 

- .„ ,1 j.(J^ .V .{x — «)» 

in (a: — «)», then it will appear that the term ot — 




which is independent of {x — «)» is P . — ^ > and that 

d X" 

fill • fl# 

all the other terms of —, being either ssero or infinity when 

d x^ 
X ^= <t, or, if finite, introducing, through the medium of the 
factorial function by which they are multiplied, multiple values 
which are greater in number than those contained in u , must 
be rejected, as forming no part of the developement. It will of 
course follow, that the function P will become, under such cir- 
cumstances, a function of h, and if we represent it by P', and 
denote its values, and those of its successive differential coeffi- 
iiients, when h = 0, by p, p\ p", p"', &c., we shall find 


none of which become ssero or injinity, in as much as P does not 
vanish when x ■= a. 

If there exist other terms in m of a similar kind, such as 

m' m" 

Q,{x — by, R (ar — cy^ , &c., the same observations will apply- 
to them. Such terms will correspond to values of x, which 
make radical expressions of any kind zero or infinity, and the 
form of the function u must be modified when necessary, so 
that such radicals may present themselves in single terms of 

the form V {x — «)». The same observations will apply to ne- 
gative as well as positive values of — , unless we suppose — a 

negative whole number. The principle of the exception in this 

last case may be readily inferred from the remarks in the note, 

d~^ 1 
p. 211, on the subject of the values of -3 — zTr • — 7> when « is a 

whole number. If we suppose, therefore, u to involve terms 

such as P (a; — «)», Q (w — Z>)»', &c., the most general form 
under which its developement can be put, supposing all terms 
which become zero or infinity for specific values of x to be 
rejected, will be as follows : 

m m VI 

a — a p, d« {x — o)» h" 

(^^ rfj ■r(i + ^) 

m' m! m' 

h — h „, dn' {x — 6)"' Jin' 

d x^' 

(> - 5) 


, du J d^u h^ d^u .h^ ^ 

«=« + ^^' + ^^r:2 + rf^ 17273+ ^*^" 


h — h ( Ifi \ Vi 

+ &c. 
We have introduced the discontinnoiis sigtis or factors : 

a — a 

g46 THIRD REPORT — 1833. 

^ ~ &c., which become equal to 1 when x = a or x =: b, &c., 
X -^ o 

but which are ^ero for all other values of x, to show that the 

terms into which they are multiplied disappear from the deve^ 

lopement in all cases except for such specific values of x. 

The existence of the terms of the series for u is hypothetical 
only, and the equation which must be satisfied, as the essential 
condition of the existence of any assigned hypothetical term, at 
once directs us to reject those terms which would lead to infi- 
nite values of the differential coefficients, as well as those which 
possess multiple values which are incompatible with those con- 
tained in «/. It is quite obvious that upon no other principle 
could we either reject such infinite values, or justify the con- 
nexion of a series of terms with the general form of ?/, which 
have no existence except for specific values of x. The con- 
clusion obtained is of considerable importance, in as much as it 
shows that the series of Taylor, if considered and investigated 
as having a contingent, and not a necessary existence, may be 
so exhibited as to comprehend all those cases in which it is 
commonly said to fail : and it will thus enable us to bring under 
the dominion of the differential calculus many peculiar cases in 
its different applications which have hitherto required to be 
treated by independent methods. 

Thus, if it was required to determine the value of the fraction 

(2 2\^ 

— " ^" , when x=sa,we should find it to be, 

x^ {x — ay 


. (x^ — a-f 


x^ {x — of 


{x + af.-~-,'{x-nf 

d x^ 

a conclusion which would be justified by the developement of 
the numerator and denominator of this fraction by the complete 
form of Taylor's series, when x = a. 

Many delicate and rather obscure questions in the theory of 
maxima and minima, particularly those which Euler has deno- 


minated maxima and minima of the second species, and others 
also relating to the singular or critical points of curve lines, 
must depend for their dilucidation upon this more general view 
of Taylor's series, as connected with the consideration of ge- 
neral differential coefficients *. 

* Euler has devoted an entire chapter of his Calculus Differentialis to the 
examination of what he terms the differentials of functions in certain peculiar 
cases. It is well known that he adopted Leibnitz's original view of the prin- 
ciples of the differential calculus, and considered diiferentials of the first and 
higher orders as infinitesimal values oi differences of the first and higher orders. 
Such a principle necessarily excludes the consideration of differential coefficients 
as essentially connected with determinate powers of the increment of the inde- 
pendent variable, which may be said to constitute the essence of Taylor's 
theorem, and which must be the foundation of all theories of the differential 
calculus, which make its results depend upon the relation of forms, and not 
upon the relation of values. As long, however, as the independent variable 
continues indeterminate, the symbolical values of the differentials are the same 
upon both hypotheses. But when we come to the consideration of specific va- 
lues of the independent variable which make differential coefficients above or 
below a certain order, infinite or zero, then such a view of the nature of dif- 
ferentials necessarily confounds those of different orders with each other. Thus, 

ii y=. a^ -\- {x — a)^, Euler makes, when a; = a, d y = (d a;)^, instead of 

(5 \ = 3 = (d x)^. Iiy = 2ax — x"-\- a^/ (a^ — a;3), he makes, 
y) t ^"^ 

when X = a, d y = a \/ — 2 a . d x^, instead of 
,j^ ff"^ (d «)■? 

These examples are quite sufficient to make manifest the inadequacy of 
merely arithmetical views of the principles of the differential calculus to ex- 
hibit the correct relation which exists between different orders of differentials, 
and, a fortiori, therefore, between different orders of differential coefficients. 

M. Cauchy, in his Lecons sur le Calcul Infinitesimal (published in 1823), has 
attempted to conciliate the direct consideration of infinitesimals with the purely 
algebraical views of the principles of this calculus, which Lagrange first securely 
established ; and it may be very easily conceded that no attempt of this able 
analyst, however much at variance with ordinary notions or ordinary practice, 
would fail from want of a sufficient command over all the resources of analysis. 
He considers all infinite series as fallacious which are not convergent, and that, 
consequently, the series of Taylor, when it takes the form of an indefinite series, 
is not generally true. It is for this reason that he has transferred it from the 
differential to the integral calculus, and exhibits it as a series with a finite 
number of terms completed by a definite integral. It is very true that M. Cauchy 
has perfectly succeeded in dispensing with the consideration of infinite series in 
the establishment of most of the great principles of the differential and integral 
calculus ; but I should by no means feel disposed to consider his success in over- 
coming difficulties which such a course presents as a decisive proof of the expe- 
diency of following in his footsteps. The fact is, that if the operations of algebra 
be general, we must necessarily obtain indefinite series, and if the symbols we 
employ are general likewise, it will be impossible to determine, in most cases, 

248 THIRD REPORT — 1833. 

Signs of discontinuity are those signs which, in conformity 
with the general laws of algebra, are equal to 1 between given 
limits of one or more of the symbols involved, and are equal to 
zero for all their other values. If merely conventiofial signs 
were required, we might assume arbitrary symbols for this 
purpose, attaching to them far greater clearness as diventical 
marks, the limits of the symbol or symbols between which the 
sign of discontinuity was supposed to be applied. Thus, we 
might suppose ^T)a to denote 1, when x was taken between 
and a, to denote zero for all other values ; ■^Da+^j, to denote 1, 
when X was taken between a and a + b, and zero for all other 
values ; and similarly in other cases. 

Thus, if 2/ = a X + /3 and y =^ of x + /3' were the equations 
of two lines, and if we supposed that the generating point whose 
coordinates are x and y was taken in the first line between the 
limits and a, and in the second line between the limits a and b, 
then we should have generally, 

y = -D; (a ^ + ^) + -D/ {«.' X + /30 (1.) 

the couvergency or divergency of the series which result. It is only, therefore, 
when we come to specific values that a question will arise generally respecting 
the character of the series : and it is only when we are compelled to deduce the 
function which generates the series from the application of the theory of limits 
to the aggregate of a finite number of its terms, that its convergency or diver- 
gency becomes important as afi^ecting the practicability of the inquiry : in short, 
it must be an erroneous view of the principles of algebra which makes the result 
of any general operation dependent upon the fundamental laws of algebra to be 
fallacious. The deficiency should in all such cases be charged upon our power 
of interpretation of such results, and not upon the results themselves, or upon the 
certainty and generality of the operations which produce them : in short, the 
rejection of diverging series from analysis, or of such series as may become 
divergent, is altogether inconsistent with the spirit and principles of symbolical 
algebra, and would necessarily bring us back again to that tedious multipli- 
cation of cases which characterized the infancy of the science. A verj' instruc- 
tive example of the consequences of adopting such a system may be seen in the 
researches of M. Liouville, which have been noticed in the note at p. 217. 

Lagrange in his Theorie des Fonctions Analytiques, and in his Calcul des 
Fonctions, has given theorems for determining the limits between which the 
remainder of Taylor's series, after a finite number of terms, is situated : and 
the same subject has been very fully discussed in a memoir by Ampere, in the 
sixth volume of the Journal de I'Ecole Poly technique. Such theorems are ex- 
tremely important in the practical applications of this series, but they in no 
respect affect either the existence or the derivation of the series itself. It is a 
very common error to confound the order in which the conclusions of algebra 
present themselves, and to connect difficulties in the interpretation and appli- 
cation of results with the existence of the results themselves : and it is the in- 
fluence of this prejudice which has induced some of the greatest modem ana- 
lysts, not merely to deny the use, but to dispute the correctness of diverging 

Messrs. Swinburne and Tylecote, the joint authors of a Treatise on the true 


Thus, if in the triangle A C B, we 
draw C D, a perpendicular from the '\f,, 

vertex to the base, and if we suppose 
A D = «, A B = 6, A the origin of m,.-' 

the coordinates, A B the axis of x, 
y =. a. X the equation of the line A C, 


and y = a' .r + /3' the equation of the ■* ^ ^ ^ 

line B C, then we should find that the value of y represented 
by the equation 

y = -^Da" . a ^ + ^Dj" (a' a; + /3') * (2.) 

would be confined to the two sides A C and B C of the triangle 
ABC, excepting only the point C, which cori-esponds to the 
common limit of the discontinuous signs. For if we suppose 
'Do° and ■^Dj" to be true up to their limits, we shall find, when 
X =■ a, that ■^D«° + '^Dj" = 2. If we replace, however, 

^D; by ^D; - ^, and ^D," by ^D*" - |^^, 

Developement of the Binomial Theorem, Avhich was published in 1827, have 
contended vigorously for the restriction of the meaning of the sign = to simple 
arithmetical equality, and would reject its use when placed between a function 
and its developement, unless its complete remainder, after a finite number of 
terms, should replace the remaining terms of the series ; or unless, when the 
indefinite series was supposed to be retained, the value or the generating func- 
tion of this remainder could be assigned. In conformity with this principle 
they have assigned the remainder in tlie series for (a -\- x)", which they exhibit 
under the following form : 

(^ + .,)n = «„ + ,,a„-l^.4.....«(«-_l)---(»->-+l).an-r;er 

+ -«'-^'(« + ^)"{(^+V^ + 

('• + 1) 

(r + 1) (r + 2) . . ■ (n - 1) 

T • • 

(a + xy + i ' 1 (a + xy + ^ 


1 . U . . . (n — r—l) ■ (a + .r)"/ ' 

the remainder being (a + j;)« jC + 1 multiplied into n — r terms of the deve- 

1 1 

lopement of -77 — ; — ;^ •>..,, or of — r-;. 

^ {(a -f x) — aj'^+i' a»- + i 

The method which they have employed for this purpose, which is extremely 
ingenious, succeeds for integral values of n, whether positive or negative, but 
fails to assign the law when the index is fractional. But my own views of the 
principles of symbolical algebra would, of course, induce me to attach very little 
value to results which were exhibited in such a form as to be incapable of being 
generalized, a defect under which the formula given above evidently labours. 

• The conventional sign '^Dj'* might be replaced, though not with perfect 

1 /»o 

propriety, by the definite integral _ ^ / dx. 

250 THIRD REPORT — 1833. 

and if we make, therefore, 

y=^'D."-^«^„+^W-i^*^(»'.+^') (3.) 

the equation will be true for the ordinate of every point of the 
sides A C and C B of the triangle ABC. 

More generally, if we suppose y = f^ x, y ■=. (p^x,y ■=■ (^^x, 
y = (p^x, &c., to be the equations of a series of curves, then the 
equation of a polylateral curve composed of the several portions 
of the separate curves corresponding to values of x, included 
between the limits a and b, b and e, c and d, &c., would be, 

+ ('D/-f-^,)fcx + &c.; (4.) 

the value of the ordinate at each successive limit being replaced 
by that of the succeeding curve. In this manner, if we should 
grant the existence of the sign of discontinuity, we should be 
enabled to represent the equations of polygons, and of poly- 
lateral curves of every description. 

It remains to consider the nature of the expressions which 
are competent to express ^Dj". 

The expressions which have been generally proposed for this 
purpose are either infinite series, or their equivalent definite 
integrals. Le Comte de Libri, however, a Florentine analyst 
of distinguished genius, has proposed* a finite exponential ex- 
pression which will answer this purpose. The examination of 
the expression 

would readily show that its value is 1 when x is greater than 
a, and that it is when x is equal to or less than a. It will 
therefore follow that the product 

^(logO) e^^^s")'^- ") ^ ^(logO)e^'^e*'>(*-^) 

is equal to 1 between the limits a and b, and is equal to at 
those limits, and for all other values. And, in as much as 

* Memoires de Mathematique et de Physique, p. 44. Florence 1829. The 
author has since been naturalized in France, and has been chosen to succeed 
Legendre as a member of the Institute : he has made most important additions 
to the mathematical theory of numbers. 


p(iogO) _ Q^ yfQ jyjay replace the preceding product by the equi- 
valent expression 

This expression, which is equivalent to ^Db" '■ j, 

has been applied by Libri to the expression of many important 
theorems in the theory of numbers *. 

The definite integral / — sin r x has been shown by 

Eulerf and many other writers, to be equal to -^ when x is 

positive, to when x is 0, and to -^ when x is negative. It 
follows, therefore, that 

2 r^dr . (b-a) C (a + b)l 

— / — sm ^^ — Tz — - r cos ■< x — ^^ — 7^ — - r > 

vJo r 2 L 2 J 

= — / ^ — sin (^ — a) r H / — sin (^ — 6) r 

vJo r ^ ' TTi/o »* 

is equal to 1 , when x is between the limits a and 6, to ^, when x is 

at those limits, and to zero, for all other values. If we denote the 

definite integral — / — sin — - — - r cos -s x — ^ — -- — '- > r 
^ -K JQ r a L 2J 

by Cj", we shall get, 

xT\ « n«i ^ ^ 1 " — b 

^' -^* +2(a:-a) + 2]F=T)' 

and consequently the equation of a polylateral curve, such as 
that which is expressed by equation (4.), will be, 

y = Cb . <P]X -{■ C/ . 9.2 -^ + Cd . ^3 X +&C., 

in as much as at the limits we have <Pi {b) = p^ (b), (p^ (c) s= a^ (<?), 

and consequently for such limits Cj" ^j (b) + C/ f^ (^) = f 1 (^) 
= ip^ (b), and not 2 (p^ (b). 

All definite integrals which have determinate values within 
given limits of a variable not involved in the integral sign, may 
be converted into formulae which will be equal to 1 within those 

• Cr elk's Journal for 1830, p. 67. 

t Inst. Calc. Integ., torn. iv. ; Fourier, Theorie de la Chaleur, p. 442. ; Frul- 
loiii, Memorie della Societa Italiana, torn. xx. p. 448. ; Libri, Memoires de Ma- 
thematique et de Physique, p. 40. 

252 THIRD REPORT — 1833. 

limits and also including the limits, and to zero for all other 
values *. But the expressions which thence arise, though fur- 
nishing their results in strict conformity with the laws of sym- 
bolical combinations, possess no advantage in the business of 
calculation beyond the conventional and arbitrary signs of dis- 
continuity which we first adopted for this purpose : but though 
it is frequently useful and necessary to express such signs ea;- 
plicitly, and to construct formulfE which may answer any as- 
signed conditions of discontinuity, yet such conditions will be 
also very commonly involved implicitly, and their existence and 
character must be ascertained from an examination of the pro- 
perties of the discontinuous formulae themselves. We shall now 
proceed to notice some examples of such formulas. 
The well known series f 

X 1 1 . 1 . 

r TT + -3- = sin j; — ^ sin 2 j: + -?r sin 3 x — -p sin 4.r + &c. (1.) 

is limited to integral values of r, whether positive or negative, 
and to such values oirit •\- -^ as are included between -^ and 

— -^ : the value of r, therefore, is not arbitrary but condi- 

* If a definite integral (C) has n determinate values »i, «2, . . . un, within 
the limits of the variable a and b, and no others, the values at those limits 
being included, and if C be equal to zero for all values beyond those limits, 
then we shall find 

«D « = _ (C — eti) ( C — eta) • • • (C — »n) ^ j. 
oil X oCq X • . ■ cin 

thus in the case considered in the text, we get 

= _ 2 (C — 1) ^C — i-) + 1 = — 2 C2 + 3 C. 

t The principle of the introduction of r w in equation (1.) by which it is ge- 
neralized, will be sufficiently obvious from the following mode of deducing it : 

= log {i + e'^~' } - log {1 + e-' ^^} 


1 I '*' 

1 +e 

, -X V-l ; ^ . - , ■ ( S'/^ -JfV-A 

= log e =. X a/— I + 2r t V — 1 = Ve — e / 

and, therefore, dividing by 2 V* — 1» and replacing the exponential expressions 

by their equivalent values, we get 

X . 1 1 . 1 . 

r IT + — = sin a; sin 2 a; + — sin 3 a; sin 4 a; + &c., 

2 2 3 4 

where x upon the second side of the equation may have any value between 
•+ 00 and — 00 . 



tional. If we successively replace, therefore, a" by -^ + t and 

— — x,yvQ shall get 

•It X 1 . ^ 1 „ 

rir + -J- + "o = cos X + -^ &\a'ax — -^ cos 3 x 
— :j- sin 4 X + &c. 

/«• + -; 7r- = COS a: yr sin2x ^ cos 3 jr 

4 2 » 3 

H — J- sin 4 d7 + &c. 

Adding these two series together and dividing by 2, we get 

(f J- f^\ •B' 1 1 

^-^ — '- * + -J- = cos ^ IT- cos 3 a: + -^ cos 5 X — &c. {2.) 

It It 
If X be included between g- and —, then r = and r' = 0, 

and we get 

-r = COS X ^ cos S X + -p- cos 5 j: — &c. (3.) 

4 3 5 ^ •' 

■jt 3 iz* 

If J? be included between -g- and -^, then r = — 1 and r' = 0, 

and we get 

w 1 1 

— -r = cos j: -^ cos 3 ^ + ^r cos 6x — &c. (4.) 

4 3 5 ^ ' 

If the limits of x be -^ and -^, —5- and -^, -^ and 

— -3-, ^ and — '-^, we shall obtain values of the series 

nt Tt 

(2.), which are alternately -7- and — -r-. 

X . 1 . 

Again, if in equation (1.), or rir + -tj- = sin x — -^sinS^r 

+ -q- sin S X -7- sin 4 t + &c., we replace x by tt — x, we 

shall get 

»•' ff -I -^ — = sin or + -^ sin 2 x + -q- sin 3 J? + -j- sin 4 j? + &c. 

Adding these equations together and dividing by 2, we get 



(r + r') IT 1 1 
Tf" + -J- = sm X + -TT sin 3 z + — sin 5 jr + &c. (5.) 



which may be easily shown to be equal to -j- and j- altera 

4 4 

nately, in the passage of x from to v, from ^ to 2 tt, from 2 it 
to 3 It, &c., or from to — tt, from — it to —2it, &c.: its 
values at those limits are zero. 

The series (2.) and (5.) have been investigated by Fourier, in 
his TMorie de la Chaleur *, by a very elaborate analysis, which 
fails, however, in showing the dependence of these series upon 
each other and upon the principles involved in the deduction 
of the fundamental series : and they present, as we shall now 
proceed to show, very curious and instructive examples of dis- 
continuous functions. 

The equation y 


is that of an indefinite straight line, 

Q A P, making an angle with the axis of x, whose tangent is 

■g-j and which passes through the origin of the coordinates : 
whilst the equation 

y sz sinx zr- &m2x + -jr- sin 3 a; ;i- sin 4 .r + &c. 

» o ^ 

is that of a series of terminated straight lines, d' c, dC,T> C, 
&c., passing through points a, A, A', &c., which are distant 
2 It from each other : the portion d C alone coincides with the 

primitive line, whose equation is y == -^. 

Again, the line whose equation is y = ~^, is parallel to the 



G' d, C 

* From page IG/" to ipo ; also 267 and 3')6. 





axis of X at the distance -j- above it : the line whose equation is 


— -J-, is also parallel to the axis of x, at the distance -j- below 

it : the line whose equation is 

y = cos X «- cos S X + -^ cos 5 x — &c. 

consists of disco7itinuous portions of the first and second of 
those lines, whose lengths are severally equal to tt. The values 

of y at the points B and b, corresponding to a- = -^ and — ^, 

are equal to srero, since the equidistant points D and C, c and d, 
are common to both equations at those points. 

It would appear, therefore, in the cases just examined, that 
the conversion of one member of the equation of a line into a 
series of sines and cosines would change the character of that 
equation from being continuous to discontinuous, the coinci- 
dence of the two equations only existing throughout the ex- 
tent of one complete period of circiUation of the trigonometrical 
series : and more generally, if, in any other case, we could ef- 
fect this conversion of one member of the equation of a curve 
into a series of sines or cosines, it is obvious that the second 
equation must be disco7iti7iuous, and that the coincidence 
would take place only throughout one period of circulation, 

whether from to tt or from — -^ to -^. It remains therefore 

to consider whether such a conversion is generally practicable. 
Let us take ti equidistant points in the axis of the curve 
whose equation is 7/ = 4> x, between the limits and tt, those 
limits being excluded : if we denominate the corresponding 
values of the ordinate by i/^, i/c^, . . . . i/„, and if it be proposed 
to express the values of these ordinates by means of a series 
of sines (of 7i terms) such as 

o, sin a; + ag sin 2 x + «3 sin 3 4? -f- . . . . + a„ sin 71 x, 

then we shall get the following n equations to determine the n 
coefficients a^, %> ^3 • • • • ^«* 


yj = ttj sm + «2 sm 

^2 = «i sin -^ — ^ + O2 sin 

y^ = a^ sin - + flTg sm 

277 , . 37r 



n + \ ^ 71+1 


47r . Gtt 

— ; — 7 + ff sm 

7fi+ \ ^ n + \ 


. ^nn: 

71 + V 


. 3M7r 
. ffnSin — — -,, 

n + V 

256 THIRD REPORT — 1833. 

w TT , . 2 tn: . 3 tin . 1^-K 

If any assigned coefficient a^ be required to be determined 
from this system of equations, we must multiply * them seve- 
rally by 

n , m It ^ . 2mTr ^ . Smrr _. nmv 

2 sm — —T, 2 sin — --r, 2 sin ^ , ... 2 sin r, 

n + r n + l n + I n + I 

when all the coefficients except a^ will disappear from the sum 
of the resulting equations : and we shall thus find 

2 f . OTTT . . 2mTt . nrmr'X 

«», = — r~\\ Vi sin — --r + Vo sin — —^ + . • . -/„ sin — -— ;- >. 
M + 1 L n + \ ^^ « + 1 "^ n + IJ 

It would thus appear that it is always possible to determine a 
series of sines of n terms with finite and determinate coeffi- 
cients, which shall be the equation of a curve which shall have 
n points in common with the curve whose equation is ?/ =: <p x, 
within the limits corresponding to values of x between and v ; 
and it is obvious that the greater the number of those points, 
the more intimate would be the contact of these two curves 
throughout the finite space corresponding to those limits. If 
we should further suppose the number of those points to be- 
come infinitely great, then the number of terms of the trigono- 
metrical series would be infinite likewise, and the coincidence of 
the curve which it expresses with the curve whose equation is 
y =s <p X, would be complete within those limits only, producing 
a species of contact to which the texxn. finite osculation has been 
applied by Fourier f . Beyond those limits the curves would 
have no necessary relation to each other. 

It would follow, also, from the preceding view of the theory of 
finite osculations, that the curve expressed hy y = <p x might be 
perfectly arbitrary, continuous, or discontinuous. Thus, it might 
express the sides of a triangle, or of a polygon, or of a multi- 
lateral curve, or of any succession of points connected by any 
conceivable law ; for in all cases when the corresponding or- 
dinates of equidistant points are finite, we shall be enabled to 
determine values of the coefficients a^ which are finite or zero 
by the process which has been pointed out above. 

* This is the process proposed by Lagrange in his "Theorie du Son," in 
the third volume of the Turin Memoirs, as stated by Poisson in his memoir 
on Periodic Series^ &c., in the 19th cahier of the Journal de I'Ecole Polytech- 

f TMorie de la Chaleur, page 250. 


The hypothesis of n being infinite would convert the series 
for «„ into the definite integral * 

2 /»0 

2 /»0 . , 

— / ax sin mx ax. 

COS a; 

if we make r = x and r = d x : or otherwise if we 

?^ + 1 w + 1 

assume the existence of the series 

(p X ■=■ a^ sin a: + «2 sin 2 x + . . . a,n sin m « + &c., 

it may be readily shown, by multiplying both sides of the equa- 
tion by sin m x d x, that 

a,n = — / (p X sin m X d X : 

and in a similar manner, if we should assume 

<p X = Oq cos .r + «i cos x + . . . a^ cos 7n x -\- &c., 

cr^ = — / <p X cos mx a X -f . 

Thus, if we should suppose ip jp = cos x, we should find 

4 C 2 4 . 6 1 

= — \ Tj — ^c sin 2 ^ + H — p sin 4 x -f -p — = sin 6 a; + &c. I • 
»r[l.o o.o o./ J' 

a very singular result, which is of course only true between the 
limits and it, excluding those limits %. 

If we should suppose (p .r = a constant quantity — between 

the limits and a, and that it is equal to zero between a. and tt, 
we should find 

(1 — cos«) . (1 — cos2«) . _ (1 — cosoa) 
(p X = ^ ^ sin X + ^ ^ -' sm 2 ^ + ■ ^ ~^ x 

sin 3 X + &c., 
excluding the limiting value u, when the value of the series is 

only -^§. 

If we should suppose <p x =. 'Dj" . a-x + "D,^ . («' x + /3'), 
which is the equation of the sides of a triangle (excluding the 

• Poisson, Journal dc I'Ecole Polytcchniqve, cahier xix. p. 447. 
+ Fourier, Th/'oric de la Chaleiir, pp. 235 & 240. 

\ IhUL, p. 233 ; Poisson, Joimial de I'Ecole Polytechmque, cahier xix. p. 418. 
§ Fourier, Tlicorie de la Chaleur, p. 244". 
loutj. s 



limit X = h), whose base is represented by tt, then we shall 

find * 

2 , / IX f 1 r . sin 2 .r sin 3 ^ o 1 
(p X — — [a-n + [a. — c^) b) ismx ^ \- — g &c. |- 

2 , ,s fsinZ>sinar , sin26sin2j: , sin3isin3j; ^ 1 
+ — («-«) I [2 + 2^ + 32 +&c.|. 

The trigonometrical series, in this last case, would represent a 
series of triangles placed alternately in an inverse position with 
respect to each other ; and a similar observation would apply 
to the discontinuous curves which are represented by any series 
of sines and cosines. Thus, \iy = <^ xho. the equation of the 
curve P C C" Q, and if we suppose 

2/ = (p ^ = Oj sin X + ttg s''" 2x + a^ sin 3 x + &c., 
between the limits and w ; and if we make A B = w, A A' 
= 2 TT, A B' = 3 TT, &c., we sliall get a discontinuous curve, 
consisting of a series of similar arcs, C D, a C", C D', &c., 
placed successively in an inverse relation with respect to each 
other upon each side of the axis of x, of which one arc C D 
alone coincides with the primitive curve. 

If we should suppose the same curve to be expressed be- 
tween the limits and tt by a series of cosines or 

2/ = ^ ar = «o + ^1 cos X + a^ cos 2 x + &c., 

and if we make A B = tt, A 6 = — tt, A A' = 2 tt, A B' = 3 w, 

&c., then the trigonometrical equation will represent a discon- 
tinuous curve c? C D C D', of which the portions C D and C d, 

• Fourier has given a particular case of this series, p. 246. 


C' D' and C D will be symmetrical by pairs ; but one portion 
only, C D, will necessarily coincide with the primitive curve. 

The theory of discontinuous functions has recently received 
considerable additions from a young analyst of the highest pro- 
mise, Mr. Murphy, of Caius College, Cambridge. In an admi- 
rable memoir on the Inverse Method of Definite Integrals *, he 
has given general methods for representing discontinuous func- 
tions, of one or a greater number of breaks, by means which are 
more directly applicable to the circumstances under which they 
present themselves in physical problems than those which have 
been proposed by Fourier, Poisson, and Libri. Mr. Murphy 
had already, in a previous memoir f , given a most remarkable 
extension to the theory of the application of Lagrange's theo- 
rem to the expression of the least root of an equation, which 
we shall have occasion to notice hereafter ; and he has shown 
that if (p {x) be an integral function of x then the coefficient of 

— in the developement of — log ^— will represent the least 

root of the equation <p x = 0. We thus find that the least 
of the two quantities « and (3 will be represented by the coeffi- 

cient of — m the series for log ^ ^-^ ~, which is 

(«+iy + ot47G- (^-±^) 

and if we re])lace « and B by — and -^, the least of the two 

^ cc p 

quantities — and -^, or the greatest of the tv/o quantities « and 
^, will be represented by 

• Transactions of the Philosophical Society of Cambridge, vol. w. p. 374. 

t Ihid. p. 125. 

X If we represent the series (2.) by S, we shall get 

d«-> S 1 „ 

__ — or 0, 

(— l)»-i T{n) d«»-i 
according a,s « is greater or less than (i : thus — — - would represent the at- 
traction within and without a spherical shell, which is or — , where « is 

the distance from the centre. 


260 THIRD REPORT — 1S33. 

Thus, if y — a .r — /3 = and y — «' .r — /3' = be the equa- 

A B p p- 

tions of two lines B C and D C, forming a triangle with a por- 
tion B D of the axis of x, then the system of lines which they 
form will be expressed by the product 

(y_«^_^) (2,_«/^-_^')^0. (3.) 

Now it is obvious that if common ordinates P M, P M' be 
drawn to the two lines, the least of them will belong to the sides 
of the triangle BCD; if we denote, therefore, P M and P M' 
by'^i and y,^, the equation 

y = 

y\ ya 
yi + ys 


1 . 1 


• (H^f 


2 J V 2 

"will become the equation of the sides of the triangle BCD, 
when yi and y^ are replaced by their values ; for y will denote 
P M for one side and 2^ "* for the other. 

In order to express a discontinuous function <p, which as- 
sumes the successive forms ipj, ip,,, f^, &c., for different values 
of a variable which it involves between the limits « and j3, /3 
and y, y and 8, &c., Mr. Murphy assumes S («i 2;), S (/Sj z), 

. . 1 . 

S (yi ss), &c., to denote the coefficient of — in the several series 


and supposes 


„ </ S («, «) , /> (/ S (/3, z) 

+ /3 

d S (y, ^) 

+ &c. 

doL '-''' d^ '-''■'' dy 
If a. be less than z or 2 greater than «, then S (a, ^) = «, 

, , t. d S (u, z) 1 ■r■a^ 1 i.u 4-1 '■^ S (^, s) 

and therefore r"^— ^ = 1 = up be less than z, then ^-^ — - 

d « 


1 : if Y be less than z, then injll = 1 and so on; con- 

' d y 


sequeiitly, in the first case we have <p = /i = <Pi : 
in the second, 

9 =/, +/2 = ?2, and therefore/^ = (^2 - ^, : 
in the third, 

<P =fi +A+f3 = <P3> and therefore /a = (p^ - ^3. 
It appears therefore that 

is a formula which is competent to express all the required 
conditions of discontinuity *, 

Equivalent forms may be considered as permanent within 
the limits of continuity, and no further, unless the requisite 
signs of discontinuity, whether implicit or explicit, exist upon 
both sides of the sign = : thus, the equation 

4 r ^ 4 6 "I 

•'■D/cos^ = —\ T-^ sin 2 a; + -^^— psin4^ + -=— =sm6a: + &c. \ 

11.0 0.0 0,1 I 

is permanent within the limits indicated by the sign '*'D^° and 
no further, and similarly in most of the cases which have been 
considered above. The imprudent extension of such equivalent 
forms, which has arisen from the omission of the necessary 
signs of discontinuity, has frequently led to very erroneous 
conclusions : thus, the equation 

.r» 1 1 r.r2 - 1 1 r* - 1 1 ^^ - 1 . )t 

-D.« log a: = 2-|^^— I + -J . (^-.qpT)^+ y • (:,qri)a+ &«• \ 

which is true for all values of x between and co , has been 
extended to all values of x between — 00 and + co , and has 
thus been made the foundation of an argument for the identity 
of the logarithms of the same number, both when positive and 

There are two species of discontinuity which we have consi- 
dered above, one of which may be called instantaneous and the 
oi\\ev finite : the first generally accompanies such changes of 
form as are consequent upon the introduction of critical values 

* These formulae would require generally a correction at their limits, in 
order to render them symbolically general. The nature of these corrections 
may in most cases be easily applied from the observations which we have 
made above. 

t This series is given by M. Bouvier in the 14th volume of Gcrgoniie's 
Aimalcs ties Malhemutifjiies. The conclusion referred to in the text assumes 
the identity of the logarithms of x" and of ( — .r)^, which is in fact the wholo 
(jucstion in dispute. 

262 THIRD KEPORT — 1833. 

of the variables, when the corresponding equivalent form no 
longer exists, or when the conditions which determined its exist- 
ence no longer apply ; the second restricts the existence of the 
equivalent form to limits of the variable which have a finite dif- 
ference from each other. In neither case, if we suppose the con- 
ditions of the discontinuity to be implicitly involved, or if we 
suppose the explicit signs of discontinuity to be assumed con- 
formably to the general laws of algebra, can we consider the 
law of the permanence of equivalent forms to be violated. It is 
only when a continuous formula is assumed to be equivalent to 
a discontinuous formula, without the introduction of the requi- 
site sign of discontinuity to limit the extent of the continuous 
formula, that we can suppose this fundamental law to be vio- 
lated or the asserted equation between such expressions to be 
false. Many important errors have been introduced into ana- 
lysis from the neglect of those conditions. 

The identity of the values of powers of 1, whose indices 
are general whole numbers, and also of the sines and cosines 
of angles which differ from each other by integral multiples of 
360°, is a frequent source of error in the generalization of equi- 
valent forms, when the symbols which express those indices or 
multiples are no longer whole numbers. A very remarkable 
example of both these sources of error has occurred in the for- 

{2 cos x)*" = cos 7n {2 r TT + x) + m cos (/« — 2) {2 rv + x) 

-\ \ — ^ — cos {m — 4') {2rTr + x) + ike. 

+ i/^l { sin 7n {2rTf + x) + m sin {m — 2) (2 r if + x) 
-,. '^'1^-^ sin {m - 4) (2 r TT -}- x) + &c. (1.) 

If we suppose m to be a whole number, this equation degene- 
rates into 

{2 cos x)"" = cos mx + m cos {m — 2) x -\ ^j — -^ — -' 

cos (w — 4) X -I- &c. (2.) 

the series first discovered by Euler, and which he assumed to 
be true for all values of m. li, however, we should suppose tn 

to be a fraction of the form —> we should have g values of 

. 9 
the first member of the equation (2.), and only one of the second. 

And if we should confine our attention to the arithmetical 
value (/} of the first, it would not be equal to the second, un- 


less m was a whole number ; for if we should denote the series 
of cosines 

cos {m2rir + x) + w cos (m — 2) (2 r * + .r) + &c., by Cr, 
and the series of sines 

sin m {2rTf + X) + m sin {tn — 2) (2rTr + x) + &c., by Sr, 
we should find, when cos x is positive, 

Cr ^r 

^ "" COS 2 m rir ~ sin 2 m rv' 
and when cos x is negative. 


^ cos m {2 r + \) ir sin w (2 r + 1) ** 

It will follow, therefore, that when r is not a whole number, p 
will be expressible indifferently by a series of cosines or of 
sines, unless cos 2 mrie = or s'm 2m r if =■ 0, when cos x is 
positive, or cos m (2 r + 1) * = 0, or sin ?n {2r + 1) w = 0, 
when cos x is negative. 

In a similar manner, assuming 

1 . 2 ... 6 

r («2-P) 3 

X = n < cos X — ^:j — g — ^ cos'^ j:* 

we shall find 

cos 7« (2 »• TT + x) = cos n ( 2 r + Y ) '^ • ^ 

+ cos (w - 1) (2 r + 4") 'T . X'. 

If we suppose r to be equal to zero, this equation will become 

nif ^ (n — 1)* ^, 
cos n X = cos -^ . X + cos 5 . A', 

which is the form which has been erroneously assigned by La- 
grange* and Lacroixf as generally true for all values of w. 
Many other examples of similar undulating functions, ex- 

* Cakttl den Funciio)'s, chap. xi. 

t TVaite du Caktil Diff. ct lu/erj., torn. i. p. 261. 

204 ^ THIRD REPORT — 1833. 

pressing the various relations between the cosines and sines of 
multiple arcs and the powers of simple arcs, whether ascending 
or descending, have been given by Lagrange * and other writers 
as general, which are either degenerate forms of the coi'rect 
and more comprehensive eqviations, or altogether erroneous. 
Poisson had pointed out some of the inconsistencies to which 
some of these imperfect equations lead, and had slightly hinted 
at their cause and their explanation ; and the discussion of such 
cases became soon afterwards a favourite subject of speculation 
with many writers in the Mathematical Journals of France f and 
Germany % ; but the complete theory and correction of these 
expressions was first given by M. Poinsot in an admirable me- 
moir which was read to the Academy of Sciences of Paris in 
1823, and published in 1825. They form a most remarkable 
example of expressions extremely simple and elementary in 
their nature, which have escaped from the review and analysis 
of the greatest of modern analysts, in forms which were not 
merely imperfect, but in some cases absolutely erroneous. 

The difficulties which have presented themselves in the 
theory of the logarithms of negative numbers, as compared with 
those of the same numbers with a positive sign, have had a 
very similar origin. If we consider the signs of quantities as 
yflc/or,y of their arithmetical values, and if we trace them through- 
out the whole course of the changes which they undergo, we 
shall find many examples of results which are identical when 
considered in their final equivalent forms, but which are not 
in every respect identical when considered with respect to their 
derivation: thus (+ aY is identical with {— a)^, when consi- 
dered in their common result + a^, but not when considered 
with respect to their derivation. Let us now consider their se- 
veral logarithms, the common arithmetical value of the logarithm 
of a being denoted by p : 

Iog(+a)2 = log(l)2a2 = 4r7r \/'~i +2p (1.) 

log (- af = log (- ly^ a^- = (2r + 1)2 nf \/'^l + 2p (2.) 

log a^ = log 1 . o- = 2 /• Tf V' ^ + 2 p (3.) 

It thus appears that the values of log ( + «)^ and log ( — a)- are 
included amongst those of log «^, but not conversely; and also 
that the values of log (+ «)^ and log(— a)'^, the arithmetical 
value being excepted, are not included in each other. 

* Correajjottdence sur I'Ecole Fulyirvlin'iqHo, torn. ii. p. 212. 

■\ In Gergoniie's-yJ«««/pi' des Matlu'inutiqncs, torn. xiv. xv. xvi. xvii. 

I In Creile's Journal fur die rcine uiid aiigewandie Muthemafik. Berlin. 


Again, if we consider — a" as originating from (— 1) (+ a) 
Me shall set 


\og — a"" = (2 r + 2 7)1 r' + I) Tf \/ — I + m p* : 

if we suppose /k = — , r = and r' = — 1, we shall get 

log - x/a =^ p = ^ log « = log i/ a ; 

or the logarithm of a negative quantity will be identical with 
the logarithm of the same quantity with a positive sign. In a 

similar manner, if we suppose 7)i = i~-~, where p is prime to n, 

r' = — n and r = ^^-^^, then 2 r + 2 7n r' + 1=0, and the 

corresponding logarithm of — a™ will coincide with the arith- 
metical logarithm of a"'. We should thus obtain possible loga- 
rithms of negative numbers in those cases in which we should 
be prepared to expect them from the ordinary definition f of 

In the absence of all knowledge of the specific process of de- 
rivation of quantities, such as a'" and — a"*, we should consider 
their logarithms as identical with those of l"", A and (—1) 1 *". A, 
where A is the arithmetical value of a"' : and in considering 
the different orders of logarithms which correspond to the same 
value of a"^ or of — «*", they will be found to diflfer from each 
other by the logarithms of V" and (— I) 1'" only, which are 
2mr'K V — \ and {2 r -\- 2 m r' -{- 1) * -/ — 1 respectively. The 
logarithms in qviestion are Napierian logarithms whose base is e. 
If we should suppose the logarithms to be calculated to any 
other base, we should replace the Napierian logarithms of 1"* 
and (— 1) 1"' by the logarithms of those qviantities (or signs) 
multiplied by the inodiiliis INl : the same remarks will apply to 
such logarithms which have been made with respect to Na- 
. pierian logarithms. 

The question of the identity of the logarithms of the same 
number, whether positive or negative, was agitated between 
Leibnitz and Bernoulli, between Euler and D'Alembert, and 
has been frequently resumed in later times. The arguments in 

* Peacock's Algebra, p. 569- 

t The logarithm being defined to be the index of the power of a given base 

which is equal to a given number, it would follow, since ai = + n, that— ;j 

is eijually the logarithm of + n and — h. The same remark aj plies to all in- 
dices or luyarilhmv which are rational fractions with even dentrainatorb. 

266 THIRD REPORT — 1833. 

favour of the affirmative of this proposition, which were for the 
most part founded upon the analytical interpretation of the pro- 
perties of the hyperbola and logarithmic curve, were not en- 
titled to much consideration, in as much as they were not drawn 
from an analysis of the course followed in the derivation of the 
symbolical expressions themselves and from the principles of 
interpretation which those laws of derivation authorized. A 
very slight examination of those principles, combined with a re- 
ference to those upon which algebraical signs of affection are in- 
troduced, will readily show the whole of the very limited nvmi- 
ber of cases in which such a proposition can be considered to 
be true *. 

* In the 15th volume of the Annales des Mathematujues of Gergonne, there 
is an ingenious paper by M. Vincent on the construction of the logarithmic 
and other congenerous transcendental curves. Thus, if y = e^^ there will be in 
the plane of x y a continuous branch such as is commonly considered, and a 
discontinuous branch corresponding to those negative values of y which arise 
from values of x, which are expressible by rational fractions with even deno- 
minators : thus, if we suppose the line between x z= and a- = 1 to be di- 
vided into an even number 2 p of parts, (where p is an odd number,) the 
values of* will form a series effractions, 

J_ ± 3 2 2p-l 

2p p Zp p 2p 

which have alternately odd and even denominators, and which correspond 
therefore to values of y which are alternately single and double. If we may 
suppose, therefore, a curve to be composed of the successive apposition of 
points, the complete logarithmic curve will consist of two symmetrical 
branches, one above and the other below the axis of x, one of which, in cor- 
responding parts of the curve, will have double the number of points with the 
other. The inferior curve, therefore, may in this sense be considered as dis- 
continuous, being composed of an infinite number of conjugate points, forming, 
in the language of M. Vincent, une h-anche pointillee. 'J he same remark ap- 
plies to other exponential curves, such as the catenary, &c. 

It was objected to this theory of M. Vincent by M. Stein, another writer 
in the same journal, that every fractional index in this interval might be con- 
verted into an equivalent fraction with an even denominator, which would 
give a double possible value of the ordinate, which would be different from 
that given by the fractional index in its lowest terms ; and that consequently 
there would necessarily be a double ordinate for every point of the axis, and 
therefore also a double number, one positive and the other negative, corre- 
sponding to every logarithm. In reply to this objection, it is merely neces- 

m mp m mp 

sary to observe that the values of a" and o"^ or of 1 " and 1 "^ are in every 
respect identical with each other, the n p values in the second case consisting 
merely of p periodical repetitions of those in the first. 

In a paper in the Philosophical Transactions for 1829, Mr. Graves has given 
a very elaborate analysis of logarithmic formulee, and has arrived at some 
conclusions of great generality which it is difficult to reconcile with those 
which have been commonly received. Amongst some others may be men- 
tioned the formula which he has given for the Napierian logarithms of 1, 


Convergency and Divergency of Series. — The subject of di- 
vergent series, their origin, their interpretation and their use 
in analysis, is one of great importance and great difficulty, and 
has been and continues to be the occasion of much controversy 
and doubt. I shall feel it necessary, for such reasons, to notice 
it somewhat in detail. 

If the operations of algebra be considered as general, and 
the symbols vi'hich are subject to them as unlimited in value, 
it will be impossible to avoid the formation of divergent as 
well as of convergent series : and if such series be consi- 
dered as the results of operations which are definable, apart 
from the series themselves, then it will not be very important 
to enter into such an examination of the relation of the arith- 
metical values of the successive terms as may be necessary to 
ascertain their convergency or divergency ; for, under such 
circumstances, they must be considered as equivalent foi'ms 
representing their generating function, and as possessing, for 
the purposes of such operations, equivalent properties. Thus, 
if they result from the division of the numerator of an alge- 
braical fraction by its denominator, then they will ^jrorfwce the 
numerator when multiplied into the denominator or divisor : if 
they result from the extraction of the square or cube root of 
an algebraical expression, then their square or cube will pro- 
duce that expression ; and similarly in other cases, no regard 

, 2r v 

which is not 2 tt v — 1, but ^z /^^., which, thouarh it includes the 

former, is not included by it. It appears to me, however, that there exists a 
fundamental error in the attempt which has been made by Mr. Graves to 
generalize the ordinary logarithmic formulie upon the same principles which 
have been applied by Poinsot to the generalization of the trigonometrical series 
which have been noticed in the text. He assumes / {&) ^= cos 6 + -v^l sin 3 
:= e and makes the series for/(^) and/"' (&), combined with the equa- 

tion/ (x &) — a. value of/ {&f, and therefore/"' / ^ = 2 r 5r + ^, the foun- 
dation of his logarithmic developements : in other words, he makes e '^~' a 
periodic quantity the base of his system of logarithms, an assumption which 
is essential to the truth of the formula/"'/^ = 2 ?• ar -f and to the gene- 
ralization of the series for/"' 6 by means of it; an hypothesis which is al- 
together at variance with our notions of logarithms as ascertained by the ordi- 
nary definition. The logarithms of + 1 and of (+ 1)"' alone, for very obvious 
reasons, can be considered as possessing such a character. 

Though I have felt myself called upon to state my objections to the fun- 
damental principle assumed in this memoir of Mr. Graves, and consequently 
to many of the conclusions which are founded upon it, yet I think it right at 
the same time to observe that it displays great skill and ingenuity in the con- 
duct of the investigations, and is accompanied by many valuable and ori- 
ginal obscrvationb upon the general principles of analysis. 

268 THIRD REPOrwT — 1833. 

being jjaid in such cases to terms which are at an infinite di- 
stance from the origin. 

It is this last condition, which, though quite indispensable, 
is rather calculated to oftend our popular notions of the values 
of series as exhibited in their sums. We speak of series as 
having sums when the arithmetical values of their terms are 
considered, and when the actual expression for the sum of n 
terms does not become infinite when n is infinite, or when, in 
the absence of such an explicit expression, we can show from 
other considerations that its value is finite. In all other cases 
the sex'ies, arithmetically speaking, may be considered as di- 
vergent, and thei'efore as having no sum *, if the word sum be 
used in an arithmetical sense only, as distinguished from gene- 
rating function. 

We are in the habit of considering quantities which are in- 
finitely great and infinitely little as very difl^erently circum- 
stanced with respect to their relation to finite magnitude. We 
at once identify the latter with zero, of which we are accus- 
tomed to speak as if it had a real existence ; but if we subject 
our ideas of zero and infinity to a more accurate analysis, we 
shall find that it is equally impossible for us to conceive either 
one or the other as a real state of existence to which a mag- 
nitude can attain or through which it can pass. But it is the 
relation which magnitudes in their finite and conceivable state 
still bear to other magnitudes in their course of continued in- 
crease or continued diminution, which enables us to consider 
their symbolical relations when they cease to be finite ; and 
whilst quantities infinitely little are neglected as being absorbed 
in a finite magnitude, so likewise finite magnitudes are consi- 
dered as being absorbed in infinity, and therefore neglected 
when considered with relation to it. The principle, therefore, 
of neglecting terms beyond a finite distance from the origin, in 
converging series, is both safe and intelligible, whilst the case is 
very different with respect to the neglect of similar terms in a 
diverging series. Of such series it is said that they have no 
arithmetical sum ; but it may be said in the same sense of all 
algebraical series involving general symbols that they have no 
sum. But it is not the business of symbolical algebra to deal 
with arithmetical values, but with symbolical results only ; and 
such series must be considered with reference to the functions 
which generate them, and the laws of the operations employed 
for that purpose. The neglect, therefore, of terms beyond a 

* This would appear Cauchy's view of the subject : see the 6th chapter 
of his Cours d\biafyse. 


finite distance from the origin would be perfectly safe as far as 
it does not influence the determination of the series from the 
generating function, or the generating function from the series ; 
and it is upon this principle that the practice is both founded 
and justified. A few examples may make this reasoning more 

Let it be required to determine the function which generates 
the series 

a + a x + a x^ + a a^ + &c. (1.) 

Let s be taken to represent this function, and therefore 

s = a + a X + a •r'^ + a a:'^ + &c. 
= a + X {a + a X + a x^ + a x^ + &c. } 
= a + X s : consequently 


s = . 

I — x 

If the arithmetical values of the terms of this series be con- 
sidered, and if x be less than 1, then , is the stim of the 

1 — X 

series : in all other cases it is its generating function. 

We may consider, however, s (whether it expresses a sum or 
a generating function) as identical with *,, s^, s^, &c., in the 
several expi'essions 

s= a + X s^ 

s = a + a X + x^ s^ 

s = a + a X + a x^ + x^ Sg 

s = a + ax + ax'^+ . . . a x^"-^ + x"' s„, : 

for if the number of terms of the series * be expressed by n 
and if ?i be infinite, we must consider *,, s^, Sq, a . . . Sm as abso- 
lutely identical expressions ; for otherwise we must consider an 
infinite as possessing the properties of an absolute number, and 
must cease to regard infinities with finite differences as iden- 
tical quantities when compared with each other. It is for this 
reason that we assume it as a principle that no regard must be 
paid to terms at an infinite distance from the origin, whatever 
their arithmetical values may be. 
The sum of the series 

a — a-\-a — a-\- &c. 
was assigned by Leibnitz, upon very singular metaphysical 
considerations, to be — : the principle just stated would allow 
us to put 

^70 THIRD REPORT — 1833. 

s = a — {a — a + a — a + &c.) 

= a — * ; and therefore s = -r-*, 


* The same principle would show that the equation 

*• = « +/(« +f(a+f{a + . . .))) 

is identical with the equation 

x= a +f (X) ; 
and that 

* = «/(«/ (a/ (a...))) 
is identical with 

X = af(x). 
The example in the text is the most simple case of a class of periodic series, 
the determination of whose sums to infinity has been the occasion of much 
controversy and of many curious researches. The general property' of such 
series is the perpetual recurrence of the same group of terms whose sum is 
equal to zero : thus, if there should be p terms in each group, and if the num- 
ber of terms n = mp + i, their sum would be identical with that of the i first 
terms of the series ; and if we should denote those terms by aj, a.,, . . . o , 
and if we should take the successive values of this sum for all the values of i 
between 1 axidp inclusive, their aggregate value would be represented by 

pai + (p—1) a<^+ (p _ 2) a, + . . . a^. 
of which the average (A) or mean would be represented by 

pay + (p — I) a.2 + (p — 2) as + . . . a^ 

If this periodic series was continued to infinity, it was contended by Daniel 
Bernoulli, in memoirs in the 17th and 18th volumes of Novi Commmtarii 
Petropolitani, for 1772 and 1773, that its sum would be correctly represented 
by the average (A), in as much as it was equally probable that any one of 
the p values would be the true one. Upon this principle it would follow, 
that of the apparently identical series 

1 — 1 + 1 — 1 + 1 — &c. ... 

1+0— 1 + 1 + 0— 1 + 1 + &c. ... 

1-t-O + O— 1 + 1+0 + 0— 1+&C. 

12 3 

the first would be equal to — , the second to — -, and the third to — . In the 

A o 4 

same manner we should find 

1 + 1 — 1 — 1 + 1 + 1 — 1 — 1 +, &c. 
equal to 1, and 

1+1+0 — 1 — 1 + 1 + 1+0— 1 — l+I + l +&C. 

equal to — -. The same observations would apply to the series 

1 + cos X + cos 2 X -\- cos 3 .r + cos 4 a; + &c. 

1 + cos a- + + cos 2 X -\- cos 3 .r + + cos 4 .r + &c. 
where x is commensurable with 2 tt. 

These conclusions, however, though curious and probable, rested upon no 


If we consider this principle of the identity of series, whose 
terms within a finite distance from the origin are identical, as 
established, we shall experience no difficulty in admitting the 
perfect algebraical equivalence of such series, and their gene- 
secure basis founded upon the general principles of analysis, and their truth 
•was not, therefore, generally admitted amongst mathematicians. In the year 
1798, Callet, the author of the logarithmic tables which go by his name, pre- 
sented a memoir to the Institute for the purpose of showing that the sums 
of such periodic series were really indeterminate : thus, if we divide 1 by 
I + X and subsequently make x = 1, we get 

1 — 1+1 — 1 + &C. (1.) 

the value of which is — . In a similar manner, if we divide 1 + a; by 

1 + a; + a;2, we get for the quotient 

1 — x" + x3 — x^ + x^ — x^ + &c., 

which becomes the same series (1.), though the value of the generating func- 

tion under the same circumstances becomes — . The same remark applies 

to the result of the division of 1 + x + x^ + . . x"' by I + x + x- -\- . . x", 
which produces the same series (1.) when a; =: 1, though under such circum- 

stances its generatmg function becomes — . 

This memoir of Callet gave occasion to a most elegant Report upon this 
delicate point of analysis by Lagrange, who justified upon very simple prin- 
ciples the conclusion of Daniel Bernoulli. The series which results from the 
division of 1 + a; by 1 + .r + a^, if the deficient terms be replaced, becomes 

l+O.x — x^ + x^ + O.x'^— a^ + x^ + O.x'^ — x^ + &c., 

which degenerates, when x ■=■ \, into the series 

1+0—1 + 1+0—1 + 1 + &c., 

and not into the series (1.). The same remark applies to the series which 
arises from the division of 1 + a; + . . a;*" by 1 + a; + . . . . x", n "Z m; 
which becomes, when x z= I, 

1+0 + + + &c. — 1+0 + + &c. +1+0 + &c„ 

•which is equal, by Bernoulli's rule, to — . 


But it is not necessary to resort to this expedient for the purpose of deter- 
mining the sums of such series ; for the series 

Oi + On X + 03 a- + . . a x^~ + a, a'' + &c. 

is a recurring series resulting from the developement of 

a, + Oo a; + 03 x- -\- . . a x^" 

1 - xP 
which becomes — when a: = 1. If we replace x by —, this fraction will 

272 THIRD REPORT — 1833. 

rating functions. Foi* the same principle would justify us in 
rejecting remainders after an infinite number of terms, whatever 
their arithmetical values may be ; for such remainders can in- 
fluence no terms at a finite distance from the origin, and there- 
fore can in no respect affect any reverse operation, by which 
it may be required to pass from the series to any expression 
dependent upon the generating function. Thus, if 

1 =a + ax + ax^+ ixc = *, 

we shall get 

a = (I — x) s = a, 

if we reject remainders after an infinite number of terms ; and 
similarly in other cases. It would thus appear that algebraical 
equivalence is not necessarily dependent upon the aritlametical 
equality of the series and its generating function. 

It is, however, an inquiry of the utmost importance to be able 
to ascertain when this arithmetical equality exists ; or, in other 
words, to ascertain under what circumstances we can determine 
the sum of the series, either from our knowledge of the law of 
formation of its successive terms, or approximate, to any required 

eti z^ + a^ z^~ + . . a z 

which becomes by the application of the ordinary rule of the differential cal- 
culus, when s; = 1 or .r = 1, 

p Oi + (p — I) a^ + . . Op 

~ f 

which is the average or mean value determined by Bernoulli's rule. 

The discussion of the values of these periodic series has-been resumed by 
Poisson in the twelfth volume of the Journal de I'Ecole Poly technique. He 
considers them as the limits of these series when considered as converging 
series, a view of their origin and meaning which is almost entirely coincident 
with that of Lagrange. Thus, the gum of the series 

sin q + p sin (x + q) + p" sin (2 .r + q) + &c. 

is equal to 

sin q -{- p sin (x — q) 
1 — 2 p cos X -\- p^ ' 

when|) is less than 1, an expression which degenerates, when^ = 1, into 
— sinq + ~ cos q cot -j' 

which may be considered, therefore, as the limit of the sum of the series 
sin q + sin (.r + ?) -|- sin (2 x + </) + &c. in wfiii. 


degree of accuracy, to its value by the aggregation of a finite 
number of those temns. Many tests of the siimmability of series 
(considered as different from the determination of their gene- 
rating functions,) have been proposed, possessing very different 
degrees of certaintj' and apphcabiUty. The geometrical series 
which we have just been considering is convergent or divergent, 
that is, summable or not, according us x is greater or less than 1 ; 
and it is convenient, for this and for other reasons, to assume it 
as the measure of the convergency or divergency of other series. 
If it can be shown that a converging geometrical series can be 
formed whose terms within a finite and assignable distance from 
the origin become severally greater than those corresponding 
to them of the assigned series, then that series is convergent. 
And if it can be shown that a divergent geometric series can be 
formed whose terms within a finite and assignable distance from 
the origin are severally less than those corresponding to them 
of the assigned series, then that series is divergent*. Such 
tests are certain, as far as they are applicable ; but there may 
be many cases, both of divergent and convergent series, which 
they are not sufficiently delicate to comprehend. 

It would appear from the preceding observations that di- 
verging series have no arithmetical sums, and consequently 

• Peacock's Algebra, Art. 324, and following. Cauchy, Cours d' Analyse 
Algebrique, chap. vi. This last work contains the most complete examination 
of the tests of convergency with which I am acquainted. 

The measure of convergency mentioned in the text, which was first sug- 
gested and applied by D'AIembert, will immediately lead to the following: 

" If u^ represent the m"^ term of a series, it is co?ivergent (or will become so) 

if the superior limit of (ii )" be less than 1, when 7i is infinite; divergent in 

the contrary case." 

" If the limit of the ratio tr^, j to 7/^^ be less than 1, the series is convergent, 
and divergent in the contrary case." 

Many other consequences of these and other tests are mentioned by Cauchy 
in the work above referred to. 

M. Louis Olivier, in the second volume of Crelle's Journal, has proposed the 
following test of convergencj'. " If the limit of the value of the product n u^ 
\)B finite or zero when n is infinite, then the series is divergent in the first case, 
and convergent in the second." This principle, however, though apparently 
very simple and elementary, has been shown by Abel, in the same Journal, to 
be not universally true. Thus, the series 

2 log 2 ^ 3 log 3 ^ 4 log 4 ^ ■ ■ ■ ■ ^ ?j log « 

may be shown to be infinite, though the product n n^ is equal to zero when w is 
infinite. The same acute and original analyst has shown that there is no func- 
tion of n whatever which multiplied into u^ will produce a re.sult which is zero 
or finite when « is infinite, according as the series is convergent or divergent. 

1833. T 

274 THIRD REPORT — 1833. 

admit of no arithmetical interpretation. And it will be after- 
wards made to appear that such series do not include in their 
expression, at least in many cases, all the algebraical conditions 
of their generating functions. Before we proceed, however, to 
draw any inferences from this fact, it may be expedient in the 
first instance to give a short analysis of some of the circum- 
stances in which such series originate. 
The series 

] 1 6 62 

+ -2 + 73 + &C- 

a — b a a^ a^ 

is convergent or divergent according as a is greater or less 
than b. As this series is incapable, from its form, of receiving 
a change of sign corresponding to a change in the relation of 
a and b to each other, it would evidently be erroneous in the 
latter case if it admitted of any arithmetical value, in as much 
as it would then be equivalent to a quantity which is no longer 
arithmetical. In this case, therefore, the series may be replaced 
by the symbol go , which is the proper sign of transition, (see 
page 237,) which indicates a change in the constitution of the 
generating function, of such a kind as to be incapable of being 
expressed by the series which is otherwise equivalent to it. 
The same observations apply to the equation 

r, b n(n-\) b^ 

(a-bf = a'' < 1 - rt — + — ; ^ . -2 

^ ^ l_ a \ . 2 or 

~ 1.2.3 • ^ + ^""'J' 

as we have already stated in our remarks upon signs of transition, 
in page 237. It will be extremely important, however, to examine, 
both in this and in other cases, the circumstances which attend 
the transition from generating functions to their equivalent 
series, in as much as they will serve to explain some difficulties 
which have caused considerable embarrassment. 
The two series 

If, 2b , 3b^ 4 63 -| 

a^ [^ a a'' a'' J 

I If, 2b . 3b'' 4*3 

(« - bf 


J _ If 2 a 3a^ 4 a^ \ 

- af~ b^X '^ b + ~b^ "^ 63 + *'^-/ 


will be divergent in one case, and convergent in the other, 
whatever be the relation of a and b, though they both equally 


braically, as well as arithmetically, equivalent to each other. It 
might be contended, therefore, that in this instance the sign oo , 
which replaces one of the two series, is no indication of a change 
in the constitution of the generating function which is conse- 
quent upon a change of the sign of a — b or b — a. But 
though a^ — 2 a b + b'^is equal to (« — b)^, and b^ — 9, ab -\- c? 
to {b — (if ; and though a^ — 2 a 6 + Z>^ is identical in value 
and signification with b"^ — 2 ab + a^ when they are considered 
without reference to their origin, yet we should not, on that 
account, be justified in considering (a — hf and {b — af as 
algebraically identical with each other. The first is equal to 
(+ 1)^ (a — by, and the second to (— 1)^ (« — hf ; or the first 
to (- \f {b — af, and the second to (+ 1)^ {b — af. But the 
signs (+1)^ and (—1)^ are not algebraically identical with each 
other, though identical when considered in their common result, 
in as much as their square and other roots and logarithms are 
different from each other*. It follows, therefore, that there is 

a symbolical change in the quantity denoted by -; j-^ in its 

passage through infinity, which is indicated by the infinite value 
of the equivalent series, in as much as it is not competent to ex- 
press, in its developed form, the algebraical change which its 
generating function has undergone. The same remarks will 
apply to the series for {a — 6)" and (b — a)", in all cases in 
which w is a negative even number. When w is a negative odd 
number, the change of constitution of the generating function 
is manifest, and requires no explanation. 
The two series 


If, b b^ b^ b^ „ -1 
a L « « a a* J 

1 _ 1 /, a a^ a^ a'' . \ 

correspond to the same generating function, though one of 
them is divergent, and the other convergent. But the divergent 
series, whose terms are alternately positive and negative, cannot 
be replaced by the symbol oo , in as much as it does not indicate 

• Thus, if a denote a line, (+ af and (— «)« can only be considered as 
identical in their common result a'. When (+ a)2 and ( — a)' are considered 
with reference to each other, they are not identical quantities, though equal to 
each other. 

T 2 

^76 THIRD REPORT 1833. t 

any change in the constitution of the generating function. They 
may both of them, therefore, be considered as representing the 
vahie of this function, though in one case only can we approxi- 
mate to its arithmetical value by the aggregation of any number 
of its terms *. 

Similar observations would apply to the series 

/ , i\n « r I n b « (w — 1 ) 6^ . o ~1 
(a + 5)" = «» |l -f - + -hV-^ + ^^-j 

when n is not a positive whole number. In all such cases, the 
developement will sooner or later become a series, whose terms 
are alternately negative and positive, and which will be di- 
vergent or convergent, according to the relation of a and b to 
each other. More generally we might assume it as a general 
proposition, " that divergent series which correspond to no 
change in the constitution of the generating function, will have 
their terms or groups of terms alternately positive and nega- 
tive :" and conversely, " that divergent series which correspond 
to a change in the constitution of the generating function, will 
have all their terms or groups of terms affected with the same 
sign, whether -f- or — , and the whole series may be replaced 
by the symbol oo ." 

In both these propositions the change of which we speak is 
that which corresponds to those values of the symbols which 
convert the equivalent series from convergency to divergency, 
and conversely. 

I am not aware of any proof of the truth of these important 
propositions which is more general than that which is derived 
from an induction founded upon an examination of particular 
cases. But such or similar conclusions might be naturally ex- 
pected to follow from the fundamental principles and assump- 
tions of symbolical algebra. If the rules of algebra be perfectly 
general, all symbolical conclusions which follow from them must 
be equally true : and those rules have been so assumed, that 
when the symbols of algebra represent ai'ithmetical quantitiesj 
the operations with the same names represent arithinetical 
operations, and become symbolical only when the correspond- 
ing arithmetical operations are no longer possible. It will be 
essential, therefore, to the perfection of algebraical language 
that it should be competent to express fully its own limitations. 

* The equations s = and « = -r 7- will equally give us 

s ;= in one case, and s = r in the other, whatever be the relation 

, a + b a + 

of and b. 


Such limitations will be conveyed by the introduction of signs 
of affection, of signs of transition, or of signs of discontinuity, 
which may be involved either implicitly or explicitly. It is for 
such reasons that all those signs must be considered in the 
interpretation of algebraical formulie, and their occurrence will 
at once suggest the necessity of such an examination of the 
circumstances of their introduction as may be required for their 
correct explanation*. 

We thus recognise two classes of diverging series, which are 
distinct in their origin and in their representation. The first 
may be considered as involving the symbol or sign co implicitly, 
and as capable, therefore, of the same interpretation as we give 
to the sign when it presents itself explicitly. The second re- 
presents finite magnitudes, which in their existing form are 
incapable of calculation by the aggregation of any number of 
their terms. Such series are in many cases capable of trans- 
formations of form, which convert them into equivalent con- 
verging series ; and in some cases, where such a transformation 
is not practicable, or is not eflPected, tlie approximate values of 
the generating functions may be determined, from indirect con- 
siderations, supplied by very various expedients. 

The well known transformation of the series 

ax — ba:"' + cx^ — dx^-\-ex^— fx^ + &c., 
which Euler has given f, into the equivalent series 

f "v** y'^ "j^ 

a — Ti—. — -To .^a + T^i— — To .A^a — 71:—. — vt • ^^ a + &c. 

1 + x"' (l + x)2-""'-(l + ^)3-- " {l + xy 

would be competent to convert a great number of divergent 
series of the second class into equivalent convergent series, or 
into such as would become so. In this manner the Leibnitzian 

1 - 1 -h 1 - 1 -f &c. 

may be shown to be equal to -^. The series 

1 - 3 + 6 - 10 + 15 - 21 + &c. 

♦ The essential character ot arithmetical division is that the quotient should 
approximate continual)}' to its true value, and that the terms of the quotient 
which are introduced by each successive operation should be less and less con- 
tinually. In the formation, therefore, of the quotient of j and , 

the analogy between the arithmetical and algebraical operation would cease to 
exist, unless a was greater than b, or unless the several terms in the quotient 
went on diminishing continually. 

f Jnstitulioncs Calculi Dlfferenlialis, Pars posterior, cap. i. 

278 THIRD REPORT— 1833. 

of triangular numbers to -5-. The series 

1 - 4 + 9 - 16 + 25 - &c. 
of square numbers to 0. The series of tabular logarithms 
log 2 — log 3 + log 4 — log 5 + &c., 

would be found to be equal to •0980601 nearly. If we should 
suppose X negative and greater than 1, the original and the 
transformed series would become divergent series of the first 

The series 

, ,,,(«- \f {a -If {a- \y , 5 

log a = (a - 1) - ^ g ^ + ^ 3 ^ - —^ + &C-- 

is divergent when a is greater than 2, and convertible by Euler's 
formula into the convergent series 

(«-l) , 1 (« - If ,1 {a-\f \ {a- \f . 

or by the method of Lagrange into the series 

„ (^a _ 1) _ |. (^a _ 1)2 + |- (^_ 1)3 _ &c., 

which may be made to possess any required degree of con- 
vergency. But it is not necessary to produce further examples 
of such transformations, which embrace a very great part of the 
most refined artifices which have been employed in analysis. 

One of the most remarkable of these artifices presents itself 
in a series to which Legendre has given the name of demicon- 
vergent*. The factorial function 1^(1 + a:) is expressed by the 
continuous expression 

(^) '(2*a:)*R, 

where R is a quantity whose Napierian logarithm is expressed 

A B_ C 

\ .2.x 3 . 4 . x2 ''■ 5 . 6 . ^i" - &c-> 

where A, B, C, &c., are the numbers of Bernoulli. The law of 
formation of these numbers, as is well known, is extremely 

• Fonetions ElUptiques, torn. ii. chap. ix. p. 425. 


irregular, and after the third term they increase with great 
rapidity. The series under consideration, therefore, even for 
considerable values of s, becomes divergent after a certain 
number of terms. But an approximate value of the series will 
be obtained from the aggregation of the convergent terms only : 
and it has been proved by a German analyst* that the error 
which is thus made in the value of the generating function will 
in this case be less than the last of the convergent, or the first 
of the divergent, terms. 

It has been usual amongst some later mathematicians of the 
highest rank to denominate diverging series, without any di- 
stinction of their class, as false, not merely when arithmetical 
values are considered, but also when employed as equivalent 
forms, in purely symbolical processes. The view of their ori- 
gin and nature which we have taken above would explain the 
sense in which they might be so considered in relation both to 
arithmetical processes and to the calculation of arithmetical 
values. It seems, however, an abuse of terms to apply the term 
false to any results which necessarily follow from the lavys of 
algebra. M. Poisson, perhaps the most illustrious of living 
analysts, has referred, in confirmation of this opinion, to some 
examples of erroneous conclusions produced through the me- 
dium of divergent series f ; and as the question is one of great 
importance and of great difficulty, I shall venture to notice them 
in detail. 

Let it be required to express the value of 

»+i d X 


{{\-2ax + a^){\ -2bx + b^)Y 


by means of series. 

Assuming K = (1 - 2 « x + «2)- 4 and K' = (1 - 2 5 r + b^ *, 
let us suppose K and K' developed according to ascending and 
descending powers of a and b respectively ; or, 

K = 1 + a X, + «'■' Xg f a^ X3 + &c. 

K' = 1 + 6 Xj + 62 Xg + b^ X3 + &c. 

K' = l + ^Xi + lx,+ lx3+&c. 

* Erchinger in Schrader's Commentatio de Siimmatione Seriei, Src. Weimar 
t Journal de I'Ecole Polylechnique, torn. xii. 

280 THIRD REPORT 1833. 

The coefficients Xj, Xj, Xg, &c., are reciprocal* functions, pos- 
sessing the following remarkable property, that / X„ X„ rf .r 

= 0, in all cases, unless n = m, in which case / X„ X„ </ a; 

1 ^-' 

~2n + r . 

The knowledge of this property will readily enable us to de- 
termine the following four different values of s : 

«1 = 1 + -g- + -^ + -y- + &C. 

1 (J/ fll ft 

"^2 ^ T + ST^ "^ 5^ "*■ 76^ + ^''• 

% = — + Q^2 + K-73 + i?^ + &C. 

"4 — 

a ' 3 a2 -1^ 5 a3 ^ 7 a^ 

1 1 _1_ _1_ 

a 6 ^ 3 a2 62 -T- 5 ^3 ^3 -T^ ,j, ^4 ^4 

Whatever be the relation of a and b to each other and to 1 , 
two of these four series are convergent, and two of them di- 
vergent. But it appears from the examination of the finite in- 

tegral / K K' c? a:, that one only of these two convergent 

series gives the correct value of z, being that which arises from 
the combination of the two convergent developements of K and 
K"*, whilst the incorrect value arises from the combination of a 
convergent developement of K with a divergent developement 
of K', or conversely. The conclusion which is drawn from this 
fact is, that the introduction of the divergent developement of 
K or of K' vitiates the corresponding value of z, even though 
that value is expressed by a convergent series. Let us now 

/» + ' 
examine how far the definite integral of / K K' c? a: will jus- 
tify such an inference. 

If we denote K K' by — , we shall easily find, 

* Functions which possess this property have been denominated reciprocal 
functions by Mr. Murphy, in a second memoir on the Inverse Method of Defi- 
nite Integrals, in the fifth vohnneof the Transactions of the Philosophical Society 
of Cumhridge, in which general methods are given for discovering all species of 
such functions, and where one very remarkable form of them is assigned. The 
functions referred to in the text were first noticed by Legendre, in his first me- 
moir on the Attraction of Ellipsoids, and subsequently, at great length, in the 
Fifth Part of his Exercices du Calcul Integral. Cauchy has used the term recipro- 
ea/ function in a different sense; see Exercices des Mathematiques, torn. ii. p. HI. 


p-^d^^ _^=.log {P-P- + 9,p ^Th) + const.; 

and if we denote by r and r' the extreme values of p, when a: 
= — 1 and X — + 1, we shall find, 

r + ^dx _ 1 ^ 2r\/ab + ^ab-{a + h){\ + ah )'} 

^~J -\ "y ~ 4 s/ah ^^i2r^ //ah-^ah- (a + 6)(l+a6)y 

inasmuch as^-^ is 4 a 6 — (« + A) (1 + « 6) in one case, and 
a X 

— A^ah — {a + h){\ + « 6) in the other. It will appear like- 
wise that r and r' will have the same sign, whether + or — , 
in as much as p will preserve the same sign throughout the whole 
course of the integration. If, therefore, r' = + (1 + «) (1 + a), 
then r = + (1 - a) (1 - 6) ; and if / = - (1 + a) (1 + h\ 
then r = — (1 — a) (1 — h). It thus appears that (1 — «) (1 — h) 
must have the same sign with (1 + a) (1 + 6), and consequently 
if a 7 1, and h 7 1, we shall have, 

. = _L^ loff (<?-!) (^> - 1) j/^ + 4a 6 - (a 4- 6) (1 + «6) 
4 ./a 6 (a + l)(6 + l) v'"«6-4a6-(« + 6)(l +«*) 

= - — =- log . Iv «_ + _j_ (stj.i]jinrr out the common divisor 

_ 1 - a/« 6 + 1 , 

2^ah-a-h)^ ^^^ log ^^T^^^^Y = ± -4- 

If a 2: 1 and 6 l\, we shall find r = (1 - a) (1 - h), and 

« = === log I 7= ) = ± «i. 

2 A/a 6 ^ Vl - A/a 6/ 

If o Z 1 and h 7 1, we shall find r = (1 — a) (1 — h), and 

1 , ( Vh + a/«^ 

X = 

1 ( Vh + A/a\ , 

log ( ~T ^ ) = + ^2. 

2 -/afi 
If a 7 1 and 6 Z 1, we shall find r = (a — 1) (1 — h), and 

2 \/ 

It would thus appear that the definite integral would furnish 
erroneous values of z if no attention was paid to those values 
of the factors of r and »•', which the circumstances of the inte- 
gration require : and it may be very easily shown that an atten- 
tion to the developements of K and K' will, with equal certainty, 
enable us to select the proper devclopement for ^r. Thus, if a 7 I 

282 THIRD REPORT— 1833. 

and 6 7 1, we have r = {a — 1) (b — I) = 

and the value of s (z^) is determined by the combination of the 
two last developements. In a similar manner, if a Z 1 and 6 Z 1, 
SI (^i) will be formed by the combination of the two first. If 

« Zl and 6 7 1, then r = (1 - «) (6 - 1) = ^~ ^— ^ : 

and the value of s (^g) is formed by the combination of the first 
and third developement. And if a 7 1 and 6 Z 1, then the value 
of s (sg) will be formed by the combination of the second and 
third developements : in other words, the selection of the de- 
velopements is not arbitrary, in as much as {{l — af}-i and 
{(a — 1)*}~* ought not to be considered, as we have already 
shown, as identical quantities. 

These combinations of the convergent and divergent series 
form all the four values of s, of which it appears that one value 
alone is correct for any assigned relation of a and b to \, being 
that which arises from the combination of the convergent series 
for K and K' only. The considerations, however, which deter- 
mine the selection of the correct developement of z are as de- 
finite and certain when the general series are employed as when 
that value is determined directly from the definite integral 
which expresses the value of ;£. It would appear to me, there- 
fore, that not only was the employment of divergent series 
necessary for the determination of all the values of z, but that 
when the theory of their origin is perfectly vmderstood they 
are perfectly competent to express all the limitations which are 
essential to their usage. The attempt to exclude the use of 
divergent series in symbolical operations would necessarily im- 
pose a limit upon the universality of algebraical formulae and 
operations which is altogether contrary to the spirit of the 
science, considered as a science of symbols and their combina- 
tions. It would necessarily lead to a great and embarrassing 
multiplication of cases ; it would deprive almost all algebraical 
operations of much of their certainty and simplicity ; and it 
would altogether change the order of the investigation of results 
when obtained, and of their interpretation, to which I have so fre- 
quently referred in former parts of this Report, and upon which 
so many important conclusions have been made to depend. 

Elementary Works on Algebra. — There are few tasks the 
execution of which is so difficult as the composition of an ele- 
mentary woi'k ; and very few in which, considering the immense 
number of such works, complete success is so rare. They re- 
quire, indeed, a union of qualities which the class of writers 
who usually undertake such works are not often competent to 


furnish. Great simplicity in the exposition and exempHfication 
of first principles, a perfect knowledge of the consequences to 
which they lead, and great forbearance in not making them an 
occasion for the display of the peculiar opinions or original re- 
searches of their authors. 

There is, in fact, only one elementary work which is entitled 
to be considered as having made a very near approach to per- 
fection. The Elements of Euclid have been the text-book of 
geometers for two thousand years ; and though they labour 
under some defects, which may or may not admit of remedy, 
without injury to the body of the work, yet they have not re- 
ceived any fundamental change, either in the propositions them- 
selves, or in their order of succession, or in the principles of 
their demonstrations, in the propriety of which geometers of 
any age or country have been found to acquiesce. It is true that 
both the objects and limits of the science of geometry are per- 
fectly defined and understood, and that systems of geometry 
must, more or less, necessarily approach to a common arrange- 
ment, in the order of their propositions, and to common prin- 
ciples as the bases of their demonstrations. But even if we 
should make every allowance for the superior simplicity of the 
truths to be demonstrated, and for the superior definiteness of 
the objects of the science to be taught, and also for the superior 
sanction and authority which time and the respect and accept- 
ance of all ages have assigned to this remarkable work, we may 
well despair of ever seeing any elementary exposition of the prin- 
ciples of algebra, or of any other science, which will be entitled to 
claim an equal authority, or which will equally become a model to 
which all other systems must, more or less, nearly approximate. 

There are great difficulties in the elementary exposition of 
the principles of algebra. As long as we confine our attention 
to the principles of ai'ithmetical algebra, we have to deal with 
a science all whose objects are distinctly defined and clearly un- 
derstood, and all whose processes may be justified by demon- 
strative evidence. If we pass, however, beyond the limits which 
the principles of arithmetical algebra impose, both upon the re- 
presentation of the symbols, and upon the extent of the opera- 
tions to which they are subject, we are obliged to abandon the 
aid which is afforded by an immediate reference to the sensible 
objects of our reasoning. In the preceding parts of this Report 
we have endeavoured to explain the true connexion between 
arithmetical and symbolical algebra, and also the course which 
must be followed in order to give to the principles of the latter 
in their most general form such a character as may be adequate 
to justify all its conclusions. But the necessity which is thus 

284 THIRD REPORT — 1833. 

imposed upon us of dealing with abstractions of a nature so 
complete and compreliensive, renders it extremely difficult to 
give to the principles of this science such a form as may bring 
them perfectly ^vithin the reach of a student of ordinary powers, 
and which have not hitherto been invigorated by the sevei'e dis- 
cipline of a course of mathematical study. 

The range of the science of algebra is so vast, and its appli- 
cations are so various, both in their objects and in their degrees 
of difficulty, that it is quite impossible to fix absolutely the 
proper proportion of space which should be assigned to the 
developement of its difterent departments. If a system of al- 
gebra could be confined to the statement of fundamental prin- 
ciples, and to the establishment of fundamental propositions 
only, it might be possible to approximate to a fixed standard, 
which should possess the requisite union of simplicity and of 
sufficient generality. But it is a science which cannot be taught 
by an exposition of principles and their general consequences 
only, but requires a more or less lengthened institution of ex- 
amples of many of its different applications, in order to produce 
in the student mechanical habits of dealing with symbols and 
their combinations. The extent also to which such develope- 
ments are necessary will vary greatly with the capacities of dif- 
ferent students, and it would be quite impossible to determine 
any just mean between diffuseness and compression which shall 
be best adapted to the wants of the general average of students, 
or to the systems of instruction followed by the general average 
of teachers. 

In the early part of the last century the Algebra of Maclaurin 
was almost exclusively used in the public education of this 
country. It is unduly compressed in many of its most essential 
elementary parts, and is also undvdy expanded in others which 
have reference to his own discoveries. It was written, however, 
in a simple and pure taste, and derived no small part of its 
authority as a text-book from the great and well-merited repu- 
tation of its author. It was subsequently, in a great measure, 
superseded, in the English Universities at least, by the large 
work of Sanderson, which was composed by this celebrated 
teacher to meet the wants of his numerous pupils. It was, in 
consequence, swelled out to a very unwieldy size by a vast 
number of examples worked out at great length ; and it laboured 
under the very serious defect of teaching almost exclusively 
arithmetical algebra, being far behind the work of Maclaurin 
in the exposition of general views of the science. At the latter 
end of the last century Dr. Wood, the present learned and 
venerable master of St. John's College Cambridge, in conjunc- 


tion with the late Professor Vince, undertook the pubHcation 
of a series of elementary works on analysis, and on the appU- 
cation of mathematics to different branches of natural philo- 
sophy, principally with a view to the benefit of students at the 
Universities. The works of the latter of these two writers have 
already fallen into very general neglect, in consequence partly 
of their want of elegance, and partly in consequence of their 
total unfitness to teach the more modern and improved forms 
of those different branches of science. But the works of his 
colleague in this undertaking have continued to increase in 
circulation, and are likely to exercise for many years a consi- 
derable influence upon our national system of education ; for 
they possess in a very eminent degree the great requisites of 
simplicity and elegance, both in their composition and in their 
design. The propositions are clearly stated and demonstrated, 
and are not incumbered with unnecessary explanations and 
illustrations. There is no attempt to bring prominently forward 
the peculiar views and researches of the author, and the dif- 
ferent parts of the subjects discussed are made to bear a proper 
subordination to each other. It is the union of all these qua- 
lities which has given to his works, and particularly to his 
Algebra, so great a degree of popularity, and which has se- 
cured, and is likely to continue to secure, their adoption as 
text-books for lectures and instruction, notwithstanding the 
absence of very profound and philosophical views of the first 
principles, and their want of adaptation, in many important 
particulars, to the methods which have been followed by the 
great continental writers. 

In later times a great number of elementary works on algebra, 
possessing various degrees of merit, have been published. 
Those, however, which have been written for purposes of in- 
struction only, without any reference to the advancement of 
new views, either of the principles of the science, or to the ex- 
tension of its applications, have generally failed in those great 
and essential requisites of simplicity, and of adequate, but not 
excessive, illustration, for which the work of Dr. Wood is so 
remarkably distinguished ; whilst other works, which have pos- 
sessed a more ambitious character, have been generally devoted 
too exclusively to the developement of some peculiar views of 
their authors, and have conseqviently not been entitled to be 
generally adopted as text-books in a system of academical or 
national education. There are, however, many private reasons 
which should prevent the author of this Report from enlarging 
upon this part of his subject, who is too conscious that there 
are few defects which he could presume to charge upon the 

^86 THIRD REPORT— 1833. 

works of other authors from which he could venture to exempt 
his own. 

The elementary works on algebra and on all other branches 
of analytical and physical science which have been published 
in France since the period of the Revolution, have been very 
extensively used, not merely in this country, but in almost 
every part of the continent of Europe where the French lan- 
guage is known and understood. The great number of illus- 
trious men who took part in the lectures at the Normal and 
Polytechnic Schools at the time of their first institution, and 
the enlarged views which were consequently taken of the prin- 
ciples of elementary instruction and of their adaptation to the 
highest developement of the several sciences to which they 
lead, combined with the powerful stimulus given to the human 
mind in all ranks of life, in consequence of the stirring events 
which were taking place around them, at once placed the scien- 
tific education of France immensely in advance of that of the 
rest of Europe. The works of Lagrange, particularly his Calcul 
des Fonctions and his Theorie des Fonctions Analytiqiies, which 
formed the substance of lectures given at the Ecole Polytech- 
nique, exhibited the principles of the differential and integral 
calculus in a new light, and contributed, in connexion with his 
numerous other works and memoirs, which are unrivalled for 
their general elegance and fine philosophical views, to fami- 
liarize the French student with the most perfect forms and 
with the most correct and at the same time most general prin- 
ciples of analytical science. The labours of Monge also, upon 
the application of algebra to geometry, succeeded in bringing 
all the relations of space, with which every department of na- 
tural philosophy is concerned, completely under the dominion 
of analysis *, and thus enabled their elementary and other 
MTiters to exhibit the mathematical principles of every branch 
of natural philosophy under analytical and symmetrical forms. 
Laplace himself gave lectures on the principles of arithmetic 
and of algebra, which appear in the Stances de V Ecole Nor- 
male and in the Journal de V Ecole Polytechniqiie ; and there are 
very few of the illustrious men of science, of that or of a subse- 
quent period, who have done so much honour to France, who 
have not been more or less intimately associated with carrying 

* The developement of the details of this most important branch of analy- 
tical science, which has been so extensively and successfully cultivated in 
France, is greatly indebted to Monge's pupils in the Polytechnic School, 
many of whom have subsequently attained to great scientific eminence : their 
results are chiefly contained in the three volumes of Correspondance sur 
I' Ecole Polytechnique. 


on the business of national education in its highest departments. 
The influence of such men has been felt not merely in the very 
general diffusion of scientific knowledge in that great nation, 
but also in the form and character of their elementary books, 
which are generally remarkable for their precision and clear- 
ness of statement, for their symmetry of form, and for their 
adaptation to the most extensive developement of the several 
sciences upon which they treat. 

The elementary works of M. Lacroix upon almost every de- 
partment of analytical science have been deservedly celebrated : 
they possess nearly all the excellences above enumerated as 
characteristic of French elementary writers, and they are also 
remarkable for the purity and simplicity of the style in which 
they are written *. The Cours des Math^matiques Pures of 
M. Francoeur possesses merits of a similar kind, being too 
much compressed, however, for the purposes of self-instruction, 
though well adapted to foi'm a basis for the lectures of a teacher. 
The works of M. Garnier are chiefly valuable for their careful 
illustration of, and judicious selection from, the writings of 
Lagrange, and are well calculated to make the general views 
and principles of that great analyst and philosopher familiar 
to the mind of a student, li^he Arithmetic, Algebra-f, and Appli- 
cation of Algebra to Geometry, of M. Bourdon are works of 
more than ordinary merit, and present a very clear and fully 
developed view of the elements of those sciences. Many other 
works have been published of the same kind and with similar 
views by Reynaud, Boucharlat and other writers. 

I am too little acquainted with the elementary works which 
are used in the different Universities of Germany to be able to 
express any opinion of their character. Those which I have seen 
have been wanting in that precise and symmetrical form which 
constitutes the distinguishing merit of the French elementary 
writers ; but they are generally copious, even to excess, in their 
examples and illustrations. The immense developement which 
public instruction, in all its departments, has received in that 
country would lead us to conclude that they possess elementary 
mathematical works, which are at least not inferior to those which 

• Before the Revolution, the Cours des Mathematiques Pures et Appliquees 
of Bezout, in six volumes, was generally used in public education in France : 
it is a work much superior to any other publication of that period of a simi- 
lar kind which was to be found in any European language. 

t A part of the Algebra of Bourdon has been translated and highly com- 
mended by Mr. De Morgan, a gentleman whose philosophical work on Arith- 
metic and whose various publications on the elementary and higher parts of 
mathematics, and particularly those which have reference to mathematical 
education, entitle his opinion to the greatest consideration. 

288 THIRD REPORT — 1833. 

exist in other languages : and the labours of Gauss, Bessel, and 
Jacobi, and the numerous and important memoirs which appear 
in their public Journals and Transactions upon the most difficult 
questions of analysis and the physical sciences, sufficiently show 
that the mathematical literature of this most learned nation is 
not less diligently and successfully cultivated than that which 
belongs to every other department of human knowledge. 

The combinatorial analysis, which Hindenburg first intro- 
duced, has been cultivated in Germany with a singular and 
perfectly national predilection *; and it must be allowed that it 
is well calculated to compress into the smallest possible space 
the greatest possible quantity of meaning. In the doctrine of 
series it is also frequently of great use, and enables us to ex- 
hibit and to perceive relations which would not otherwise be 
easily discoverable. Without denying, however, the advantages 
which may attend either the study or the use of the notation of 
the combinatorial analysis, it may be very reasonably doubted 
whether those advantages form a sufficient compensation for 
the labour of acquiring an habitual command over the use and 
interpretation of a conventional symbolical language, which is 
necessarily more or less at variance with the ordinary usage and ' 
meaning of the symbols employed and of the laws of their com- 
binations. These objections would apply, if such a conven- 
tional use of symbolical language was universally adopted and 
understood ; but they acquire a double force and authority, 
when it appears that they are only partially used in the only 
country f in which the combinatoi'ial analysis is extensively 
cultivated, and that, consequently, those works in which it is 
adopted are excluded from general perusal, in consequence 
of their not being written in that peculiar form of symbolical 
language with which our mathematical associations ai'e indis- 
solubly connected. 

Trigonometry. — The term Trigonometry sufficiently indicates 
the primitive object of this science, which was the determina- 
tion, from the requisite data, of the sides and angles of trian- 
gles : it was in fact considered in a great degree as an inde- 

* See Eytelwein's Grundlehre der hohern Analysis, avery voluminous work, 
which contains the principal results of modern analysis and of the theory of 
series exhibited in the language and notation of this analysis. 

\ Professor Jarrett, of Catherine Hal], Cambridge, in some papers in the 
Transactions of the Philosojihical Society of Cambridge, and in a Treatise on 
Algebraical Developement, has attempted to introduce the use of the lan- 
guage of the combinatorial analysis. The great neglect, however, which has 
attended those speculations, which are very general and in some respects 
extremely ingenious, is a sufficient proof of the difficulty of overcoming those 
mathematical habits which a long practice has generated and confirmed. 


pendent science, and not as auxiliary to the application of al- 
gebra to geometry. It is to Euler* that we are indebted for 
the emancipation of this most important branch of analytical 
science from this very limited application, who first introduced 
the functional designations sin z, cos z, tan ^, &c., to denote the 
sine, cosine, tangent, &c., of an arc ^, whose radius is 1, which 
had previously been designated by words at length, or by simple 
and independent symbols, such as a, b, s, c, t, &c. The intro- 
duction of this new algorithm speedily changed the whole form 
and character of symbolical language, and greatly extended 
and simplified its applications to analysis, and to every branch 
of natural philosophy. 

The angles which enter into consideration in trigonometry 
are generally assumed to be measured by the arcs of a cn-cle 
of a given radius, and their sines and cosines are commonly de- 
fined with reference to the determination of these arcs, and not 
with refei'ence to the determination of the angles which they 
measure. It is in consequence of this defined connexion of 
sines and cosines with the arcs, and not immediately with the 
angles which they measure, that the radius of the circle upon 
which those arcs are taken must necessarily enter as an element 
in the comparison of the sines and cosines of the same angle 
determined by different measures : and though they Vvcre ge- 
nerally, at least in later writers, reduced to a common standard, 
by assuming the radius of this circle to be 1, yet formulae were 
considered as not perfectly general unless they were expressed 
with reference to any radius whatsoever-}-. In the application, 
likewise, of such formulae to the business of calculation, the 
consideration of the radius was generally introduced, producing 
no small degree of confusion and embarrassment ; and even in 
the construction of logarithmic tables of sines and cosines the 

• Introductio in Analysirn Injinitoruw, vol. i. cap. viii. " Quemadmodum 
logarithmi peculiarem algorithmum vequinint, cujus in universa analyst summiis 
extat usus, ita quantitates circulares ad certain quoque algorithmi normam 
perduxi : ut in calculo aeqiie commode ac logarithmi et ipsae quantitates alge- 
braicae tractari possent." — Extract from Preface. 

t We may refer to Vince's Trigonometry, a work in general use in this 
country less than a quarter of a century ago, and to other earlier as well as 
contemporary writers on this subject, for examples of formulse, which are uni- 
formly embarrassed by the introduction of this extraneous element. Later writers 
have assumed the radius of the circle to be 1, and have contented themselves 
with giving rules for the conversion of the resulting formulae to those which 
would arise from the use of any other radius. It is somewhat remarkable that 
the elementary writers on this subject should have continued to encumber their 
formulae with this element long after its use had been abandoned by Euler, 
Lagrange, Laplace, and all the other great and classical mathematical writers 
on the Continent. 

1833. u 

290 THIRD REPORT — 1833. 

occurrence of negative logarithms was avoided by a fiction, 
which supposed them to be the sines and cosines of arcs of a 
circle whose radius was 10'". 

A very slight modification of the definition of the sine and 
cosine would enable us to get rid of this element altogether. 
In a right-angled triangle, the ratio of any two of its sides will 
determine its species, and conse- 
quently the magnitude of its angles. 
If we suppose, therefore, a point P 
to be taken in one (A C) of the two 
lines A C and A B containing the 
angle BAG (5), and P M to be 
drawn perpendicular to the other 
line (A B), then we may define the 

. PM 

sine of fl to be the ratio . p , and the cosine of 9 to be the 

ratio -J— Ti. By such definitions we shall make the sine and 
A P •' 

cosine of an angle depend upon the angle itself, and not upon 
its measure, or upon the radius of the circle in which it is taken : 
and upon this foundation all the formulae of trigonometry may 
be established, and their applications made, without the neces- 
sity of mentioning the word radius*. 

If we likewise assume the ratio of the arc which subtends an 
angle to the radius of the circle in which it is taken, and not 
the arc itself, for the measure of an angle, we shall obtain a 
quantity which is independent of this radius. In assuming, 
therefore, the angle 9 to be not only measured, but also repre- 
sented by this ratio, we shall be enabled to compare sin & and 
cos Q directly with 9, and thus to express one of them in terms 
of the other. It is this hypothesis which is made in deducing 
the exponential expressions for the sine and cosine, and the 
series which result immediately from them-j-. 

* See A Syllabus of a Course of Lectures upon Trigonometry, and the Appli- 
cation of Algebra to Geometry, published at Cambridge in 1833, in which all 
the formulae of trigonometi-y are deduced in conformity with these definitions. 

t If we should attempt to deduce the exponential expressions for sin 6 and 
cos 6 from the system of fundamental equations, 

cos" d -\- sin^ ^ = 1 (1.) 

cos 6 = cos (— 6) (2.) 

sin ^ = — sin (— 6) (3.) 

we should find, 

aoaAH , g-A('\/^ gA^A/irT_g-A^v'irT 

COS 6 = ^^^— ' and sin ^ = ;;== 

2 2 V — 1 

in which the quantity A, in the absence of any determinate measure of the 


The sines and cosines and the measures of angles defined 
and determined as above, are the only essential elements in a 
system of trigonometry, and are sufficient for the deduction of 
all the important formulas which are required either in algebra 

angle 6, would be perfectly indeterminate. It is the assumption of the measure 
of an angle which is mentioned in the text which makes it necessary to re- 
place A by 1. 

The knowledge of the exponential expressions for the sine and cosine would 
furnish us immediately with all the other properties of these transcendents. 
Thus, if the sines and cosines of two angles be given, we can find the sines 
and cosines of their sum and difference ; and from hence, also, we can find 
the sine and cosine of any multiple of an angle from the values of the sine 
and cosine of the simple angle ; and also through the medium of the solution 
of equations the sine and cosine of its subraultiples. In fact, as far as the 
symbolical properties of those transcendents are concerned, it is altogether 
indifferent whether we consider them to be deduced primarily from the 
assumed functional equations (1.), (2.), (3.), or from the primitive geome- 
trical definitions of which those equations are the immediate symbolical con- 

X»x d X f*y d y 

If we should denote the integrals / . and / — (com- 

l/O VI— X' Jq ^/\—yi 

raencing from respectively) by d and d' respectively, then the integral of 
the equation 

dx dy 

would furnish us with the fundamental equation 

sin {& 4- ff) = sin 6 cos 6' + cos & sin &', (/3.) 

if we should replace x by sin 6, •v/l — x^ by cos 6, y by sin 6', and Vl — y* 
by cos i'. If the formulae of trigonometry were founded upon such a basis, 
they would require no previous knowledge either of circular arcs considered 
as the measures of angles, or of the geometrical definitions of the sines and 
cosines, except so far as they may be ascertained from the examination of the 
values and properties of the transcendents which enter into the equation (a.). 
In a similar manner, if we should suppose 6 and 6' to represent the integrals 

/*J" dx Py dy 

of the transcendents / —yr- — ; — ^r and / —,7t- — 9^» then the integral 

Jo V(i+*-) Jo Vii—y^) ^ 

of the equation 

d X , d y „ / ^ 

aAi + ■■') ^Ai + r) 

would be expressed by the equation 

h sin (i + 6') = h sin & X h cos &' + h cos d X h sin 6', (3.) 

if we should make x ■=: h sin 6 (the hyperbolic sine of 6), and V (1 4- «*) 
= h cos 6 (the hyperbolic cosine of 6), y ^ h sin $', and Vl -\- y^ =: h cos &', 
adopting the terms which Lambert introduced, and which have been noticed 
in the note in p. 231 ; and it is evident that it would be possible from equa- 
tion (S.), combined with the assumptions made in deducing it, to frame a 
svstem of hvperbolic trigonometry (having reference to the sectors, and not 

u 2 

292 THIRD REPORT — 1833. 

or in its applications to geometry. The terms tangent, co- 
tangent, secant and cosecant, and versed sine, which denote 
very simple functions of the sine and cosine, may be defined by 
those functions and will be merely used when they enable us to 
exhibit formulae involving sines and cosines, in a more simple 
form. By adopting such a view of the meaning and origin of 
the transcendental functions, the relations and properties of 
which constitute the science of trigonometry, we are at once 
freed from the necessity of considering those functions as lines 
described in and about a circle, and as jointly dependent upon 
the magnitude of the angles to which they correspond and of the 
radius of the circle itself. It is this last element, which is thus 
introduced, which is not merely superfluous, but calculated to 
give erroneous views of the origin and constitution of trigono- 
metrical formulae and greatly to embarrass all their applications. 

to the arcs of the equilateral hyperbola), whose formulse would bear a very 
striking analogy to the formulae of trigonometry, properly so called. 

Abel, in the second volume of Crelle's Jotirnal, has laid the foundation, of 
a species of elliptic trigonometry, (if such a term may be used,) in connexion 
with a remarkable extension of the theory of elliptic integrals. If we denote 
the elliptic integral of the first species 


'^ ^/(l-c2sin2,^) 
by 6, and replace sin \p by jc, we shall get 


d X 

or more generally 


d X 

V{(1 +e"*=) {l-<?x^)y 

If we now suppose a; = <p ^, V (1 — c^ «") =/ ^ and V (1 + e^-r") = F tf, it 
may be demonstrated that 

<p{i) + e) - r+e2^2-^2-^-7^2-^» . 

, fd.f6'—c-(p6.(pd'.Y6. Fd' 
f {0 + 6) — 1 + e2 c2 (p2 tf . (p2 tf' 

_ ¥d.¥6'-\-e^(p6.<pd' .fd.fd \ 

or if, for the sake of more distinct and immediate reference to these peculiar 
transcendents, we denote 

(p S by sin 6 (elliptic sine of ff), 

f dhy cos 6 (elliptic cosine of ff), and 

F ^ by sur 6 (elliptic sursine of 6), 


The primitive signs 4- and — , when applied to symbols de- 
noting lines, are only competent to express the relation of lines 
which are parallel to each other when drawn or estimated in dif- 
ferent directions; but the more general sign cos fl + ■v^— i sin 3, 
which has been noticed in the former part of this Report, when 
applied to such symbols, is competent to express all the rela- 
tions of position of lines in the same plane with respect to each 
other. It is the use of this sign which enables us to subject 
the properties of rectilinear figures to the dominion of algebra : 
thus, a series of lines represented in magnitude and position by 

«o, (cos 9i + -/ — 1 sin flj) a^, {cos (di + 62)+ V — 1 sin (9i-l- ^2) }«2» 

. . . {cos (fli + ^2 + • • • ^»-i) + ^^^\ sin (51+ 9.2 + . . . 9„_,)} «„_„ 
will be competent to form a closed figure, if the following equa- 
tions be satisfied : 

then these fundamental equations will become 

sin & cos ^' surs ^' + sin ^' cos & surs 6 

T ^^'^^'- 1 + e2 c2 sin2 6 sin^ $' ~' 

e e 

cos 8 COS 6' — c2 sin & sin 6' surs 8 surs i' 

f A \^ AW e e e e e e_ 

COS (» + ff; _ 1 +e2c2^2 ^-^ina ^' • 

e e 

surs d surs 6' + e'^ sin i sin & cos & cos d' 
surs C« + tf ) _ 1 + e2 c2 sin2 ^ sin2 d' 

e e 

If we add, subtract and multiply, the elliptic sines, cosines and sursines of 
the sum and difference of 6 and 6' respectively, reducing them, when necessary, 
by the aid of the fundamental relations which exist amongst these three tran- 
scendents, we shall obtain a series of formulae, some of which are very remark- 
able, and which degenerate into the ordinary formulae of trigonometry, when 
c =: and c = 1 : we shall thus likewise be enabled to express sin n 6, cos n 6, 

e e 

surs n 8, in terms of sin 8, cos 6, surs 6. The inverse problem, however, to express 

« e e e 

sin 6, cos i, surs 6, in terms of sin n 6, cos n d, surs n 6, is one of much greater 

e e e e e e 

difficulty, requiring the consideration of equations of high orders, but whose 
ultimate solution can be made to depend upon that of an equation of (w -|- 1) 
dimensions only. It is in the discussion of these equations that Abel has dis- 
played all the resources of his extraordinary genius. 

It would be altogether out of place to enter upon a lengthened statement 
of the various properties of these elliptic sines, cosines, and sursines ; their 
periodicity, their limits, their roots, and their extraordinary use in the trans- 
formation of elliptic functions. My object has been merely to notice the ru- 
diments of a species of elliptic trigonometry, the cultivation of which, even 
without the aid of a distinct algorithm, has already contributed so greatly to 
the enlargement of the domains of analysis. 

g94 THIRD REPORT 1833. 

Co + flfi cosfii + a2COs(9, +S^) + , . «„-, cos (3, + fi2 + . • fi«-i) = (1.) 

aisinfli + ff2sin(fii + fl2) + . . a„_, sin (9i + 92 + . . 9„-i) = (2.) 

91 + ^2+ ... 9„_, . = (« - 2 r) TT (3.) 

The first two of these equations may be called equations of 
figure, and the last the equation of angles, and all of them must 
be satisfied in order that the lines in question may be capable 
of being formed into a figure, along the sides of which if a point 
be moved it will circulate continually. If the values of flj, 
^2 ~ ^1' ^3 ~ ^2 • • ^«-i "" ^n-2 ^^ ^11 positive, aud if r = 1, then 
the equation of angles will correspond to those rectilineal figures 
to which the corollaries to the thirty-second proposition of the 
first book of Euclid are applicable, and which are contemplated 
by the ordinary definitions of rectilineal figures in geometry. 
If we should suppose r = 2 or 3 or any other whole number 
different from 1, the equation would correspond to stellated 
figures, where the sum of the exterior angles shall be 8, 12, or 
4 r right angles. The properties of such *^t?Z/a^e(/ figures were 
first noticed by Poinsot in the fourth volume of the Journal 
de FEcole Polytechnique, in a very interesting memoir on the 
Geometry of Situation*. 

All equal and parallel lines drawn or estimated in the same 
direction are expressed by the same symbol aflfected by the 
same sign, whatever it may be : and it is this infinity of lines, 
geometrically different from each other, which have the same 
algebraical representation, which renders it necessary to con- 
sider the position of lines, not merely with respect to each other, 
but also with respect to fixed lines or axes, through the medium 
of the equations of their generating points. In other words, it 
is not possible to supersede even rectilineal geometry by means 
of affected symbols only. We are thus led to the consideration 
of a new branch of analytical science, which is specifically de- 
nominated the Application of Algebra to Geometry, and which 
enables us to consider every relation of points in space and the 
laws of their connexion with each other, whatever those laws 
may be. It is not our intention, however, to enter upon the 
discussion of the general principles of this science, or to notice 
its present state or recent progress. 

A great number of elementary works on trigonometry have 
been published of late years in this counti'y, many of which are 
remarkable for the great simplicity of form to which they have 
reduced the investigation of the fundamental formulae. Such 
works are admirably calculated to promote the extension of 

• See also Peacock's Algebra, p. 448. 


mathematical education, by placing this most important branch 
of analytical science, the very key-stone of all the applications 
of mathematics to natural philosophy, within the reach of every 
student who has mastered the elements of geometry and the 
first principles of algebra. 

We have before had occasion to notice the work of the late 
Professor Vince upon this subject, which was generally used 
in the Universities of England for some years after the com- 
mencement of the present century. Its author was a mathema- 
tician of no inconsiderable powers, and of very extensive know- 
ledge, but who was totally destitute of all feeling for elegance 
in the selection and construction of his formulae, and who had 
no acquaintance with, or rather no proper power of appreciating, 
those beautiful models of symmetry and of correct taste which 
were presented by the works of Euler and Lagrange. But 
though this treatise was singularly rude and barbarous in its 
form, and altogether inadequate to introduce the student to a 
proper knowledge either of the objects or of the powers of this 
science, yet it was greatly in advance of other treatises which 
were used and studied in this country at the period of its pub- 
lication. Amongst these may be mentioned the treatise on Tri- 
gonometry which is appended to Simson's Euclid, which was 
more adapted to the state of the science in the age of Ptolemy 
than at the close of the eighteenth century*. 

The Plane and Spherical Trigonometry of the late Professor 
Woodhouse appeared in 1810, and more than any other work 
contributed to revolutionize the mathematical studies of this 
country. It was a work, independently of its singularly oppor- 
tune appearance, of great merit, and such as is not likely, not- 
withstanding the crowd of similar publications in the present 
day, to be speedily superseded in the business of education. 
The fundamental formulae are demonstrated with considerable 
elegance and simplicity ; the examples of their application, both 
in plane and spherical trigonometry, are well selected and very 
carefully worked out ; the uses of trigonometrical formulas, in 
some of their highest applications, are exhibited and pointed 

• Similar remarks might be applied to treatises upon trigonometry which 
were published both before and after the appearance of Professor Wood- 
house's Trigonometry. The author of this Report well recollects a treatise of 
this kind which was extensively used when he was a student at the Univer- 
sity, in which the proposition for expressing the sine of an angle in terms of 
the sides of a triangle, was familiarly denominated the hlaclc triangle, in con- 
sequence of the use of thick and dark lines to distinguish the primitive tri- 
angle amidst the confused mass of other lines in which it was enveloped, for 
the purpose of obtaining the required result by means of an incongruous 
combination of geometry and algebra. 

296 THIRD REPORT — 1833. 

out in a very clear and striking form; and, like all otlier 
woi'ks of this author, it is written in a manner well calculated 
to fix strongly the attention of the student, and to make him 
reflect attentively upon the particular processes which are fol- 
lowed, and upon the reasons which lead to their adoption. 

The circumstances attending the publication and reception of 
this work in the University of Cambridge were sufficiently re- 
markable. It was opposed and stigmatized by many of the older 
members, as tending to produce a dangerous innovation in the 
existing course of academical studies, and to subvert the pre- 
valent taste for the geometrical form of conducting investiga- 
tions and of exhibiting results which had been adopted hy 
Newton in the greatest of his works, and which it became us, 
therefore, from a regard to the national honour and our own, 
to maintain unaltered. It was contended, also, that the primary 
object of academical education, namely, the severe cultivation 
and discipline of the mind, was more effectually attained by 
geometrical than by analytical studies, in which the objects of 
our reasoning are less definite and tangible, and where the 
processes of demonstration are much less logical and complete. 
The opposition, however, to this change, though urged with 
considerable violence, experienced the ordinary fate of attempts 
made to resist the inevitable progress of knowledge and the 
increased wants and improving spirit of the age. In the course 
of a few years the work in question was universally adopted. 
The antiquated fluxional notation which interfered so greatly 
with the familiar study of the works of Euler, Lagrange, La- 
place, and the other great records of analytical and philoso- 
phical knowledge, was abandoned * ; the works of the best 
mathematical writers on the continent of Europe were rapidly 
introduced into the course of the studies of the University ; and 
the secure foundations were laid of a system of mathematical 
and philosophical education at once severe and comprehensive, 
which is now producing, and is likely to continue to produce, 
the most important effects upon the scientific character of the 

Theory of Equations. 1 . Composition of Equations. — The 
first and one of the most difficult propositions which presents 
itself in the theory of equations is to prove " that all equations 
under a rational form, and arranged according to the method 

* The continental notation of the differential calculus was first publicly 
introduced into the Senate House examinations in 1817- Though the change 
■was strongly deprecated at the time, it was very speedily adopted, and in 
less than two years from that time the fluxional notation had altogether dis- 


of Harriott, the significant terms forming one membei', and 
zero the other, are said to be resolvible into simple or quadratic 
factors." It is only another form of the same proposition to 
say, " that every equation has as many roots as it has dimen- 
sions, and no more; those roots being either real* or ima- 
ginary ;" that is, being quantities which are expressible by 
symbols denoting real magnitudes affected by such signs as 
are recognised in algebra. 

We have before said that it is impossible to assign before- 
hand an absolute limit to the possible existence of signs of 
affection different from those which are involved in the sym- 
bolical values of (1)" and ( — 1)"; and when it is said that every 
equation is resolvable into factors of the form a; — a, we presume 
that a is either a real magnitude, or of the form a + (3 v' — 1, 
where « and jS are real magnitudes. If we should fail in esta- 
blishing this proposition, it would by no means necessarily fol- 
low that there might not exist other forms of factors like x — a, 
where a denoted a real magnitude affected by some unknown 
sign different from +, — , or cos fi + V — 1 sin 9, which might 
satisfy the required conditions : at the same time its demonstra- 
tion will show that our recognised signs are competent to de- 
note all the affections of magnitude which are subject to any 
conditions which are reducible to the form of an equation. 

If we assume in the first instance the composition of equa- 
tions to be such as we have stated in the enunciation of the 
fundamental proposition, we can at once ascertain the composi- 
tion of the several coefficients of the powers of x in the equa- 

and we can complete the investigation of all those general pro- 
perties of equations which such an hypothesis would lead to. 
All such conclusions, when established upon such a foundation, 
are conditional only. It is not expedient, however, to make 
the fate of any number of propositions, however consistent with 
each other, and however unquestionable their truth may appear 
to be from indirect or from a posteriori considerations, depend- 
ent upon an hypothesis, when it is possible to convert this hypo- 
thesis into a necessary symbolical truth. Using such an hypo- 
thesis, therefore, as a suggestion merely, let us propose the 

* It is convenient in the theory of equations, for the purpose of avoiding 
repetition, to consider symbols denoting arithmetical magnitudes and affected 
with the signs + or — , unreal; and quantities denoted by symbols affecttd 
with the" sign cos + V— l sin 6, as imaginary. 

^^S THIRD REPORT — 1833. 

following problem, and examine all the consequences to which 
its solution will lead. 

" To find n quantities x, x^, x^, . . . x„_i, such that their sum 
shall be equal to ^9,, the sum of all their products two and two 
shall be equal to p^, the sum of all their products three and 
three shall be equal to p^, and so on, until we arrive at their 
continued product, which shall be equal to p„." 

The quantities x, a-j, . . . x„^i, are supposed to be any quan- 
tities whatever, whether real or affected by any signs of affec- 
tion whether known or unknown. It is our object to show that 
the only sign of affection required is cos 6 + -/^ sin 9, taken 
in its most general sense. 

It is very easy to show that the solution of this problem will 
lead to a general equation, whose coefficients are p^, p^, . . . jo„ : 
for if we suppose the first of these quantities x to be omitted, 
and Pj, Pg, . . . P„_i to be the quantities corresponding to p^, 
pc^, . . . pn when there are (w — 1) quantities instead of n, then 
we shall get 

a; + Pj =/?„ 
a: Pi + Pa = p<i, 

XV^ + P3=^3, 


^ P»-2 + P»-l — pn-l> 
^ Pn-1 = Pn- 

If we multiply these equations from the first downwards by the 
terms of the series x"-', x"-^, . . . x'^, x, 1, and add the first, 
third, fifth, &c., of the results together, and subtract the second, 
fourth, sixth, &c., we shall get the general equation 

x"~23i a:"-' + j)^ x"-^ -... + (- 1)«^„ = 0. (1.) 

In as much as />,, p^, . . . p„ may represent any real magni- 
tudes whatever, zero included, it is obvious that we may consi- 
der this equation as the result of the solution of the problem in 
its most general form. And in as much as x may represent any 
one of the n quantities involved in the problem, we must equally 
obtain the same equation for all those n quantities : it also fol- 
lows that every general solution of this equation must compre- 
hend the expression of all the roots. 

By this mode of presenting the question we are authorized in 
considering the spnbolical composition of the coefficients of 
everi/ equation as known, though the ultimate symbolical form 
of the roots is not knoivn ; and our inquiry will now be properly 
limited to the question of ascertaining whether symbols repre- 


senting real magnitudes afFected by the recognised and Icnown 
signs of aiFection only, are competent, under all circumstances, 
to answer the required conditions of the problem. 

If the value of one root can be ascertained, and that root be 
real, the problem can be simplified, and the dimensions of the 
equation depressed by unity ; for the coefficients of the reduced 
equation Pj . P^ . P«-i, which are also real, can be successively 
determined. If more real roots than one can be found, the 
dimensions of the equation can be depressed by as many unities 
as there are real roots. If the root determined be not real, and if 
a similar process for depressing the dimensions of the equation 
be adopted, the coefficients of the new equation would not be 
real, and the conditions of the problem with respect to the re- 
maining roots would be changed. But if we could ascertain a 
pair of such roots, such that their sum = x + x^ and their pro- 
duct = X x^ should be real, then the dimensions of the equation 
might be depressed by two unities, without changing the con- 
ditions of the problem with respect to the remaining roots; for 
if we supposed Q^, Q^, Qg, &;c., to represent the coefficients of 
the reduced equation, we should find, 

.r + 0-1 + Qi = pi, 
xx^ + (x + x{) Qi + Q2 = P2, 

^ ^1 Ql + (a- + X^) Q2 + Qg = jOg, 

X a-1 Q„_4 + {x + x^) Q„_3 + Q„_2 = p„_i, 

X X^ (:)J„_2 = Pnf 

from which equations we can determine successively rational 
values of Qj, Q^, . . . Q„_2. It remains to show, therefore, 
that in all cases we can find pairs of roots which will answer 
these conditions. 

If the number of quantities x, x^, . . . .r„, be odd, it is very 
easy to p)rove that there is always a real value of one of them, x, 
which will satisfy the conditions of the general equation (I.) *, 
and that consequently the dimensions of the equation may be 
depressed by unity, and our attention confined therefore to 
the case where the dimensions of the equation are even. If m, 
therefore, be any odd number, the form of n may be either 2 m, 
2^ m, 2^ m, 2" ?«, and so on. Let us consider, in the first place, 
the first of these cases. 

The number of combinations of 2 m, things taken two and two 
together, is m {2 m — \,) and therefore an odd number : these 

* This may be easily proved without the necessity of making any hypothesis 
respecting the composition of the equation. See the Article ' Equations' in 
the Supplement to the Encyclopaedia Britannica, written by Mr. Ivory. 

300 THIRD REPORT — 1833. 

combinations may be either the sums of every two of the quanti- 
ties, X, x^,... T„_i, such as X + x^, x + x^^, &c., or their products, 
such as X Xi, or other rational linear functions of those quanti- 
ties, involving two of them only, such as, x + x^^ + x x^, x + x^ 
•^ 2 X x^, ov X + x-^ + k X x■^, where k may be any given num- 
ber whatsoever. If we take any one of these sets of combina- 
tions, we can form rational expressions for their sum, for the 
sum of their products, two and two, three and three, and so on, 
in terms of the coefficients p^, p.^, . . . pn, of the original equa- 
tion (1.), by means of the common theory of symmetrical func- 
tions *, and consequently, we can form the corresponding equa- 
tions of m {2 m — 1) dimensions which will have rational and 
known coefficients. Such equations being of odd dimensions 
must have at least one real root ; or, in other words, there must 
exist at least one real value of one of the sums of two roots, 
such as a; + x^, of one of the products, such as x x^, of one 
of the functions, x + x^ + x x^, or a: + a;] + k x Xy If the 
symbols which form the real sum x -\- x^ are the same with those 
which form the real value of the product x x^ then, under such 
circumstances, x and x^ are expressible by real magnitudes af- 
fected with the ordinary signs of algebra f . We shall now pro- 
ceed to show that this must be the case. 

If we form the equations successively whose roots are x -\- x^ 
+ k X x-^, corresponding to different values of k, we shall have 
one real root at least in each of them. If we form more than 
m (2 m — 1), such equations for different values of k, we must 
at least have amongst them the same combination of x and x^^ 
forming the real root, in as much as there are only m {2 m — \) 
such combinations which are different from each other. Let k 
and X'l be the values of k which give such combinations, and 
let «' and (6' be the values of the real roots corresponding ; then 
we must have 

X + x^ -\- k X x^ ^= u.' 
X + .r, + k^ X x^ = /3' 

* The formation of symmetrical combinations of any number oi symbolical 
quantities x, x^, . . . a-n-i, and the determination of their symbolical values 
in terms of their sums {px), their products two and two (pq), three and three 
(JO3), and so on, involves no principle which is not contained in the direct 
processes of algebra, and is altogether independent of the theory of equations. 
The theorems for this purpose may be found in the first chapter of Waring's 
Meditationes Algebraicce, in Lagrange's Traite sur la Resolution des Equations 
Ni/meriques, chap. i. and notes 3 and 10, and with more or less detail in 
nearly all treatises on Algebra. 

t If J + •'^1 = a and xx^ = /i . , where « and /3 are real magnitudes, then 

j; = -^- + s'' s " — /3 ? the values of which are either real or of the form 
(cos (I + \^ — 1 sin 6) ^^/3, where the modulus \//3 is real. 


and therefore 

X J-, 

k - X-i 

"*" + "^1 ~ ' k^-k ' 

There are therefore necessarily two roots of the equation or two 
values of the symbols .r, a:{, x^, . . . Xn-\, such that x + x^ and 
X Xj are real ; and therefore it is always possible, in an equation 
whose dimensions are impariter par, to depress them by two 
unities, so that the reduced equation may still possess rational 

If the number of symbols involved in the original problem be 
2^ m, then the number of their binary combinations must be 
9, m (2^ m — 1 ) or impariter par. It will immediately follow, from 
what we have already proved, that there are two values of the 
sum and product of the same symbols, which are either real or 
of the form a -\- ^ V — \ \ and consequently the symbols them- 
selves will admit of expression under a similar form *. 

If the dimensions of the original equation be 2^ m or £'* m, or 
any one in an ascending series of orders oi parity, it may be re- 
duced down to the next order of parity in a similar manner : and 
under all circumstances it may be shown that there must be two 
roots which are reducible to the form « + /3 -/ — 1 , where « and /3 
real or zero ; and also in any equation of even dimensions, we 
can reduce its dimensions successively by two unities, thus pro- 
ducing a series of equations of successive or decreasing orders of 
parity, in which we can demonstrate the existence of successive 
pairs of roots of the required form until they are all exhausted. 

This mode of proving the composition of equations differs 
chiefly from that which was noticed by Laplace, in his lectures 
to the Ecole Normals in 1795f , in the form in which the ques- 
tion is proposed. A certain number of symbols, representing 
magnitudes with unknoum affections, are required to satisfy 

• I-et X + x' = r (cos tf + V — 1 sin ^) 

X x' = J (cos <p + V — 1 sin (p) 

X -\- x\- — A xx' ^= R-(cos 2;/' + \^ — 1 sin 2 ^l>) 

or *■ — x' =i R (cos ^Z' + -v^ — 1 sirn^) 

r cos tf + R cos ip (jr sin ^ + R sin ip) i 

X — - • H 2 V — 1 

= r' (cos ic + -v^ — 1 sin x) 

x' = r' (cos X — V — 1 sin x)- 
t Lemons de I'Ecole Normale, torn. ii. 

302 THIRD REPORT — 1833. 

certain real conditions : those conditions are found to be iden- 
tical with those which the unknown quantity, or, in other woi'ds, 
the root in an equation of n dimensions, is required to satisfy. 
The object of the proof above given is to show that it is always 
possible to find n real magnitudes with known aiFections which 
ai*e competent to satisfy these conditions ; and those quantities, 
therefore, are of such a kind that the equation, whose roots 
they are, is always resolvible into real quadratic factors ; a most 
important conclusion, which the greatest analysts have laboured 
to deduce by methods which have not been, in most cases at 
least, free from very serious objections. 

There are two classes of demonstrations which have been 
given of this fundamental proposition in the theory of equations. 
The first class comprehends those in which the form of the 
roots is determined from the conditions which they are required 
to satisfy ; the second class, those in which the form of the 
roots is assumed to be comprehended under different values of 
p and fl in the expression f (cos 9 + V^ — 1 sin 6), and it is shown 
that they are competent to satisfy the conditions of the equa- 
tion. To the first class belongs the demonstration given above ; 
those given by Lagrange in notes ix. and x. to his Resolution des 
Equations Niim^riques ; the first of those given by Gauss in the 
Gottingen Transactions ior 1816*; and by Mr. Ivory in his 
article on Equations in the Supplement to the Encyclopcedia 
Britannica. To the second class belongs the second demon- 
stration given by Gauss in the same volume of the Gottingen 
Transactions; by Legendre in the 14th section of the first 
Part of his Theorie des Nomhres ; by Cauchy in the 18th 
cahier of the Journal de VEcole Polytechniqiie ; and subse- 
quently under a slightly different form in his Cours d' Analyse 

The first of the demonstrations given by Gauss, like many 
other writings of that great analyst, is extremely difficult to 
follow, in consequence of the want of distinct enunciations of 
the propositions to be proved, and still more from their not 
always succeeding each other in the natural order of investi- 
gation. It requires the aid likewise of principles, or rather of 
processes, which are too far advanced in the order of the re- 
sults of algebra to be properly employed in the establishment 
of a proposition which is elementary in the order of truths, 
though it may not be so in the order of difficulty. If we may 

* There is another demonstration by Gauss, published in 1799, which I 
have never seen. In his Preface to his Demonstratio Nova Altera he speaks 
of its being founded partly on geometrical considerations, and in other re- 
spects as involving very different principles from the second. 


be allowed, however, to consider it apart from such considera- 
tions, it would appear to be complete and satisfactory, and 
very carefully guarded against any approach to an assumption 
of the proposition to be proved, a defect to which most of the 
demonstrations of this class are moi*e or less liable*. It extends 
to equations whose dimensions involve different or successive 
orders of parities, nearly in the same manner as in the demon- 
stration which we have given above. 

The demonstration given by Mr. Ivory is different from any 
other, and the principles involved in it are such as naturally 
present themselves in such an investigation ; and it will be re- 
commended to many persons by its not involving directly the 
use, or supposing the necessary existence of, imaginary quan- 
tities. It is not, however, altogether free from some very serious 
defects in the form under which it at present appears, though 
most of them admit of being remedied without any injury to 
the general scheme of the demonstration, which is framed with 
great skill, and which exhibits throughout a perfect command 
over the most refined and difficult artifices of analysis. 

Lagrange has devoted two notes to his great work on the 
Resolution of Numerical Equations to the discussion of the 
forms of the roots of equations. In the first of these notes, 
after examining the very remarkable observations of D'Alem- 
bert on the forms of imaginary quantities, he proceeds to con- 
sider the case of an equation such as f (x) + V = 0, where 
y (x) is a rational function o£ x ; if for different values a and b 
of the last term of this equation, where a ^b,we may suppose 
a root which is not real for values of V between those limits, to 
become real at those limits, he then shows that for values 
of V between those limits, and indefinitely near to them, the 
corresponding root of the equation must involve -/— 1, or 
•v/ — 1, or \/ — l, and so on; or, in other words, that the roots 
of the equation in the transition of their values from real 
to imaginary (whatever may be the affection of magnitude 
which re nde rs them imaginary), will change in form from a to 
7W -f- « -/ — 1 . He subsequently shoAvs that the same result will 
follow for any values of V between a and b, and consequently, 

* I do not venture to speak more decidedly; for though I have read it en- 
tirely through several times with great care, I do not retain that distinct and 
clear conviction of the essential connexion of all its parts which is necessary 
to compel assent to the truth of a demonstration. It is unfortunately fre- 
quently the character of many of the higher and more difficult investigations 
connected with the general theory of the composition and solution of equa- 
tions to leave a vague and imperfect impression of their truth and correctness 
even upon the minds of the most laborious and best instructed readers. 

304 THIRD REPORT — 1833. 

that in every instance, when roots of equations cease to be real, ' 
they will assume the form m + ?? V^— 1. 

This demonstration is not merely indirect, but it does not 
arise naturally from the question to be investigated. It seems 
likewise to assume the existence of some algebraical form which 
expresses the value of the root in terms of the coefficients of 
the equation, an assumption which, as will afterwards be seen, 
it M'ould be difficult to justify by any a priori considerations. 
The illustrious author hmiself seems to have felt the full force 
of these objections, and he proceeds therefore in the following 
Note to prove that every polynomial of a rational form will ad- 
mit of rational divisors of the first or second degree. The de- 
monstration which he has given is founded upon the theory of 
symmetrical functions, and shows that the coefficients of such 
a divisor may be made to depend severally upon equations all 
whose coefficients are rational functions of the coefficients of 
the polynomial dividend. Whatever be the degree of parity of 
the number which expresses the dimensions of this polynome, 
he shows the possibility of the coefficients of this quadratic di- 
visor, which is the capital conclusion in the theory. It ought 
to be observed, however, that the whole theory of the compo- 
sition of equations is so much involved in the different steps of 
this investigation, or, at all events, that so little provision is 
made in conducting it to guard against the assumption of 
this truth, that we should not be justified in considering this 
demonstration as perfectly independent or as furnishing an 
adequate foundation for so important a conclusion. If we view 
it, however, simply with reference to the problem for exhibiting 
the nature of the law of dependence which connects the coeffi- 
cients of the polynomial factor with those of the original poly- 
nomial dividend, it must still be considered as an investigation 
of no inconsiderable importance, as bearing upon the general 
theory of the solution and depression of equations. 

The second of the proofs given by Gauss, the proof of Le- 
gendre, and both of those which have been given by Cauchy, 
belong to the second class of demonstrations to which we 
have referred above. Assuming the root to be represented 
by p (cos 6 + 'S/ — 1 sin 9), the equation is reduced to the form 
P + Q ^/^-i, or ^(P2 + Q2) . (cos <p + V^l sin (f); and the 
object of the demonstration is to show that there exist neces- 
sarily real values of p and 9, which make P^ + Q'^ = 0. This 
is effected by Gauss by processes which are somewhat syn- 
thetical in their form, and such as do not arise very natu- 
rally or directly from the problem to be investigated ; and the 


essential part of the demonstration requires a double integra- 
tion between assigned limits, a process against which serious 
objections may in this instance be raised, independently of its 
involving analytical truths and principles of too advanced an 

The demonstration of Legendre depends upon the possible 
discovery, by tentative or other means, of values of g and &, 
which render P and Q very small ; and subsequently requires 
us, by the application of the ordinary processes of approxima- 
tion, to find other values of g and S, subject to repeated correc- 
tion, which may render P and Q smaller and smaller, and ulti- 
mately equal to zero. The objection to this demonstration, if 
so it may be called, is the absence of any proof of the necessary 
existence of values of § and & ; and if they should be shown to 
exist, it seems to fail in showing that the subsequent correc- 
tions of their values which this process would assign would really 
and necessarily increase the required approximations. 

The demonstrations of Cauchy are formed upon the general 
scheme of that which is given by Legendre, at the same time 
that they seem to avoid the very serious defects under which 
that demonstration labours : he shows that (P- + Q^) must ad- 
mit of a minimum, and that this minimtim value must be zero. 
The second of the demonstrations differs from the first merely 
in the manner of establishing the existence and value of this 
minimum : they both of them appear to me to be quite com- 
plete and satisfactory. 

It is not very difficult to estabhsh this fundamental propo- 
sition by reasonings derived from the geometrical representa- 
tion of impossible quantities. This was done, though imper- 
fectly, by M. Argand, in the fifth volume of Gergonne's An- 
nates des Mathe??iatigt(es*, and has been since reconsidered by 
M. Murey, in a very fanciful work upon the geometrical in- 
terpretation of imaginary quantities, which was published in 
1827. It seems to me, however, to be a violation of propriety 
to make such interpretations which are conventional merely, 
and not necessary, the foundation of a most important symbo- 
lical truth, which should be considered as a necessary result of 
the first principles of algebra, and which ought to admit of de- 
monstration by the aid of those principles alone. 

General Solution of Equations. — The solution of equations 
in its most general sense would require the expression of its 
roots by such functions of their coefficients as were competent 

* In the fourth volume of the same collection there are demonstrations of 
this fundamental proposition, given by M. Dubourguet and M. Encontre, 
which do not appear, however, to merit a more particular notice. 
1833. X 

306 THIRD REPORT — 1833. 

to express them, when those coefficients were general symbols, 
though representing rational numbers. Such functions also 
must equally express all the roots, in as much as they are all of 
them equally dependent upon the coefficients for their value ; 
and they mvist express likewise the values of no quantities which 
are not roots of the equation. 

The problem, in fact, is the inverse of that for the formation 
of the equation which is required to satisfy assigned condi- 
tions. And as we have shown that there always exist quanti- 
ties expressible by the ordinary signs of algebra which will fulfil 
the conditions of any equation with rational coefficients, so like- 
wise we might appear to be justified in concluding that there 
must exist explicable functions of those coefficients which in all 
cases would be competent to represent those roots. 

A very little consideration, however, would show that such a 
conclusion was premature. In the first place, such a function 
must be irrational, in as much as all rational functions of the 
coefficients admit but of one value ; and they must be such ir- 
rational functions of the coefficients as will successively insulate 
the several roots of the equation, — for they must be equally ca- 
pable of expressing all the roots, — and they must be capable 
likewise of effecting this insulation without any reference to the 
specific values of the S3rmbols involved, or to the relation of the 
values of the roots themselves ; for otherwise they could not be 
said to represent the general solution of any equation whatever 
of a given degree. The question which naturally presents it- 
self, after the enumeration of such conditions, is, whether we 
could conclude that any succession of operations which are, pro- 
perly speaking, algebraical, would be competent to fulfil them. 

If it be further considered that those successive operations 
must be assigned beforehand for every general equation of an 
assigned degree ; that every one of these operations can give 
one real value only, or at the most two ; and that the result of 
these operations, which must embrace all the coefficients, must 
express the n roots of the equation and those roots only ; it 
will readily be conceded that the solution of this great pro- 
blem is probably one which will be found to transcend the 
powers of analysis. 

The solutions of cubic and biquadratic equations have been 
known for nearly three centuries ; and all the attempts which 
have hitherto been made to proceed beyond them, at least in 
equations in which there exists no relation of dependence 
amongst the several coefficients, and no presumed or presuma- 
ble relation amongst the roots, have altogether failed of success : 
and if we consider that this great problem has been subjected to 


the most scrutinizing and laborious examination by nearly all 
the greatest analysts who have lived in that period, we may be 
justified in concluding that this failure is rather to be attributed 
to the essential impossibility of the problem itself than to the 
want of skill or perseverance on the part of those "-ho have 
made the attempt. But in the absence of any compete and 
uncontrovertible proof of this impossibility, the question cannot 
be considered as concluded, and will still remain open to spe- 
culations upon the part of those with whom extensive and well- 
matured knowledge, and a deep conviction founded upon it, have 
not altogether extinguished hope. 

The different methods which have been proposed for the 
resolution of cubic and biquadratic equations, and the conse- 
quences of the extension of their principles to the solution of 
equations of higher orders, have been subjected to a very de- 
tailed analysis by Lagrange, in the Berlin Memoirs for 1770 
and 1771, and in the Notes xiii. and xiv. of his Traits sur la 
Resolution des Equations Numiriques ; and it would be diffi- 
cult to refer to any investigations of this great analyst which are 
better calculated to show the extraordinary power which he 
possessed of referring methods apparently the most distinct to 
a common principle of a much higher and more comprehensive 
generality. In the subsequent remarks which we shall make, 
we shall rarely have occasion to proceed beyond a notice of the 
general conclusions to which he has arrived, and to show their 
bearing upon some later speculations upon the same subject. 

A very slight examination of the principles involved in the 
solution of the equations of the third and fourth degrees will 
show them to be inapplicable to those of higher orders. A no- 
tice of a very few of such methods vdll be quite sufficient for 
our purpose. 

Thus, the ordinary solution of the cubic equation 
a^-Sqx + 2r-Q* 
is made to depend upon that of the following problem : 

" To find two numbers or quantities such that the sum of 
their cubes shall be equal to 2 r and their product equal to q." 

If we represent the required numbers by u and v, we readily 
obtain the equation of reduction 

u^ — 2r u^ + ^"^ = 0, 

• This equation may be considered as equally general with 
a^_Aa;2 + B«— C = 0, 
in as much as we can pass from one to the other by a very easy transforma- 
tion ; and the same remark may be extended to equations of higher orders. 
Such a change of form, however, will determine the applicability or inappli- 
cability of many of the methods which are proposed for their solution. 


308 THIRD REPORT — 183^. 

which gives, when solved as a quadratic equation, 


= r + >^ if — (f), 

and consequently, 

and therefore 

^^q_^ 9' 

u {r + v/ {r^ — (f')Y 

If we call 1, «, o?, the three cube roots of 1, or the roots of 
the equation s^ — 1 = 0, and if we assvime a to represent the 
arithmetical value of u, we shall obtain the following three 
values of ?< + v, which are 

a + -^s a a ■] 2_ « «^ -| 2-. 

a a u aw' 

These values, though derived from the solution of an equation 
of six dimensions *, are only three in number, and form, there- 
fore, the roots of a cubic equation. A little further inquiry will 
show that they are the roots of the cubic equation 

x^ — 3qx + 2r=0: 

for it may readily be shown, in the first place, that their sum 
= ; that the sum of their products two and two =: — 3 q; and 
that their continued product = 2 r; or in other words, that 
they are the roots of an equation which is in every respect iden- 
tical with the equation in question-}-. 

* There are six values of u, in as much as the values of u and v are inter- 
changeable, from the form in which the problem was proposed ; but there are 
only three values of u + v. 

t Since q ^ i 

it is usual to express the roots of the equation x^ — 3q x -\- 2r = 0, by the 

x={r+ V(r^-q^)}^ + {'•- V(»-=-?^)}i (1.) 

which is in a certain sense incorrect, in as much as it admits of nine values 
instead of three. The six additional values are the roots of the two equations 

x^ — 3ctqx + 2r = 0, 
x^ — 3a'^qx + 2r=0, 
and the formula (1.) expresses the complete solution of the equation 

(«« — 2 rf — 27 </3 a;3 = 0, 
which is of 9 dimensions. It is the formula 

« = M + -|-, where u= {r + V (r^ — q^) }i 

and has the same value in both terms of the expression, which corresponds to 
the equation x^ — 3qx + 2r — 0. 


This mode of effecting the solution of a cubic equation would 
altogether fail if the original equation possessed all its tex-ms : 
and though the absence of the second term of a cubic equation 
cannot be said, in a certain sense at least, to affect the gene- 
rality of its character, yet it would lead us to expect that the 
method which we had followed was of so limited a nature as not 
to be appliccible to general equations of a higher order. Thus, 
if it was proposed to find two quantities, u and r, the sum of 
whose «"^ powers was equal to 2 r, and whose product was equal 
to q, we should find 

u = {r + ^(r^ - ?")}«; 

1 ^ 

where m + ^? is the root of the equation 
x^-nq ^"- + '^^^ f a;«- - n {n - ^A)^{n- ^) ^ ^„_,^ 

+ &c. = 2 r*. 

The form of this equation is of such a kind as to prevent its 
being identified with any general equation whatever, beyond a 
cubic equation wanting the second term ; a circumstance which 
precludes all further attempts, therefore, to exhibit the roots of 
higher equations by radicals f of this very simple order : but 
it is possible that there may exist determinate functions of the 
roots of higher equations (not symmetrical functions of all of 
them, which are invariable as far as the permutations of the roots 
amongst each other are concei'ned,) which may admit of triple 
values only, and which will be expressible, therefore, by means 
of a cubic equation, and consequently by the general formula 
for its solution. 

Thus, if x^, x^, oTg, x^, were assumed to represent the roots of 
a biquadratic equation 

• This equation was first solved by Demoivre in the Philosophical Trans- 
actions for 1737, and it was readily derived from the theorem which goes by 
his name. It was afterwards shown to be true, by a process, however, not al- 
together general, by Euler, in the sixth volume of the Comment, ^cad. Petrop., 
p. 226. See also Abel's " Me'moire sur une Classe particuliere d'Equations 
r^solubles algebriquement," in Crelle's Journal, vol. iv. 

t Abel has used the term radicality to designate such expressions. To 
say, therefore, that the root of an equation is expressible by radicalities, is 
the same thing as to say that the equation is solvable algebraically. It is 
used in contradistinction to such transcendental functions, whether of a known 
or unknown nature, as may, possibly, be competent to express those roots, 
when all general algebraical methods fail to determine them. 

310 THIRD REPORT — 1833. 

x'* — p x^ + q x^ — r X + s = 0, (1.) 

sixch functions would be x^ x^ + x^ x^ and (ic, + x^ — x^ — x^^, 
which admit but of three different values, and which may seve- 
rally form, therefore, the roots of cubic equations, whose coeffi- 
cients are expressible in terms of the coefficients of the original 
equation. Such a function also would be {x-^ + x^^, if we should 
suppose p or the coefficient of the second term of equation (1.) 
to be zero *. The function (Xj + ^2) (■'^3 + ^4) would give 
three values only under all circumstances. The functions x^ 
+ ^2 + ^3 ^^^ ^1 ^2 •''^3 ^^^ capable of four different values, 
and therefore do not admit of being expressed by a determina- 
ble equation of lower dimensions than the primitive equation. 
Functions of the form x^ Xo admit of six values, and require for 
their expression equations of six dimensions, which are reduci- 
ble to three, in consequence of being quasi recurring equations -j-. 
Innumerable functions may be formed which admit of 12 and of 
24 values, and one alternate function which admits of two values 
only J. 

The success of such transformations in reducing the dimen- 
sions of the equation to be solved, would naturally direct us to 
the research of similar functions of the roots of higher equa- 
tions than the fourth, which admit of values whose number is 
inferior to the dimensions of the equation. We may presume 
that, if such functions exist, they are rational functions, for 
if not, their irrationality/ would increase the dimensions of 
the reducing equation, and would tend to distribute its roots 
into cyclical periods ; and what is more, it has been very 
clearly proved that if equations admit of algebraical solution, 
all the algebraical functions which are jointly or separately in- 
volved in the expression of their roots, will be equal to rational 

* The first of these transformations involves the principles of Ferrari's, some- 
times called Waring's, solution of biquadratic equations ; the second that of 
Euler ; and the third that of Des Cartes. See the third chapter of Meyer 
Hirsch's Sammhmg von Aiifgaben aus der Theorie der algehraischen Gleickuyigen, 
which contains the most complete collection of formulae and of propositions 
relating to symmetrical and other functions of the roots of equations with 
which I am acquainted. The combinatory analysis receives its most advan- 
tageous and immediate applications in investigations connected with the 
theory of such functions. See also Peacock's Algebra, note, p. 6I9. 

■f The form of its roots being u and — , they are reducible by the same me- 
thods as are applied to recurring equations. 

I See Cauchy, Cokvs d' Analyse, chap. iii. and noteiv. The use of such al- 
ternate functions in the elimination of the several unknown quantities from n 
simultaneous equations of the first order, involving « unknov/n quantities, 
will be noticed hereafter. 


functions of these roots ; and consequently, if irrational func- 
tions of those roots are employed in the formation of the re- 
ducing equation, the roots of the eqviation must enter into the 
final expression of the required roots, in a form where that ir- 
rationality has altogether disappeared *. If we assume, there- 
fore, that such functions are in all cases rational, the next ques- 
tion will he, whether they are discovei'able in higher equations 
than the fourth. 

This inquii'y was undertaken by Paolo Ruffini, of Modena, 
in his Teoria delle Equazione Algebraiche, published at Bo- 
logna in 1799, and subsequently in the tenth volume of the 
Memorie delta Societa Italiana, in a memoir on the impossibi- 
lity of solving equations of higher degrees than the fourth. 
He has demonstrated that the number of values of such func- 
tions of the roots of an equation ofw dimensions must be either 
equal to 1 , 2 . 3 . , . ??, or to some submultiple of it ; and that 
when n = 5, there is no such function, the alternate function 
being excluded, which possesses less than 5 values. The pro- 
cess of reasoning which is employed by the author for this pur- 

* This proposition has been proved by Abel, in his Beweis der Unmbglich- 
keit algebraische Gleichungen von hoheren Graden als dem vierten allgemein 
Aufzidosen, in the first volume of Crelle's Journal : the same demonstration 
was printed at Paris, in a less perfectly developed form, during his residence 
in that capital. This proof applies to algebraical solutions only, excluding 
the consideration of the possibility of expressing such roots by the aid of un- 
known transcendents. After defining the most general form of algebraical 
functions of any assigned degree and order ; and after demonstrating the pro- 
position referred to in the text, and analysing the demonstrations of Ruffini and 
Cauchy, and showing their precise bearing upon the theory of the solution 
of equations, he proceeds to show that the hypothesis of the existence of 
such a solution in an equation of five dimensions will necessarily lead to an 
equation, one member of which has 120 values and the other only 10 ; an ab- 
surd conclusion. It is quite impossible to exhibit this demonstration in 
a very abridged form so as to make it intelligible ; and though some parts of 
it are obscure and not perfectly conclusive, yet it is, perhaps, as satisfactory, 
upon the whole, as the nature of the subject will allow us to expect. 

It is impossible to mention the name of M. Abel in connexion with this 
subject, without expressing our sense of the great loss which the mathematical 
sciences have sustained by his death. Like other ardent young men, he com- 
menced his career in analysis by attempting the general solution of an equa- 
tion of five dimensions, and was for some time seduced by glimpses of an 
imagined success ; but he nobly compensated for his error by furnishing the 
most able sketch of a demonstration of its impossibility which has hitherto 
appeared. His subsequent discoveries in the theory of elliptic functions, 
which were almost simultaneous with those of Jacobi, have contributed most 
materially to change the whole aspect of one of the most difficult branches of 
analytical science, and furnish everywhere proofs of a most vigorous and in- 
ventive genius. He died of consumption, at Christiania in Norway, in 1827, 
in the 27th year of his age. 

312 THIRD REPORT 1883. 

pose is exceedingly difficult to follow, being unnecessarily en- 
cumbered with vast multitudes of forms of combination, and 
requiring a very tedious and minute examination of different 
classes of cases ; and it was, perhaps, as much owing to the 
necessary obscurity of this very difficult inquiry as to any im- 
perfection in the demonstration itself, that doubts were ex- 
pressed of its correctness by Malfatti* and other contemporary 
writers. The subject, however, has been resumed by Cauchy in 
the tenth volume of the Journal de VEcole Pohjtechnique , who 
has fully and clearly demonstrated the following proposition, 
which is somewhat more general than that of Ruffini : " That 
the number of different values of any rational function of n 
quantities, is a submultiple of 1 . 2 . 3 . , . n, and cannot be re- 
duced below the greatest prime number contained in n, without 
becoming equal to 2 or to 1." If we grant, therefore, the truth 
of this proposition, it will be in vain to seek for the reduction 
of equations of higher dimensions than the fourth, by transfor- 
mations dependent upon rational functions of the roots. 

The establishment of this proposition forms an epoch in the 
history of the progress of our knowledge of the theory of equa- 
tions, in as much as it so greatly limits the objects of research 
in attempts to discover the general methods for their solution. 
And if the demonstration of Abel should be likewise admitted, 
there would be an end of any further prosecution of such in- 
quiries, at least with the views with which they are commonly 

Lagrange, in liis incomparable analysis of the different me- 
thods which have been proposed for the solution of biquadratic 
and higher equations, has shown their common relation to each 
other, and that they all of them equally tend to the formation of 
a reducing equation, whose root is 

Xi + u jTg + «^ .Tg + 0,^X4+ &c. 

where x\, x^, x^, &c., are the roots of the primitive equation, 
and where « is a root of the equation 

a"-' + «»-2 ^ ^«-3 -f . . . a + 1 = 0, 

where n expresses the dimensions of the equation to be solved. 
He then reverses the inquiry, and assuming this form as 
correctly representing the root of the reducing equation, he 
seeks to determine its dimensions. The beautiful process which 
he has employed for this purpose is so well known f that it is 
quite unnecessary to describe it in this place ; and the result, 

* Mcmnrie della Sor. Hal., torn. xi. 

t Resolution des Eqiiafions Ntimeriques, Note xiii. 


as might be expected, perfectly agrees with the conclusions 
which are derived from more direct, and, perhaps, more ge- 
neral considerations. If n, or the number of roots ^j, .Tg, x.^, 
&c., be a prime number, then the dimensions of the final re- 
ducing equation will be 1 . 2 ... (w — 2) ; and if w be a compo- 
site number := mp, then the dimensions of the final reducing 
equation will be 

1.2. ..w 1.2...ra 


{m- 1) m . (1 . 2 . . . J})'" {p -\)p.{\.2... m)P' 

according as we arrive at it, by grouping the terms of the ex- 

A'l + U X^-\- 0^ Xq + &c. 

into m periods of^ terms, or into jo periods of »« terms. It thus 
appears, that for an equation of 5 dimensions, the final reducing 
equation is of 6 dimensions ; for an equation of 6 dimensions, 
the final reducing equation is of 10 dimensions in one mode of 
derivation and 15 in the other ; and the higher the dimensions 
of the equation are, the greater will be the excess of the dimen- 
sions of the final reducing equation. And in as much as there 
exist no periodical or other relations amongst the roots of these 
reducing equations, it is obvious that the application of this 
process, and therefore also of any of those primary methods 
which lead to the assumption of the form of the roots of the 
reducing equation, must increase instead of diminishing the 
difficulties of the solution which was required to be found. 

It was the imagined discovery of a cyclical period amongst 
the roots of this reducing equation which induced Meyer Hirsch, 
a mathematician of very considerable attainments, to believe 
that he had discovered methods for the general solution of equa- 
tions of the fifth and higher degrees. Amongst the different 
methods which Lagrange has analysed in the Berlin Memoirs 
is that which Tschirnhausen proposed in the Acta Eruditorum 
for 1683, It proposed to exterminate, by means of an auxiliary 
equation, all the terms of the original equation except the first 
and the last, and thus to reduce it to a binomial equation. 
Thus, in order to exterminate the second term of x^ -\- a x 
+ 6 = 0, we must employ the auxiliary equation y -\- K ■\- x 
= 0, and then eliminate x. To exterminate simultaneously 
the second and third terms of the cubic equation a;^ -J- « x'^ 
+ b X + c = 0, we must employ the auxiliary equation y + A 
+ B X + x'^ = 0, and then eliminate x ; and more generally, to 
destroy all the intermediate terms of an equation of « dimen- 

x" + «j .I'"-' + a^ x"-' -{- . . . a„ = 0, 

314 THIRD REPORT — 1833. 

we must employ the auxiliary equation 

7/ + A + AjX + Ac^x^ + . . . ;r«-' = 0, 

whose dimensions are less by 1 than those of the given equation. 

Such a process is apparently very simple and uniform and 
equally applicable to all equations ; and so it appeared to its 
author. But it will be found that the equations upon which 
the determination of A, Aj, Ag, depend, in an equation of the 
fourth degree, will rise to the sixth degree, which are subse- 
quently reducible to others of the third degree ; and that for 
an equation of the fifth degree, it will be impossible to reduce 
them below the sixth degree. Such was the decision of La- 
grange, who has subjected this process to a most laborious 
analysis, and who has actually calculated one of the coefficients 
of the final reducing equation, and shown the mode in which 
the others may be determined *. 

Meyer Hirsch, however, though fully adopting the conclu- 
sions of Lagrange to this extent, attempted to proceed further ; 
and, deceived by the form which he gave to his types of combina- 
tion, imagined that he had discovered cyclical periods amongst 
the roots of this final equation, by which it might be resolved 
into two equations of the third degree. If such a distribution 
of the roots was practicable in tlie case of the final equation cor- 
responding to equations of the fifth degree, it would be practi- 
cable in that corresponding to equations of higher degrees. 
But some consequences of this discovery, and particularly the 
multiplicity of solutions which it gave, would have startled an 
analyst whose prudence was not laid asleep by the excitement 
consequent upon the expected attainment of a memorable ad- 
vancement in analysis, which had eluded the grasp even of 
Lagrange. Its author, however, was too profound an analyst 
to continue long ignorant at once of the consequences of his 
error and of the source from which it sprung. In the Preface 
to his Integraltafeln, an excellent work, which was published 
in 1810, within two years of the announcement of his discovery, 
he acknowledges with great modesty and propriety, that he 
had not succeeded in effecting general solutions of equations 
in the sense in which the problem was understood by Euler, 
Lagrange, and the greatest analysts. 

The well known Hoene de Wronski, in a short pamphlet pub- 
lished in 181 1, announced a method for the general resolution of 
equations. He assumes hypothetical expressions for the roots of 
the given equation in terms of the n roots of 1, and of the {m — V) 

* In the Berlin Memoirs for 1771. p- 170 : it forms a work of prodigious 
labour, such as few persons would venture to undertake or to repeat. 


roots of a reduced equation of (w — 1) dimensions, and employs 
in the determination of the coefficients of this reduced equation 
if ~^ fundamental eqaaiions, designated by the Hebrew letter {^, 
and ?«"~^ others designated by the Greek letter i2. It is un- 
necessary, however, to enter upon an examination of the truth 
of processes which the author who proposes them has left un- 
demonstrated ; and in as much as the application of his method 
to an equation of 5 dimensions would require the formation of 
625 fundamental equations of the class Aleph and 125 of the 
class Omega, and the determination of the greatest common 
measure of 2 polynomials of 24 and 30 dimensions respectively, 
it was quite clear that M. Wronski might in perfect safety retire 
behind an intrenchment of equations and operations of this 
formidable nature. And this was the position which he took 
in answer to M. Gergonne, who, in the third volume of the An- 
nales de Mathematiques, in the modest form of doubts, showed 
that the form of the roots which he had assumed was not essen- 
tially different from those which Waring, Bezout, and Euler, 
had assumed, and which Lagrange had shown to be incompa- 
tible with the existence of a final reducing equation of the di- 
mensions assigned to it*. 

The process given by Lagrange for determining the dimen- 
sions and nature of the final reducing equation has been the 
touchstone by which all the methods which have been hitherto 
proposed for the solution of equations have been tried, and will 
probably continue to serve the same purpose for all similar at- 
tempts which may be hereafter made. Its illustrious author, 
however, hesitated to pronounce a decisive opinion respecting 
the possibility of the problem, contenting himself with demon- 
strating it to be so, with reference to every method which had 
been suggested, or which could be shown to arise naturally out 

* The works of Hoene de Wronski were received with extraordinary favour 
in Portugal, where the Baron Stockier, a mathematician of considerable at- 
tainments, and other members of the Academy of Sciences became converts 
to his opinions. There is, in fact, a bold and imposing generality, and appa- 
rent comprehensiveness of views in his speculations, which are well calculated 
to deceive a reader whose mind is not fortified by the possession of an extensive 
and well digested knowledge of analysis. In the year 1817, the Academy of 
Sciences at Lisbon proposed as a prize, " The demonstration of Wronski 's 
formulae for the general resolution of equations," which was adjudged in the 
following year to an excellent refutation of their truth by the academician 
Evangelista Torriani : it chiefly consists in showing, and that very clearly, 
that the coefficients of the reducing equation of (« — 1) dimensions, assuming 
the form of the roots of the equation which Wronski assigned to them, can- 
not be symmetrical functions of those roots, and therefore cannot be expressed 
by the coefficients of the primitive equation, whatever be the number, nature 
and derivation of the fundamental equations }^ and -Q which arc interposed. 

316 THIRD REPORT — 1833. 

of the conditions of the problem itself. But even if we should 
assume the impossibility of the problem, to the full extent of 
Abel's demonstration, it is still possible that there may exist 
solutions by means of undiscovered transcendents. It is, in fact, 
quite impossible to attempt to limit the resources of analysis, or 
to demonstrate the nonexistence of symbolical forms which may 
be competent to fulfil every condition which the solution of this 
problem may require. In conformity with such views, we may 
consider the numerical roots of equations as the only discover- 
able values of such transcendental functions ; but it is quite 
obvious that such values will in no respect assist us in deter- 
mining their nature or symbolical form, in the absence of any 
knowledge of the course of successive operations upon all the 
coefficients of the equation which were required for their de- 

Though we may venture to despair, at least in the present 
limited state of our knowledge of transcendental functions, of 
ever effecting the general resolution of equations, in the large 
sense in which that problem is commonly proposed and under- 
stood, yet there are large classes of equations of all orders 
which admit of perfect algebraical solution. The principal pro- 
perties of the roots of the binomial equation ^"—1=0, had 
long been ascertained by the researches of Waring and La- 
grange, and its general transcendental solution had been com- 
pletely effected. Its algebraical solution, howevei*, had been 
limited to values of n not exceeding 10 ; and though Vander- 
monde in some very remarkable researches *, which were con- 
temporary with those of Lagrange, had given the solution of 
the equation x" — 1 = 0, as a consequence of his general me- 
thod for the solution of equations, and had asserted that it 
could be extended to those of higher dimensions, yet his solu- 
tion contained no developement of the peculiar theory of such 
binomial equations, and was so little understood, that his dis- 
covery, if such it may be termed, remained al^arren fact, which 
in no way contributed to the advancement of our analytical 

The appearance of the Disquisitiones Arithmeticee of the 

* Memoires de I'Academie de Paris for 1771. The result only of this solu- 
tion was given, the steps of the process by which it was obtained being omitted. 
This result has been verified by Lagrange in Note xiv. to his Traiie szir la 
Resolution des Equations Nameriqties. Poinsot, in a memoir on the solution 
of the congruence x" — 1 := M {p), which will be noticed in the text, has at- 
tempted to set up a prior claim in favour of Vandermonde for Gauss's memo- 
rable discovery ; in doing so, however, he appears to have been more influ- 
enced by his national predilections in favour of his countrymen, than by a 
strict regard to historical truth and justice. 


celebrated Gauss, in 1801, gave an immense extension to our 

knowledge of the theory and solution of such binomial equa- 

tions. It was well known that the roots of the equation — ; — r- =0, 

where w is a prime number, could be expressed by the terms of 
the series 

r + r'^ -\- r^ + . . . r"~^, 

where r represented any root whatever of the equation, and 
where, consequently, the first term r might be replaced by any 
term of the series. But in this form of the roots there is pre- 
sented no means of distributing them into cyclical periods, nor 
even of ascei'taining the existence of such periods or of determin- 
ing their laws. It was the happy substitution of a geometrical 
series formed by the successive powers of a primitive root* oin, 
in place of the arithmetical series of natural numbers, as the in- 
dices of r, which enabled him to exhibit not merely all the dif- 


- 1 

ferent roots of the equation j = 0, but which also made 

manifest the cyclical periods which existed amongst them. 
Thus, if a was a primitive root of n, and n — I — mk, then in 
the series 

**, r , r , r , . , , r , . , . r , 

the m successive series which are formed by the selection of 
every k^^ term, beginning with the first, the second, the third, 
and so on successively, or the k successive series which are 
formed in a similar manner by the selection of every m^^ term, 
are periodical ; and if the number m or k of terms in one of 
those periods be a composite number, they will further admit of 
resolutions into periods in the same manner with the complete 
series of roots of the equation. The terms of such periods will 
be reproduced in the same order, if they are produced to any 
extent according to the same law, it being understood that the 
multiples of n which are included in the indices which succes- 
sively arise, are rejected, for the purpose of exhibiting their 
values and their laws of formation in the most simple and ob- 
vious form. If two or more periods also are multiplied together, 
the product will be composed of complete periods or of 1 , or of 
multiples of them, the rules for whose determination are easily 

• There are as' many primitive roots of n as there are numbers less than 
n — 1 ■which are prime to it. Euler, who first noticed such primitive roots 
as determined by Fermat's theorem, determined them by an empirical pro- 
cess. Mr. Ivory, in his admirable article on Equations, in the Supplement 
to the Encyclopccdia Britannka, has given a rule for finding them directly. 

318 THIRD REPORT — 1833. 

framed * ; and it arises from the application of such rules that 
we are enabled to determine the coefficients of an equation of 
which those periods are the roots, and thus to depress the 
original binomial equation to one whose dimensions are the 
greatest prime number, which is a divisor of w — 1. 

It follows, therefore, that if the highest prime factor of » — 1 
be 2, the resolution of the binomial equation a;" — 1 = will 
be made to depend upon the solution of quadratic equations 
only, and consequently to depend upon constructions which 
can be effected by combinations of straight lines and circles, 
and therefore within the strict province of plane geometry : 
this will take place whenever n is equal to 2* + 1 and is also 
a prime number. Thus, if A' = 4 we have « = 17, a prime 
number, and therefore the solution of the equation a:'^ — 1 = 
will be reducible to that of four quadratic equations. Similar 
observations apply to the equations 

^2V 1 _ 1 = and x'^^^+ 1-1=0. 

The same principles which enable us to solve algebraically 
binomial equations, under the circumstances above noticed, will 
admit of extension to other classes of equations, whose roots 
admit of analogous relations amongst each other. Gauss f has 
stated that the principles of his theory were applicable to func- 

tions dependent upon the transcendent /^yj\ 4\> which de- 

fines the arcs of the lemniscata, as well as to various species of 
congruencies ; and he has also partially applied them to certain 
classes of equations dependent upon angular sections, though 
in a form which is very imperfectly and very obscurely deve- 
loped. Abel, however, in a memoir X which is remarkable for 
the generality of its views and for its minute and careful ana- 
lysis, has not merely completed Gauss's theory, but made most 
important additions to it, particularly in the solution of exten- 
sive classes of equations which present themselves in the theory 
of elliptic transcendents §. Thus he has given the complete 

* Symmetrical functions of these periods will be multiples of the sum ( — 1) 
of these periods and of 1. This conclusion follows immediately from the re- 
placement of the arithmetical by the geometrical series of indices, according 
to the general process of Lagrange, without any antecedent distribution of 
the roots into periods. See Note xiv. to the Resolution des Equations Num.e- 
riques. It follows from thence that the coefficients of the reducing equations 
will be whole numbers. 

■f Disqiiisitiones Arithmetic(B, pp. 595, 645. 

I " Sur une Classe particuliere d'Equations resolubles algebriquement," — 
Crelle's Journal, vol. iv. p. 131. 

§ Crelle's Journal, vol. iv. p. 314, and elsewhere. 


algebraical resolution of an equation whose roots can be repre- 
sented by 

X, 6 X, &^ X, . . . . 6^-' X, 

where &^ x = x, and where 6 is a rational function of 'x and of 
known quantities ; and also of an equation where all the roots 
can be expressed rationally in terms of one of them, and where, 
if 9 ^ and d^ x express any other two of the roots, we have like- 

It is impossible, however, within a space much less than that 
of the memoir itself, to give any intelligible account of the pro- 
cess followed in the demonstration of these propositions, and 
of many others which are connected with them. We shall con- 
tent ourselves, therefore, with a slight notice of their applica- 
tion to circular functions. 

o -J. 

If we suppose a = — , the equation whose roots are cos a, 
cos 2 a, cos 3 «, . . . cos /x a is 

^^_|.^^-2 + ^./i^ii:_l)^^-4. .. =0 (1.) 

which may be easily shown to possess the required form and 
properties ; — for, in the first place, cos m a =■ ^ (cos or), where 9 
is, as is well known, a rational function of cos « or a:* ; and, 
in the second place, if 9 a' = cos m a and 9, a; = cos m^ a, then 
likewise 9 9i a: = cos mm-^a =■ cos ^Wj m a = 9i 9 x, which is the 
second condition which was required to be fulfilled. 

Let us suppose /«, = 2 « + 1, when the roots of the equation 
(1.) will be 

Stt 47r 4<nit ^ 

of which the last is 1, and the ti first of the remainder equal to 
the n last. The equation (1.) may be depressed, therefore, to 
one o{ n dimensions, which is 

x" + ^ x"-' - -T- (« — 1) x"-2 — — (n — 2) x"-3 

1 (n-2){n-3) l_ (»-3)(/»-4) _ 

+ 16' 1.2 "" ^ 32' 1.2 "" i^c.-U{4.) 

whose roots are 

2 TT 4 TT 2 M 

cos ;z r, cos p: — ^, .... COS 


2n + I 2/2 + 1 2« + 1 

320 THIRD REPORT 1833. 

•r/. 2v , 2mn /. 

If cos 7i T = X = cos a, and cos ^ -^ = 8 j; = cos »? a, 

then these roots are reducible to the form 

xjxj'^x, . . . 6"-' X, 

cos a, cos m a, cos m^ a, . . . cos m" ' « : 

and if we suppose m to be a primitive root to the modulus 
2n + I, then all the roots 

cos a, cos m a, cos m^ a, . . . cos m"~' « 

will be different from each other, and cos m" a = cos a; con- 
sequently it will follow, since the roots of the equation (2.) are 
of the form 

x,6a:,&^a:, . . . 6""' x, 

where $^x = x, they will admit, in conformity with the preceding 
theorems, of algebraical expression. 

Abel has given the general form of the expression for these 
roots, which in this case are all real ; and their determination 
will involve the division of a circle into 2 n equal parts, the 
division of an assigned or assignable arc into 2 n equal parts, 
and the extraction of the square root of 2 w + 1 ; a conclusion 
to which Gauss had also arrived, though he has not given the 
steps of the process which he followed for obtaining it*. If we 
suppose 2n = 2", we shall get the case of regvilar polygons of 
2"+i 4- 1 sides, which admit of indefinite inscription in circles 
by purely geometrical means. It will follow from the same re- 
sult that the inscription of a heptagon will depend upon that 
of a hexagon, the trisection of a given angle, and the extraction 
of the square root of 7. 

Poinsotf has given a very remarkable extension to the theory 
of the solution of the binomial equation x" — 1 = 0, by showing 
that its imaginary roots may be considered in a certain sense 
as the analytical representation of the whole numbers which 
satisfy the congruence or equation 

^« - 1 = M ip), 

whose modulus (a prime number) is p: thus, the imaginary 
cube roots of 1, or the imaginary roots of ^ — 1 = 0, are 

— 1 + V— 3 — 1 — V — 3 ^^ ^jjg whole numbers 4 and 2, 
2 ' , 2 ' 

* Disquisitiones Arithinetic<E, p. 651. 

t Journal de I'Ecole Polytechnique, cahier 18. 



which satisfy the congruence 

,r3 _ 1 = M X 7, 

- 1 + 7 + ^ — 3 + 7 

whose modukis is 7; are expressed by 


and ~ — — , which arise from adding 7 to the 

parts without and beneath the radical sign. 

The principle of this transition from the root of the equation 
to that of the congruence is sufficiently simple. We consider 
the roots of x" — 1 = as resulting from the expression for 
those of the congruence or" — 1 = M (/>), when M = ; and 
we thus are enabled to infer, in as much as M (/>), its multiples 
and powers, are involved in those formulse, whether without 
or beneath the radicals, and disappear, therefore, when M = 0, 
that some such multiples, to be determined by trial, or other- 
wise, are to be added when M (/?) is restored, or when 1 is 
replaced by 1 + M (p). When the congruence admits of in- 
tegral values of jr, which are less than p, then they can be found 
by trial : when no such integral values exist, then, amongst the 
irrational values which thus arise, those values will present them- 
selves which will satisfy the congruence algebraically, though 
they can only be ascertained by a tentative process. 

The equation of Fermat, 

^p-' — 1 = M (^), 

where /> is a prime number, will be satisfied by all the natural 
numbers 1, 2, 3, . . as far as (^ — 1) : and it follows, therefore, 
that all the rational roots of the equation 

a7» — 1 = M (j9) 

will be common to the equation 

x^-i - 1 = M (^), 

the number of them being equal to {d), the greatest common 
divisor of w and oi p — \. If <^ be 1, then all the roots except 
1 are irrational. If we suppose the equation to be 

tp - 1 = M (p), 

then all the roots will be equal to each other and to 1. It is 
unnecessary, however, to enter upon the further examination 
of such cases, which are developed with great care and sin- 
gular ingenuity in the memoir referred to. 

These views of Poinsot are chiefly interesting and valuable as 
connecting the theory of indeterminate with that of ordinary 

1833. Y 

322 THIRD REPORT — 1833. 

equations. It has, in fact, been too much the custom of analysts 
to consider the theory of numbers as altogether separated from 
that of ordinary algebra. The methods employed have generally 
been confined to the specific problem under consideration, and 
have been altogether incapable of application when the known 
quantities employed were expressed by general symbols and not 
by specific numbers. It is to this cause that we may chiefly attri- 
bute the want of continuity in the methods of investigation 
which have been pursued, and the great confusion which has 
been occasioned by the multiplication of insulated facts and 
propositions which were not referable to, nor deducible from, 
any general and comprehensive theory. 

Libri, in his Teoria del Nmneri, and in his Memoir es de 
Mathdmatiqtie et de Physique, has not merely extended the 
views of Poinsot, but has endeavoured to comprehend all those 
conditions in the theory of numbers, by means of algebraical or 
transcendental equations, which were previously understood 
merely, and not symbohcally expressed. He has shown that 
problems which have been usually considered as indeter- 
minate are really more than determinate, and he has thus been 
enabled to explain many anomalies which had formerly embar- 
rassed analysts, by showing the necessary existence of an equa- 
tion of condition, which jKiust be satisfied, in order that the 
problem required to be solved may be possible. By the aid of 
such principles the solutions of indeterminate equations, at 
least within finite limits, may be found directly, and without 
the necessity of resorting to merely tentative processes. 

A great multitude of new and interesting conclusions result 
from such views of the theory of numbers ; but the limits and 
object of this Report will not allow me to discuss them in de- 
tail, or to point out their connexion with the general theory of 
equations, and with the properties of circular and other func- 
tions. The reader, however, will find, in the second of the 
memoirs of Libri above referred to, a general sketch of the 
nature and consequences of these researches, which is unfor- 
tunately, however, too rapid and too imperfectly developed to 
put him in full and satisfactory possession of all the bases of 
this most important theory. 

On the Solution of Numerical Equations, — The resolution 
of numerical equations formed the subject of a truly classical 
work by Lagrange, in which this problem, one of the most im- 
portant in algebra, is not only completely solved, but the imper- 
fections of all the methods which had been proposed for this 
purpose by other authors are pointed out with that singular 
distinctness and elegance which always distinguish his reviews 


of the progress and existing state of the diiFerent branches of 
the mathematical sciences. In the following report we shall 
commence by a general account of the state in which the pro- 
blem was left by him, and of the practical difficulties which 
attend the use of his methods, and we shall then proceed to 
notice the important labours of Fourier and other authors, with 
a view to bring its solution within the reach of arithmetical 
processes which are at once general and easy of application. 

The resolution of numerical equations involves two principal 
objects of research : the first of them concerns the separation 
of the roots into real and imaginary, positive and negative, and 
the determination of the limits between which the real roots 
are severally placed ; the second regards the actual numerical 
approximation to their values, when their limits and nature have 
been previously ascertained. Many different methods have been 
proposed for both these objects, which dilFer greatly from each 
other, both in their theoretical perfection and in their practical 
applicability. We shall begin with a notice of the first class of me- 
thods, which have been proposed for the separation of the roots. 

If the coefficients of an equation be whole numbers or rational 
fractions, their real roots will be either whole numbers or ra- 
tional fractions, or otherwise irrational quantities, which will be 
generally conjugate* to each other and which will generally pre- 
sent themselves, therefore, in pairs. The method of divisors 
which Newton proposed, and which Maclaurin perfected, will 
enable us to determine roots of the first class, and they are also 
determined immediately and completely by nearly all methods 
of approximation. It will be to roots of the second class, there- 
fore, that our methods of approximation will require to be ap- 
plied, though such methods will never enable us to assign them 
under their finite irrational form, nor would our knowledge of 
their existence under such a form in any way aid us, unless in a 
very small number of cases, in the determination of their ap- 
proximate numerical values. 

The equal roots of equations, if any exist, may be detected 
by general methods ; and the factors corresponding to them 
may be completely determined, and the dimensions of the equar 

* An irrational real root may be conjugate to the modulus of a pair of im- 
possible roots ; and there may exist, therefore, as many irrational real roots 
which have no corresponding conjugate real roots as there are pairs of im- 
possible roots in the equation. It is not true, therefore, generally, as is some- 
times asserted, that such irrational roots enter equations by pairs. It would 
not be very diflScult to investigate the different circumstances under which 
roots present themselves, and the different conditions under which they can 
be conjugate to each other ; but the inquiry is not very important, in as much 
as the knowledge of their form would not materially influence the application 
of methods for approximating to their values. 


324 THIRD REPORT — 1833. 

tion depressed by a number of units equal to the number of 
such factors. We might suppose, therefore, in all cases, that 
the roots of the equation to be solved were unequal to each 
other ; but if it should not be considered necessary to perform 
the previous operations which are required for the detection 
and separation of the equal roots, the failure of the methods of 
approximation or other peculiar circumstances connected with 
the determination of the limits of the roots, would indicate their 
existence, and at once direct us to the specific opei'ations upon 
which their determination depends. 

If we svqjpose, therefore, the equal roots to be thus separated 
from the equation to be solved, and if we assume a quantity 
J which is less than the least difference of the unequal roots, 
then the svibstitution of the terms of the series 

k A,{Ji — \) A, . . . , 2 A, A, 0, - A, -2 A, ... . - k^ A, 

where ^ J is greater than the gi*eatest root, and — /'i A less than 
the least root *, will give a series of results, amongst which the 
number of changes of sign from + to — and from — to + will 
be as many as the number of real roots, and no more ; and v/here 
the pairs of consecutive terms of the series of multiples of A 
which correspond to each change of sign are limits to the seve- 
ral real roots of the equation. This is the principle of the me- 
thod of determining the limits of the real roots which was first 
proposed by Waring, and which has been brought into practical 
operation by Lagrange and Cauchy. It remains to explain the 
different methods which have been proposed for the purpose of 
determining the value of J. 

Waring first, and subsequently Lagrange, proposed for this 
purpose the formation of the equation whose roots are the 
squares of the differences of the roots of the given equation. If 
we subsequently transform this equation into one whose roots 
are the reciprocals of its roots, and determine a limit I greater 

than the greatest root of this transformed equation f, then— ;^ 

* A negative root is always considered as less than a positive root, unless 
the consideration of the signs of affection is expressly excluded. 

t Newton proposed for this purpose the formation of the equation whose 
roots are x — e, and v/here e is determined by trial of such a magnitude that 
all the coeflScients of the equation may become positive. In such a case e is the 
limit required. Maclaurin proved that the same property would belong to the 
greatest negative coeflBcient of the equation increased by 1. Cauchy, in his 
Cours d' Analyse, Note iii., and in his Exercices ties Mathematiques, has shown 
that if the coefficients of the equation, without reference to their sign, be 
Ai A2, . . Am, and \f n be the number of such coefficients which are different 
from zero, then that the greatest of the quantities 



will be less than the least difference of any two of the real roots 
of the primitive equation, and will consequently furnish us with 
such a value of A as will enable us to assign their limits. The 
extreme difficulty, however, of forming the equation of dif- 
ferences, which becomes nearly impracticable in the case of 
equations beyond the fourth degree*, renders it nearly, if not 
altogether, useless for the purposes for which this transforma- 
tion was intended by the illustrious analysts who first proposed 
it ; in other words, it is only in a theoretical sense that it can be 
said to furnish the solution of the problem of determining the 
limits of the real roots of an equation. 

Cauchy has succeeded in avoiding the necessity of forming 
the equation of the squares of the differences of the roots, by 
showing that a value of A may be determined from the last term 
of this transformed equation, combined with a value of a limit 
greater than the greatest root of the primitive equation. If we 
suppose H to represent this term, k to be the superior limit 
required, and a and b to represent any two roots of the equa- 
tion, whether real or imaginary, then he has shown that their 
difference a — b, or the modulus of their difference, will be 

will be a superior limit to the roots. An inferior limit (without reference to 
algebraical sign) may be readily found by the same process by the formation 
of the equation whose roots are the reciprocals of the former. 

M. Bret, in the sixth volume of Gergonne's Avnales des Mathematiques, has 
investigated other superior limits of the roots of equations, which admit of 
very easy application, and which likewise give results which are generally not 
very remote from the truth. One of these limits is furnished by the following 
theorem : " If we add to vnity a series of fractions whose numerators are the 
successive negative coefficients, taken positively, and whose denominators 
are the sums of the positive coefficients, including that of the first term, the 
greatest of the resulting values will be a superior limit of the roots of the 
equation." Thus, in the equation 

2 x^ + 11 x^ — 10 aS _ 26 a;-! + 31 a.3 + 72 a;2 — 230 x — 348 = 0, 

the number 4, which is equal to the greatest of the quantities 

1+2^,1 + ^.1 + ^,1 + ^. 

13 13 11(3' 116 

is a superior limit required ; and if we change the signs of the alternate terms, 

we shall have 1 + -— , or 7, a superior limit of the roots of the resulting 

equation : it will follow, therefore, that all the real roots of the first equation 
will be included between 4 and — 7- Other methods are proposed in the 
same memoir which are not equally new or equally simple with the one just 
given, and which I do not think it necessary to notice. 

* Waring, as is well known, gave the transformed equation of the 10th de- 
gree, whose roots were the squares of the differences of the roots of a general 
equation of the fifth degree, wanting its second term : it involves 94 different 
combinations of the coefficients of the original equation, many of them with 
large numerical coefficients. 

326 THIRD REPORT — 1833. 

greater than »(»-i) :, if n denote the dimensions of the 

equation ; and in as much as H is necessarily, when the coeffi- 
cients are whole numbers, either equal to or greater than 1, it 

will follow that «(n-i) , will furnish a proper value of J, 

where k has been determined by the methods described above, 
or in any other manner. The chief objection to the use of a 
value of J thus determined arises from its being generally much 
too small, and from the consequent necessity of making a much 
greater number of trials for the discovery of the limits of the 
roots than would otherwise be necessary. 

Lagrange has proposed different methods of determining the 
value of J, which, though much less laborious, at least for 
equations of high orders, than the equation of the squares of 
the differences, are still liable to great objections, in conse- 
quence of their being indirect, difficult of application, and likely 
to give values of J so small and so uncertain as greatly to mul- 
tiply the number of trials which are necessary to be made *. It 
is for this and other reasons that such methods have never been 
reduced to such a form as to be competent to furnish the re- 
quired limits by means of processes which are expressible in 
the form of arithmetical rules, like those which are given for the 
extraction of the square and cube root in numbers. In this re- 
spect, therefore, they have failed altogether in satisfying the 
great object proposed to be attained by their author, who con- 
sidered the resolution of numerical equations as properly consti- 
tuting a department of common arithmetic, the demonstration 
of whose rules of operation must be subsequently sought for in 
the general theory of algebraical equations +. 

The basis of all methods of solution of numerical equations 
must be found in the previous separation of the roots ; and the 
efforts of algebraists for the last two centuries and a half have 
been directed to the discovery of rules for this purpose. The 
methods, however, which have been proposed have been chiefly 
directed to the separation of the roots into classes, as positive 
and negative, real and imaginary, and not to the determination 
of the successive limits between which they are severally placed. 
The celebrated theorem of Des Cartes J gave a limit to the 
number of positive and negative roots, but failed in deter- 

* Resolution des Equations Numeriques, Note iv. 

t Jhid., Introduction. 

X The proper enunciation of this theorem is the following : "Every equa- 
tion has at least as many changes of sign from + to — and from — to + 
as it has real and positive roots, and at least as many continuations of sign 


mining the absolute number either of one class or of the other, 
in the absence of any means of ascertaining the number of ima- 
ginary roots. If the roots of the equation were all of them real, 
and could be shown to be so by any independent test, it would 
be easy to determine the limits between which the roots were 
severally placed ; for the number of changes of sign which are 
lost upon the substitution ofx + e for x would show the number 
of roots which are included between and e ; and if, therefore, 
we should assume a succession of values of e, whether positive 
or negative, such as to destroy one change of signs and rio more, 
upon the substitution of any two of these successive values, we 
should be enabled to obtain the limits of every root of the 

It was chiefly with a view to this consequence of Des Cartes's 
theorem that De Gua investigated and assigned the conditions 
of the reality of all the roots of an equation. If we suppose 
X = to be the equation, and X', X", X''', X'% X\ &c., to 
denote the successive differential coefiicients of X, then, if all 
the roots of X = be real, the roots of the several derivative 
equations X' = 0, X" = 0, X"' = 0, &c., must be real like- 
wise ; and if the roots of any one of these equations X^*"' = 
be substituted in X^''"'' and X'''+'', it will give results affected 
with different signs. If we form, therefore, a succession of 
equations in y by eliminating successively x from the equations 
t/ = XW . XC-^' and X'»-'' = 0, 

1/ - X(»->^ . X(» 3) and XC- ) = 0, 

y = Xi X"' and X" = 0, y = X X'' and X' = 0, 

the coefficients of all these equations must be positive, forming 

from + to + and from — to — as it has real and negative roots." It is very 
doubtful, notwithstanding the assertions of some authors, whether Des Cartes 
himself was aware of the necessary limitation of the application of this theorem, 
which is required by the possible or ascertained existence of imaginary roots. 

The demonstration which was given by De Gua of this theorem in the Me- 
moires de I'Academie des Sciences for 1741, founded upon the properties of the 
limiting equation or equations, has been completed by Lagrange with his 
usual fullness and elegance, in Note viii. to his Resolution des Equations Nu- 
meriques. The most simple and elementary, however, of all the demonstra- 
tions which have been given of it, and the one, likewise, which arises most 
naturally and immediately from the theory of the composition of equations, is 
that which was given by Segner in the Berlin Memoirs for 1756. The few im- 
perfections which attach to this demonstration, as far as the classification of 
the forms which algebraical products may assume is concerned, have been 
completely removed in a demonstration which Gauss has published in the 
third volume of Crelle's Journal. 

This theorem is included as a corollary to Fourier's more general theorem 
for the separation of the roots, as we shall have occasion to notice hereafter. 

328 THIRD REPORT — 1833. 

a collection of conditions of the reality of the roots of an equa- 
tion of n dimensions which are „ in number *. 

These speculations of De Gua were well calculated to show 
the importance of examining the succession of signs of these 
derivative equations, with a view to the discovery of their con- 
nexion with the nature of the roots of the primitive equation. 
The changes in the succession of signs of the coefficients of the 
equations which resulted from the substitution of a: + « and 
X + b, gave no certain indications of the nature and number of 
the roots included between a and b, unless it could be shown 
that all the roots of the primitive equation were real, a case of 
comparatively rare occurrence, and which left the general pro- 
blem of the separation of the roots, as preparatory to their 
actual calculation, nearly untouched. It was the conviction that 
all attempts to effect the solution of this problem by the aid of 
Des Cartes's theorem would necessarily fail, which led Fourier, 
one of the most pi'ofound and philosophical writers on analysis 
and physical science in modern times, to the examination of the 

* Resolution des Equations Numeriques, Note viii. The equation of the 
squares of the differences of the roots of an equation will indicate the reality 

71 \7l ^^ 1 \ 

of all the roots, if its coefficients have ^ changes of sign, or be alter- 

nately positive and negative. The succession of signs of the coefficients very 
readily furnishes the indications of the number of impossible roots in all equa-- 
tions as far as five dimensions, as has been shown by Waring and Lagrange. 

The number of conditions of the reality of the roots of an equation of five 
dimensions would appear from the formula in the text to be 10 ; but some of 
these conditions, as Lagrange has intimated, may, and indeed are, included 
in the system of the others, so as to reduce them to a smaller number. La- 
grange has assigned two conditions (not three) of the reality of the roots of 
a cubic equation ; but the first of these is necessarily included in that of the 
second, so as to reduce the essential conditions to one. Similar consequences 
are found to present themselves in the examination of these conditions for 
an equation of the fourth degree, which are three in number, and not six, as 
the formula would appear to indicate. 

Cauchy, in the 17th cahier of the Journal de I'Ecole Polytechnique, has suc- 
ceeded, by a combined examination of the geometrical properties of the curve 
whose equation is y = X (where X is a rational function of x of the form 
«" + p^ .t«-i + . . . . pn)y and of their corresponding analytical charac- 
ters, in the discovery of general methods, not merely for the determination of 
the number of real roots, but likewise of the number of positive and negative 
roots, as distinguished from each other. These methods are equally appli- 
cable to literal and numerical equations. He has applied his method to ge- 
neral equations of the first five degrees, and the results are in every respect, 
as far at least as they have been examined in common, equivalent to those 
which are derived from the equation of the squares of the differences. It is 
impossible, however, in the space which is allowed to me in this Report, to 
give any intelligible account of this most elaborate and able memoir, and I 
must content myself, therefore, with this general reference to it. 


succession of signs of the function X and its derivatives, upon 
the substitution of different values of x. The conclusions 
which have resulted from this examination, which we shall now 
proceed to state, have completely succeeded in effecting the 
practical solution of this most difficult and important problem, 
as far, at least, as real roots are concerned. 
If we suppose 

X = .r™ + a I a:""-' + a^ x"'-^ + . . .a^ = 0, 
and if we write X and its derivatives in the following order, 
XW X^'"-'\ x(™-2), . . . X", X', X, 

then the substitution of yr and ^, will give two series of re- 
sults, the terms of the first series being all of them positive, 
and those of the second being alternately positive and negative. 

The same will be the case if, in the place of -^, we put any 

limit («) greater than the greatest root of the equation X = 0, and 

if in the place of — ^ we substitute any negative value of 

a' (— /3) (to be determined by trial or otherwise) which will 
make the first terms of X, X', X", &c., considered with regard 
to numerical value only, severally greater than the sum of all 
those which follow them. In the course of the substitution of 
values of x intei'mediate to those extreme values — /3 and a, all 
the ?« changes of sign of X and its derivatives, from + to — 
and from — to + , will disappear, in conformity wath the fol- 
lowing theorems, which are capable of strict demonstration. 

1st. If, upon the substitution of any value o{ x, one or more 
changes of signs disappear, those changes are not recoverable 
by the substitution of any greater value of x. 

2nd. If upon the substitution of two values a and b of .r, 
one change of signs disappears, there is one real root and no 
more included between a and b. If under the same circum- 
stances an odd number 2 p + \ of changes of sign have disap- 
peared, there must be at least one, and there may he 2 p' + I 
(where p' is not greater than 2^) I'eal roots between a and b ; 
but if an even number 2 p of signs have disappeared in the in- 
terval, there ma?/ be 2p — 2p' real roots, and p' pairs of ima- 
ginary roots corresponding to it, where p' is not greater than p. 
If no change of sign disappears, upon the successive substi- 
tution of a and b, then no root whatever of the equation X = 
can be found between the limits a and b. 

3rd. If the substitution of a value a o{ x makes X = 0, then 
a is a root of the equation. If the substitution of the same 
value of X makes at the same time X = and X' = 0, then 

330 THIRD REPORT 1833. 

there are two real roots equal to a ; and generally, as many of 
the final functions X, X', X", &c., will disappear, under the same 
circumstances, as there are roots equal to a. 

4th. If the substitution of a value of a makes one intermediate 
function X'*"' equal to 0, and one only, and if the result be placed 
between two signs of the same kind, whether + and + or — and 
— , then there will be one pair of imaginary roots corresponding to 
this occurrence ; but if be placed between two unlike signs, + 
and — or — and +, then there will be no root corresponding to it, 
unless at the same time X = 0. If the substitution of a makes any 
number of consecutive derivative functions equal to 0, then, if 
there be an even number 2/> of consecutive zeros, there will be^ 
or (^ — 1) pairs of imaginary roots corresponding, according as 
they are placed between the same or different signs ; and if there 
be an odd number 9,p -\- \ of consecutive zeros, then there will 
be^ + \ o\ ]i pairs of imaginary roots corresponding, according 
as they are placed between the same or different signs *. 

The preceding propositions may be easily shown to include 
the theorem of Des Cartes ; for it is obvious that the substitution 
of for or in X and its derivatives will give a succession of signs 
identical with those of the successive coefficients of X, deficient 
terms being replaced by 0. If the extreme values « and — /3 
be substituted, there will be m permanences in one case and m 
changes in the second ; it will follow therefore that the number 
of real and therefore positive roots between a. and cannot ex- 
ceed the number of changes of sign corresponding to o^ = 0, or 
amongst the successive coefficients of the equation ; and that the 
number of real and therefore negative roots between — /3 and 
cannot exceed the number of permanences corresponding to 
a: = 0, or of changes between and — /3, which is also identical 
with the number of successive permanences of sign amongst the 
coefficients of the equation. 

* I have stated this rule differently from Fourier, whose rule of the double 
sign appears to me to be superfluous. If we consider the zeros as possessing 
arbitrary signs, the nature and extent of the ambiguity which they produce 
will always be determined by the circumstances of their position with respect 
to the preceding and succeeding sign. 

The rule of the double sign, when one of the derivative functions X', X", X'", 
&c., becomes equal to zero, is made use of in a memoir by Mr. W. G. Horner, 
in the Philosophical Transactions for 1819, upon a new method of solving nu- 
merical equations. This memoir, though very imperfectly developed, and in 
many parts of it very awkwardly and obscurely expressed, contains many 
original views, and also a very valuable arithmetical method of extracting the 
roots of affected equations. It makes also a very near approach to Fourier's 
method of separating the roots of equations. It is proper to state that 
Fourier's proposition was known to him as early as 1796 or 1797, as very 
clearly appears from M. Navier's Preface to \\\s Analyse des Equations Deter- 
minees, a posthumous work, which appeared in 1831. 


In order to render the preceding propositions more easily in- 
telligible, we will apply them to two examples. 

Let X = x'* — 4<r^ — 3x+ 23 = 0, and underneath X'", 
X'", X", X', X, let us write down the signs of the results of the 
substitution of 0, 1, 2, 3, 10, in the place of x, in conformity 
with the following scheme : 

X', X, 

- + 

- + 





















+ + 

For X = 0, there is a result placed between two similar 
signs ; there is therefore a pair of imaginary roots correspond- 
ing to it. Every value of x less than will give results alter- 
nately + and — , and there is therefore no real negative root. 

For X = I, there is a result placed between two dissimilar 
signs : there is therefore no pair of imaginary roots corre- 
sponding ; and since there is no loss of changes of sign in pass- 
ing from to 1, there is no real root between those values. 

For X = 2, there is a result placed between two dissimilar 
signs ; there is therefore no pair of imaginary roots correspond- 
ing, and there is no root between 1 and 2. 

For X = 3, there is a loss of one change of sign, and there is 
therefore one real root between 2 and 3. 

For jr = 10, there is a loss of one change of signs and all the 
resulting signs are positive ; there is therefore one real root 
between 3 and 10. 

The limits of the real roots are thus completely determined, 
and the substitution of the successive whole numbers, from 3 
upwards, will show the nearest whole numbers 3 and 4, between 
which the greatest root is situated. 

Let X = ^6 _ 12 > + eOx'^ + 123 .r^ + 4567 x - 89012 = 















































All tlie real roots of the equation are included between the 
extreme values — 10 and 10. 

332 THIRD REPORT — 1833. • 

One change of sign is lost in the transition from — 10 to — 1, 
and there is therefore one real root between them ; the sign of 
the last term is therefore necessarily changed from + to — . 

For X = 0, there is a result laetween two similar signs ; 
there is therefore a pair of imaginary roots cori'esponding, and 
consequently a loss of two changes of sign. 

There is no root of the equation between and 1 . 

There is a loss of three changes of sign in the transition from 
1 to 10, and therefore there are three roots corresponding, one 
or all of which may be real : the application of a subsequent 
rule will show that two of them are imaginary. 

It is obvious, in a series of derivatives, X^™^, X^"*"'', . . . X'*"' 
... X, that X'*"', X^"*"'^ may be considered as the derivatives 
of the {m — r —ly^ and {m — r — S)* order from X'*"^, as well 
as the wj"' and {tn — 1)* derivatives from X, and that the same 
rules may be applied to the separation of the roots of tliese de- 
rivatives when they become equations, whether they be consi- 
dered as belonging to the inferior or to the superior order. The 
substitution, therefore, of a and b successively for x, will show 
the number of roots of the successive derivative equations which 
are found in this interval, which will be equal successively to 
the number of changes of sign which have disappeared in the 
transition from one value of x to the other. If we now place 
under the several results of the substitution of a and b, a series 
of zeros or numbers as indices to signify that no change, or an 
indicated number of changes of signs, have disappeared, then in 
passing from the left to the right, we shall find first zero, and sub- 
sequently, whether immediately or not, the numbers, 1, 2, &c., 
which will indicate the number of roots which must be sought 
for, in that interval, in the derivative or other functions, consider- 
ed as equations, which are severally placed above them. Thus, 
i{ X. = x'^ — x^ + 4< x^ + X — 4) — 0, then from the scheme 

X'% X"', X", X', X, 
(-10) + - + _ + 

























we infer that there is one root of X = 0, and no root of any of 
the several derivative equations situated between — 10 and — 1 j 


that there is one root of X' = 0, and no root of X =: 0, between 
— 1 and ; that there is one root of X"' = 0, two roots of 
X" = 0, two roots of X' = 0, and"three roots of X = 0, situated 
between and 1. It remains to determine whether these three 
roots are all of them real, or two of them imaginary, and also 
to assign the limits, in the first case *, between v/hich they are 

In the first place, if imaginary roots exist in the derived, 
they will exist also in the primitive equation. The converse of 
this proposition is not necessarily true. 

If the succession of indices be 0, 1, 2, then the succession of 
signs corresponding to 

X", X\ X, or X('- + ^), X('^ X^'--'), 
will be 

(a) + - + or - + - 

12 12 

(b) + + + - _ _ 

There will be one real root between a and b in the equation 
X' = or X^*"^ = 0, and two roots, whether real or imaginary, 
corresponding to this interval, in X = or X^'"'^ = 0. 

In the first case, if there be two real roots between a and b, 
then the curve whose equation 
is y = X =y (a;), where o a = a, 
o b = b, a n =f(a), b m = 
f (b), will cut the axis at the 
points a and ^ between a and b. 

The curve will have no point 

of inflection between a and b, ^ ^ 

since X" preserves the same sign, whether + or — ; and there 
will be a point t, where the tangent is parallel to the axis, 
since X', in the same interval, changes from + to — , or con- 
versely, and therefore becomes equal to ^ero between those 
limits. In this case, the sum of the subtangents (considered 
without regard to algebraical signs) will be necessarily less than 
a b ; and if the interval a 6 be subdivided sufficiently, so as to 
furnish new limits a' and b', then one or both of these points 
will sooner or later be found between the points of intersection 
« and /3, and therefore/ («') and/ (6') will one or both of them 
change their signs. The analytical expression of those geo- 
metrical conditions, and therefore of the existence of two real 

roots, will be, that the sum of the subtangents or quotients *4-7^ 

./ («) 

* We seek for the limits of the real roots only ; we have no concern with 
those of the imaginary roots or of their moduli. 

834 THIRD REPORT — 1833. 

f (h\ 

+ fufi ^b — a, when no regard is paid to the sign of/' (a) 

and f (b). In this case new Kmits must be taken successively, 
intermediate to a and b, until f («') and J" (b') one or both of 
them change their sign. 

In the second case, if there be two imaginary roots cor- 
responding to the interval p- „ 
between a and b, then the ^ 
curve whose equation is 2/=X 
though similar in its other ge- 
ometrical properties to fig. 1, 
will not cut the axis between 
a and b. In this case the sum of the subtangents a n' and 
h nil will either exceed the interval a b, or will ultimately ex- 
ceed it, when the interval a 6 is sufficiently diminished. The 

corresponding analytical character will be that ^^r^. + ^irrr: 

is either greater than 6 — a, or that it may ultimately be made 
to exceed it *. 

Thus, in the example refen*ed to above, p. 332, write down 
the following scheme : 

X", X-, X", X', X, 
6 8 

(0) + _ + + _ 
13 2 3 

(1) + + + + + 

18 14 

and place above and below the indices 1 and 2, in the succes- 
sion of indices 0, 1,2, the values of X"' and X" respectively, 
without regard to sign, corresponding to a: = and x ^=\; then 

8 V . . 8 14 

we shall find -^ 7 1 and, a fortiori, therefore -pr + rn, also 

greater than 1, which is the interval between which the roots 
required are to be sought for : it consequently follows that two 
of the roots corresponding to this interval are imaginary, and 
there remains, therefore, only one real root between and 1. 
If we suppose 

X=a;5 + a;4 + a^-2x2-|-2jr-l = 0, 

* The new values a' and 6' of a and h may be made a 4- ., \~{ and b — \^ ,J . 

' f'(a) f{b)' 

which are n' and m' respectively : a second trial will generally succeed. 


the corresponding scheme will be as follows : 

X, X, 

















































If we take the interval from (— 1) to 0, we find two roots in- 

42 6 
eluded within it ; but since q^ + ht, is less than the interval, 

no certain conclusion can be drawn with respect to the nature 
of the corresponding roots. If we now consider the interval 

from H" to 0, which includes the same roots, we shall find 

9 6 1. 

^ + ^ = -^, a quantity equal to the whole interval, and we 

are consequently authorized in concluding that the correspond- 
ing roots are imaginary. In a similar manner, we find the in- 
dication of the existence of two roots between and -^ ; and 

2 1. 

in as much as -j- = -^ = the whole interval, we at once con- 
clude that the two roots in question are imaginary*. 

It thus appears that we are enabled, by the processes just 
described, to separate all the real roots of an equation and to 

• When we speak of the existence of imaginary roots between two limits, 
we do not mean that such limits comprehend the moduli of these roots, but 
merely that the real roots which would be found between those limits, if cer- 
tain conditions were satisfied, are wanting, and that there are as many ima- 
ginary roots of the equation which may be said to correspond to them which 
are sufficient to complete the required number of changes of sign which are 
lost. The theory of Fourier as given in his work, determines nothing con- 
cerning the values or limits of the moduli, or of the peculiar nature of the 
signs of affection, of such imaginary roots. 

336 THIRD KEPORT — 183^. 

assign their limits, and thus to prepare them for the certain ap- 
plication of methods of approximation. They constitute a most 
important element in the theory of numerical equations ; and 
though they do not enable us to assign the limits of the moduli 
of the pairs of impossible roots nor to determine their signs of 
affection, yet they at once indicate both their existence and their 
number, and thus form the proper pi'eparation, at least for the 
application of methods, whether tentative or not, for the deter- 
mination of their values. 

Lagrange, in the fifth chapter of his Resolution des Equa- 
tions Numeriques, has shown in what manner the equation of 
the squares of the differences may be apjilied to the deter- 
mination of these imaginary roots ; and the methods which 
thence arise are equally complete, in a theoretical sense, with 
those which are made use of, by the aid of the same equation, 
for the determination of the limits of the real roots ; and Le- 
gendre, also, has furnished tentative methods of approximating 
to then' values. But all such methods are more or less nearly 
impracticable for equations of high orders ; and the invention 
of a ready and certain method of separating the imaginary 
roots of equations, as the basis of processes for approximating 
to their values, must still be considered as a great desideratum 
in algebra. 

The method of approximating to the roots of numerical equa- 
tions, when their limits are assigned, which Lagrange has given, 
by means of continued fractions, is so well known that it is quite 
unnecessary to enter upon a detailed examination of its princi- 
ples. If there is only one real root, included between two con- 
secutive whole numbers, there will be only one positive root in 
the several transformed equations, which is greater than 1 , and 
methods which are certain and sufficiently rapid may be applied 
to the determination of the several quotients which form the 
converging fractions. If, however, there are two or more roots 
included between two consecutive whole numbers, there will be 
two or more roots of the first transformed equation, and possi- 
bly, likewise, of the transformed equations which follow which 
are greater than 1, and which may be placed between two con- 
secutive whole numbers. The separation of such roots may be 
effected by the methods of Fourier, which have been explained 
above ; but when we have once arrived at a transformed equa- 
tion which has two or more roots greater than 1, no two of which 
are included between two consecutive whole numbers, then we 
shall find the same number of sets of successive transformed 
equations, which will furnish the several sets of quotients to the 
continued fractions, which represent the roots of the primitive 


equation, which are included between two Hmits which are con- 
secutive whole numbers. The formation, however, of these 
transformed equations, and the determination of the next infe- 
rior integral limit of their roots, even when no further separation 
of the roots is required, is excessively laborious, and Lagrange 
has pointed out methods by which the operations required for 
both these objects may be greatly simplified. Legendre also, 
in the 14th section of the first part of his Theory of Numbers, 
has given a considerable practical extension to these methods 
of Lagrange. If we combine their processes for finding the 
neai'est inferior limit of the root with the theorems of Budan * 
for the formation of the transformed equations, we shall proba- 
bly have arrived at the greatest simplification which the practi- 
cal solution of numerical equations, by means of continued frac- 
tions, is capable of receiving. 

Lagrange has pointed out the principal defects of the me- 
thod of approximation to the roots of numerical equations which 
was given by Newton f. It is only under particular conditions 
that it is competent to attain the object proposed, and in no 
case does it immediately furnish a measure of the accuracy of 
the approximation. But notwithstanding these objections to 
this method, in the form under which it has been commonly 
applied, it is unquestionably that which most naturally arises 
out of the analytical conditions of the problem, and which is also 
capable of the most immediate and most simple application in 
almost every department of analysis. Lagrange had demon- 
strated that this method could only be applied with safety to 
find the greatest and least roots of an equation, and in those 
cases only in which the moduli of the imaginary roots, if any ex- 
isted, were included in value between such roots. But Fourier 
has shown, by considering the superior and inferior limits of 
every real root, and by a proper examination of certain condi- 
tions which those limits may be made to satisfy, and by insti- 
tuting the approximation simultaneously with respect to both 
those limits, that all sources of ambiguity may be removed and 
the accuracy of the approximation determined J. We shall now 
proceed to give a short notice of these researches. 

* Nouvelle Methode pour la Resolution des Equations Numiriques. It con- 
tains the exposition of exceedingly simple and rapid rules for the formation 
of the transformed equation whose unknown quantity is a; — e, where e is 
any integral or decimal number. In other respects, however, this publication, 
though announced with great pomp and circumstance, is a very superficial 
production, and is only remarkable for having received the charitable notice 
and approbation of Lagrange. 

t Resolution des Equations Numeriques, Note v. 

X Analyse des Equations determinees, livr. ii., Calcul des Racines. 

1833. z 



1. If /(^) = 0, or X = be the equation, /' (x), f" {x\ or 
X', X'' its first and second derivatives, then the limits a and h 
of one of the roots will be sufficiently near for the application of 
this method of approximation, if the three last indices (p. 332) 
be 0, 0, 1 . If this be not the case, the interval between a and h 
must be further subdivided until this last condition is satisfied. 

Under such circumstances there will be no root of the equa- 
tions/' {x) — and/^' {x) = 0, included between a and h : and 
if we suppose y — f ix) to be the equation of a paraboHc curve 
CAB, where O a=- a,Oh =^ h,am =f{a), b n ^f{b), then 

there will be no point of inflection between a and b, and no tan- 
gent parallel to the axis. The analytical conditions above men- 
tioned would show that /(«) and f{b) must necessarily have 
difi'erent signs. 

S. If we suppose b to represent the superior limit of the root 
(a), then the Newtonian approximation gives us the new su- 
perior limit b' = b — jrjj\ ; a new inferior limit will be found 

to be a = « frnl '• these limits are still superior and inferior 

limits of the root a, and are both of them nearer to it than the 
primitive limits b and a. 

If the same operation be repeated by replacing b and a in 
f{b) and/(«) by b' and a', nearer limits will be obtained, and 
it is obvious that the same process may be repeated as often as 
may be thought necessary. And in as much as we obtain both 
the inferior and superior limits corresponding to each operation, 
the difference between them will always be greater than the 
error of each approximation. If we refer to the above figure, and 
suppose n b' to be a tangent to the curve at n, and a' m to be 

drawn parallel to n b', then b b' = f'{by ^^^ « «' = f^Ty 
since/' {b) = tan w 6' 6 = tan m a a. It follows, therefore, that 


O h' and O a' are the new limits h' and d : and if ordinates h' n' 
and a' m' be drawn to the curve, and n' b" be drawn a tangent, 
and m' a" parallel to n' b", then O b" and O a" will be the new 
values b" and a" of b' and a'. The progress of the approxima- 
tion, upon the continued repetition of this process, will now be 
sufficiently manifest. 

3. If we consider the different arrangements of the signs of 

f (x), f (x), f{x), in the transition from the inferior limit a to 

the superior limit b, they will be found to be the following, it 

being kept in mind that the sign oi f{x) alone changes from 

+ to — , or conversely. 

fix) fix) fix) 

+ + - 



+ + + 

a — — + 

b _ _ _ 

+ - + 

+ - - 

a - + — 

— + + 

In the first two cases, the formulae of approximation are 

b — >, ,,, and a — ^4-Wj and commence therefore with the su- 
perior limit. In the last two cases, the formulas of approxima- 
tion are a — 4rW and b — ^-7-Kjandcommence therefore with 

the inferior limit. In other words, that limit must in all cases 
be selected which gives the same sign to /" (x) andy (^), whe- 
ther + or — . The construction of the portions of the corre- 
sponding parabolic curves included between a and b in these 
several cases, will at once make manifest the reason of the selec- 
tion of the superior or inferior limit and likewise the progress 
of the approximation itself*. 

• If, in the figure p. 338, we join the extremities m and n of the ordinates a m 

and 6 « by the chord m N n, which cuts the axis of x in the point N, we shall 

cjr^TvT f{a){b — a) , f{h){h—a) ... . 

find O N= a — •' .)' ^ ., { = 6 — %)' „/ ( » "^^^^^ g'ves a new ap- 

proximate inferior limit in the first two cases considered in the text, and a new 
superior limit in the last two. Other constructions are noticed by Fourier, 
which give similar results. 

In the M^moires de I'Academie Royale de Bruxelles for 1 826, there is a 
memoir on the resolution of numerical equations by Dandelin, in which the 
analytical conditions which must be satisfied bvthe limit, towards which the 

z 2 

•1 'Q 

340 THIRD REPORT — 1833. 

4. In tlie application of these rules some precautions are oc- 
casionally necessary. Thus, if/" (x) a.ndf{x) have a common 
measure 9 (or), and if a root (a) of ip (x) = be included between 
a and b, then there is a point of inflection of the parabolic arc 
between a and b at the point of its intersection with the axis. 
Under such circumstances, the method of approximation must 
be applied to the equation f (x) = 0, and not to tlie primitive 
equationy(a:') = 0, for the purpose of determining the value of «. 
Again, if there exists a common measure of^"' {x) andf{x), which 
becomes equal to zero, for a value of x between a and b, then 
there ai-e two or more equal roots of f{x) = in that interval, 
and the final succession of indices is no longer 0, 0, 1. Other 
precautions connected with the subdivision of the interval b — a 
are sometimes required, which the limits of this Report will not 
allow me to notice in detail. 

It remains to add a few remarks upon the rapidity of the ap- 
proximation, and upon the means by which it may be ascer- 
tained. If we express the primary and secondary intervals 
b — a and b' — «' by i and i', it may be very easily proved that 

2f'{b) ' 

wherey" {a. . .b) denotes some value whichy" (x) assumes when 
we substitute for x a quantity between a and b : and if we form 
the quotient (C) which arises from dividing the greatest value 
of/" («) and/" (6)* by the least value of 2f' («) and 2/' (6), 
and suppose k the order of the greatest articulate or subarticu- 

approximation in Newton's method must be made, are established by a com- 
bination of analytical and geometrical considerations, and in which also the 
new limits h' and a' are respectively found by what he terms the rule of tan- 
gents in one case, and by the rule of chords in the other. Tlie first is the 

subtraction of the subtangent b b' or ^ from b, as involved in the or- 
dinary Newtonian approximation when the proper limit is selected. The 
second is the determination of the value of O N, or a — f\V. ~f (\ > °^ 

f (b\ ~ f( \ ' ^^ ^^^ method taught at the beginning of this Note. It is 

evident that these conclusions involve all that is important in Fourier's re- 
searches upon this part of the subject. 

This memoir of M. Dandelin, which contains a very full and a very clear 
exposition of the whole theory of the Newtonian method of approximation, 
preceded by five years the publication of M. Fourier's work. 

• Since no root of/'" {x) = is included between a and b, it follows that 
either/" (a) or /" (b) will be the greatest value of /" (n . . .h): the same 
remark applies likewise to/' (a) and/' {l). 


late number * immediately greater than this quotient, and n the 
order of the articulate or subarticulate number which is not less 
than the difference of the limits b — a, then if we divide f{b) 
by/' {b), and continue the operation as far as the {2n + kf^ 
decimal, and increase the last digit by 1, the quotient which 
arises being subtracted from or added to, b, according as f{b) 
and/' (b) have the same or different signs, will give a result 
which will differ from the true value of the root by a quantity 

less than ( tt^ ) • And if the same operations be repeated, 

forming successively new limits by means of the results thus 
obtained, we shall obtain a series of limits which are correct as 
far as the {4>n + 3 kj^, the {8n + 7 kf^, &c., decimal place f. 

The processes of approximation which have been described 
above, as well as those which belong to all other methods, re- 
quire divisions and other operations with numbers which are 
sometimes beyond the reach of logarithmic tables, and which it 
is extremely important to abbreviate as much as possible, con- 
sistently with the determination of the accurate digits of the 
results which are required to be found. Such processes were 
taught by Oughtred and other algebraists of the seventeenth 
century, but both their theory and applications have been 
greatly and, perhaps, undeservedly, neglected in later times. 
The consideration, however, of such methods has been partially 
revived by Fourier and some other writers, the first of whom 
has given examples of what he terms ordinate division {division 
ordonn^e,) the principle of which is to conduct the division by the 
employment of a small number of the first digits of the divisor 
only, and to correct the successive remainders, augmented by 
the successive digits of the original dividend, in such a manner 
as to bring into operation the successive digits of the divisor 
when they are required for the determination of the correct 
digit of the quotient, and not before. Such processes, however, 
are incapable of being briefly described, and we can only refer to 
the original work| for the developement of the rule and for ex- 
amples of its application. 

* An articulate number is one of the series 1,10, 200, 7000, &c., where 
the first digit is followed by zeros only. A subarticulate number is one of 
the series '1, '02, '003, &c., and the number which designates the place of 
the first significant digit is supposed to be negative, 

t The course of the approximation, in order to be perfectly regular and rapid, 
would require that 2 n + h should be greater than n, or that n should be 
greater than — ^, a circumstance which might occur if A: or w was negative. 
In such a case it will be necessary, or rather expedient, to subdivide the in- 
terval b — a, until the difference of the two limits does not exceed ( — - ) , 

where « is equal to, or greater than, 1 — k. 

I Analyse des Equations determinecs, livr. ii. p. 188. 

343 THIRD UEPOIIT 1833. 

Similar processes, also, have been investigated and applied 
with remarkable ingenuity and success by Mr. Holdred *, Mr. 
Horner f, and Mr. Nicholson J. The first of these writers, a 
mathematician in humble life, vpho had formed his taste upon 
the study of the older algebraical writers of this country, gave 
very ingenious rules for finding the roots of numerical equa- 
tions. The method proposed by Mr. Horner was founded upon 
much more profound views of analysis and of the relation which 
exists between the processes of algebra and arithmetic, and he 
has not only succeeded in making a very near approximation to 
the true principles upon which the limits of the roots of numerical 
equations are assigned, but by considering the rules for extract- 
ing the roots of numbers and of aflPected numerical equations 
as founded upon common principles, he has reduced the rules 
for these purposes to a form which admits of very rapid and ef- 
fective, though not perhaps of very easy, application. Mr. Ni- 
cholson, by a combination of the methods of Mr. Holdred and 
Mr. Horner, has greatly simplified them both, and reduced them 
to the form of practical rules, which are not much more compli- 
cated than those which are commonly given for the extraction 
of the cube and higher roots of numbers. 

The Newtonian method of approximation, which we have 
hitherto considered, may be termed linear, in as much as the 
equations of a straight line combined with the general equation 
of the parabolic curve are competent to express all the circum- 
stances which characterize it. But methods of approximation 
of higher orders than the first, involving the second or higher 
powers of the unknown quantity to be determined, have likewise 
been considered by Fourier and other writers. That of the se- 
cond order, viewed with reference to the properties of curve 
lines, may be said to result from the contact of arcs of a conical 
parabola. The superior and inferior limits, thus determined, 
converge with great rapidity, the error corresponding to each 
operation being the product of a constant factor with the cube 
of the preceding error. Such methods, however, if viewed with 
reference to the facility of their practical applications, are incom- 
parably less useful than those which are founded upon linear 
approximations ; but there is much which is instrvictive in their 
theory, and particularly as furnishing the means of determining 
immediately the nature of two roots of an equation included in 
a given interval, which the application of the methods for the 

* This method is particularly noticed in Mr. Nicholson's Essay on Involu. 
tion and Evolution. I have never seen the original tract published by Mr. 

f Philosophical Transactions for 1819. 

I Essay oti Involution and Evolution. 1820. 


separation of the roots which we have previously described 
may have left in the first instance uncertain. We refer to the 
end of the second book of Fourier's Analyse des Equations 
determin^es, for a very complete examination of the theory of 
such approximations*. 

It has been a question agitated on more than one occasion, 
whether the tests of the reality of the roots of equations of finite 
dimensions which De Gua established, or rather the principles 
of the much more general theorem of Fourier, were applicable 
likewise to transcendental equations. In a discussion of the 
transcendental equation 

2/ = 1 - ^ + 22 - WTS'^ "*■ ^~7WT¥ ~ ^^■' 

which presents itself in the expression of the law of propagation 
of heat in a solid cylinder f of infinite length, Fourier ventured 
to apply the principles in question to show that all its roots were 
real ; but M. Poisson J has disputed the propriety of such an 
application, both in this case and in others : thus, if we suppose 

X = e'^— be"'', 

we shall find 

• The rule for the determination of the nature of two roots included in a 
given interval, which is given in page 333, is merely the expression of a con- 
sequence of the application of the method of linear approximation to the di- 
stinction of those roots ; and whatever difficulties in certain extreme cases 
may attend the successful application of that rule, will necessarily present 
themselves likewise in the application of the linear approximation under the 
same circumstances. This character, however, is not confined to the Newto- 
nian or linear method of approximation. If the interval of the roots be deter- 
mined, by the application of Fourier's theorem of the succession of signs of 
the original function X and its derivatives, so that no more than two roots 
may be said to exist in that interval, whose nature is unknown, whether real 
or imaginary, then the application of the method of continued fractions, as 
well as of other equivalent modes of approximation, will be competent to de- 
termine the values of those roots when real, and their nature, when imaginary. 
Such, at least, is the assertion of Fourier, who refers to the third book of his 
work on equations for its demonstration. It is unfortunate, however, that 
only two books of this work, which is full of such remarkable researches upon 
the theory of equations, were fully prepared for publication at the time of his 
death. Our knovi^ledge of the contents of the other five books, which were 
left unfinished, is derived from an Expose Synoptique prefixed to those which 
are published, and which contains a general review and analysis of their prin- 
cipal contents. It is to be hoped, however, that the materials which he has 
left behind him will be found to be sufficient at least for their partial, if not 
for their complete restoration. 

t Theorie de la Chaleiir, p. 372. 

t Journal de I' Ecole Poly technique, cahier xix. p. 381; Memoires de I'ln- 
st'Uut, torn. ix. p. 92. 

344' THIRD RErORT — 1833 


d''+^ X 

= e' — b a" e"*, 
= e*' — 6a''+' e"'. 


= e'' — b a"+2 e"'^, 

where w is any whole number, or zero. If we now suppose 

</"+^X _ 

and eliminate, by means of this equation, e", we shall get 

dx"" ~ 


and therefore 

d'^X d"+^X 
dx" ■ c?a;"+2 

: - 6 (1 - a) a» e"^, 
= 6 (1 -a)a»+'e«*, 

= - J2(l _^)3^2»+lg2«^^ 

a quantity which is negative for every real value of x. The 
conclusion which should be drawn, in conformity with Fourier's 
principles, is, that all the roots of the equation e^ — be"'' = 
are real, as well as those of its successive derivatives; whilst 
the fact is, that each of those equations has one real root, and 
an infinite number of imaginary roots, which are included un- 
der the formula 

_ log 6 a" + 2 e TT '/^A 
X — ^ ■ 

In reply to this objection, it has been urged by Fourier that 
Poisson has not very accurately stated the terms of the propo- 
sition in question as applicable to such a case *, and also that 
he has neglected to take into consideration all the roots of the 
equation. For if we suppose that the substitution of two limits 
a and b, in a function f (x) and its derivatives, gives results 
which present the same succession of signs between y*("+'') (x) 
andy^"' (x), then those extreme derivative functions, and those 

* This inaccuracy of statement is rather chargeable upon Fourier himself 
than upon Poisson, who has certainly failed to notice the necessary limitation 
of this proposition upon the occasion which gave rise to its application in 
page 373 of the T/ieorie de la Chaleur. 


also which are included between them, when considered as 
equations, will contain the same number of roots, or none, be- 
tween those limits. This proposition is true, whether the num- 
ber of derivative functions be finite, as in the case of algebraical 
equations, or infinite, as in the case of transcendental equations. 
In the first case, however, it admits of absolute application, in 
consequence of our arriving at a final derivative, from which 
the comparison of the signs of the two series of results com- 
mences. In the second case we can draw no conclusion, in the 
absence of any difference in the signs of the series of results, 
in the transition from one derivative function to another, with 
respect to the number of roots of any of those functions which 
are included betw'een the given limits: thus, \i f{x) = sin x, 
we shall have the same series of signs of sin x and of its deri- 
vatives, however far continued, upon the substitution of the 
limits a and a + 2 ■n-, although it is manifest that there are two 
real roots of sin ^ = between those limits. The general pro- 
position, therefore, will, in such a case, authorize us in con- 
cluding merely that whatever number of roots the equation 
sin a; = includes between the limits a and « + 2 tt, will be 
possessed likewise by all its derivative equations between the 
same limits *. 

There is another point of view, likewise, in which the objec- 
tion advanced by Poisson may be considered as not altogether 
applicable to the example which he puts forward. In considering 

the roots of the derivative functions , „ , ^ — -p, -r — r^ , he 

has not included those of the factor e'^, which those functions 

1 + — ) = 0, it follows that 

there are an infinite number of equal roots (where a: = — oo ) 
of e*' = 0, which equally reduce three or any number of conse- 
cutive derivative functions to zero, and to which, therefore, 
the test of De Gua is no longer appHcable. It would follow, 
therefore, that the existence of imaginary roots in the equation 
X = is no longer contradictory to Fourier's proposition, even 

• If the transcendental function denoted by / (j;) be a determinate function, 
it will always be possible to assign an interval 3, such that the derivative 
function/" {x) = contains no root, or a determinate number of roots, be- 
tween a and a + S. If such an interval or succession of intervals can be de- 
termined for any one derivative function, such as /(") (.i), it will become a 
point of departure for the determination of the number and nature of the roots 
corresponding to the same interval or intervals for all the other derivative 
functions which form the superior or inferior terms of the series. In the case 
of algebraical functions, the point of departure is that derivative function which 
ie a constant quantity. 

346 irilKi) niijfoRT — luutj. 

admitting the correctness of that form of it which Poisson ha^ 

* If we transform e*' by replacing a; by pi, we shall get the expression 

c •* , which may be easily shown as above, and also by other means, to be 
equal to zero when x' is equal to zero, and equal to 1 when x' is equal to in- 

Professor Hamilton of Dublin, in a paper in the Irish Transactions for 1830, 


has quoted the expression e •* as possessing some very peculiar properties, 
which are inconsistent with the universality of a very commonly received 
principle of analysis. It is commonly assumed that " if a real function of a 
positive variable x approaches to zero with the variable, and vanishes along 
with it, then that function can be developed in a real series of the form 

A a;" + B a^ + C ;r'' + &c. (1.) 

where ct, /3, y, &c., are constant and positive. A, B, C, &c., constant, and all 
those coefficients different from zero : but if we put the equation under the 

y.-«. e~^^ = A + B xP'-"' + C xy-''+ &c„ 

supposing 86 the least of the several indices a, /3, y, &c., then if x =: 0, we 

-~2 . 1 

shall find x~'^ e *'=OorA equal to zero ; for if we replace — by y, we 

shall get 

1 i. 

_i. =a;«e^' = y-«ey^ 
4— a 6— a 

= 3,-« + /-« + y + _! + &c., 

^ " ^1.2 1.2.3 

all whose terms are positive, and which, when a;=: or y = oo , will necessarily 

become equal to infijiity : it follows, therefore, that the function e * is not ca- 
pable of developement in a series of the assumed form (1.), The same ex- 
pression, as has been remarked by Professor Hamilton, has been noticed by 
Cauchy as an example of the vanishing of a function and of all its differential 
coefficients, for a particular value of the variable, without the function va- 
nishing for other values of the variable, thus forming an exception to another 
principle generally received in analysis. In his Lemons snr le Calcul Infinite- 
simal, Cauchy has produced this last anomaly as a sufficient reason for not 
founding the principles of the differential calculus upon the developement of 
functions, as effected by or exhibited in, the series of Taylor. 

It is possible that more enlarged views of the analytical relations of zero 
and ivfinity, and of the interpretation of the circumstances of their occurrence, 
as well as of the principles and applications of Taylor's series, may enable 
us to explain these and other anomalies, and to show that they arise natu- 
rally and necessarily out of the very framework of analysis ; but it must be 
confessed that there are many other difficulties, which are yet unexplained, 
which are connected with the developement of e" when x is negative or ima- 


Another method of approximation to the roots of equations 
by means of recurring series was proposed by Daniel Ber- 
nouUi *, and very extensively illustrated and applied by Euler f . 
If we write down m ai-bitrary numbers to form the first m terms 
of the series, and if we assume, for the scale of relation, the 
coefficients of an equation of m dimensions, and form by means 
of it and the assumed terms the other terms of the series which 
may be indefinitely continued, and if we also form a series of 
quotients by dividing each succeeding term (after the arbitrary 
terms) by that which precedes it, then the terms of the se- 
ries of quotients which thence arise, will converge continually 
towards the value of the greatest root of the equation ; and if 
we form the equation whose roots are the reciprocals of those 
of the original equation, and proceed in a similar manner, we 
shall obtain a series of quotients which will converge to the 
greatest root of this equation, whose reciprocal will be the least 
root of the original equation, considered without reference to 
its algebraical sign. 

Lagrange, in the 6th Note to his Resolution des Equations 
Numeriques, has analysed the principles of this method, and 
has shown that its success will depend upon the greatest real 
root, without reference to algebraical sign, being greater than 
the modulus of any of the imaginary roots. If this condition be 
not satisfied, the quotients will not approximate to the value of 
any root of the equation, a consequence which Euler had also 
pointed out. 

The recurring series which is formed by dividing the first 
derivative function /' {x) by / {x), which is equal to 

ginary. Some of these have been noticed in the note to p. 267, in connexion 
with our observations upon Mr. Graves's researches upon the theory of loga- 
rithms ; another is noticed by M. Clausen of Altona, in the second volume of 

Crelle's Journal, p. 287; it is stated as follows :— Since e^'"' '^"^ = 1, 

when ra is a whole number, we get g^ + ^ " '^ '^^ = e and therefore 

consequently e""*" '^' = 1, whenever w is a whole number, — a conclusion 
which M. Clausen characterizes as absurd. Its explanation involves no other 

2»i sr \/ —\ . , 

difficulty than that which is included in the equation e =1, and 

must be sought for in the circumstances which accompany the transition from 
a function to its equivalent series, when a strict arithmetical equality does not 
exist between them. It must be confessed, however, that these difficulties 
arc of a very serious nature, and are in every way deserving of a more care- 
ful examination and analysis than they have hitherto received. 

* Comment. Acad. Petrop., vol. iii. 

t IntroducHo in Analysim Infinitorum, vol. i. cap. xvii. 

348 THIRD REPORT — 1833. 

+ &C., 

when a, /3, y, &c., are the roots of the equation, whether real or 
imaginary, has been shown by Lagrange to be the series fur- 
nished by this method which is most easily formed, and to be 
likewise that which converges most rapidly and certainly to a 
geometrical series in the case of equal roots. In every case the 
terms of the series of quotients are alternately greater and less 
than the root to be determined, and consequently furnish a mea- 
sure of the accuracy of the approximation. 

This method of approximation is generally less rapid and 
certain than those which have already been considered, and, 
as commonly stated, is extremely hmited in its application. It 
is true, as has been shown by Lagrange, that a knowledge of 
the limits of the roots would enable us to apply it to the deter- 
mination of all the real roots by means of a series of transformed 
equations equal to their number, such as is required in the New- 
tonian method of approximation, and also in that of Lagrange ; 
but under such circumstances, and with such data, it is more 
convenient and more expeditious to employ those methods in 
preference to the one which we are now considering. 

Fourier has shown in what manner this method may be ap- 
plied to determine all the roots of an equation, whether ima- 
ginary or real. Let us suppose a, b, c, d, e, &c., to represent 
the roots of the equation arranged in the order of magnitude,, 
the magnitudes of imaginary roots being estimated by the mag- 
nitudes of their moduli; and let A, B, C, D, E, &c., be the 
terms of the recurring series, whose quotients furnish the value 
of the greatest root, when that root is real. Form, in the se- 
cond place, a series whose terms are AD — BC, BE — CD, 
C F — D E, &c., which is also a recurring series, whose quo- 
tients may be easily shown to approximate to the sum of the 
two first roots a + b. Again, form a series whose terms are 
A C - B2, B D - C2, C E - W, &c., which is also a recurring 
series, whose successive quotients will approximate to the value 
of the greatest product a b. In a similar manner, we may de-' 
duce from the primitive recurring series three other recurring 
series, the terms of the convergent series formed by whose quo- 
tients will form, in the first series, the sum a + 6 + c of the 
three first roots ; in the second, the sum of their products two 
or two, ox ab + a c -\- be; and in the third their continued 
product a b c : and similarly for four or a greater number of 
roots. If, therefore, we suppose the first root a to be imaginary, 
the first series will give no result ; but the values of a + 6 and 


of a b, which are given by the two first recurring series de- 
rived from the primitive recurring series, will enable us to de- 
termine their separate values : in both cases the series of quo- 
tients is convergent. 

If the third root be real, the third series of derived quotients 
is convergent ; if not, the fourth series will be so, and so on as 
far as we wish to proceed. 

These propositions have been merely announced by Fourier 
in his Introduction. The chapter of his work, which contains 
the demonstrations, has not yet been published. 

If the root of an equation be determined approximately, the 
equation may be depressed, and the general processes of solu- 
tion or of approximation may be applied to find the roots of 
the quotient of the division. Thus, in the equation 
^3 _ 3 ^ _,. 2-0000001 = 0, 

one of the roots is very nearly equal to 1, if we divide the 
equation by a; — 1, and neglect the small remainder which re- 
sults from the division, we shall get the quotient 

x^ -X -2= {x -\){x + 2) = 0, 

whose roots are 1 and — 2 ; or we may suppose one of the 
roots to be 1*000 1, the second -9999, and the third —2; or 
we may suppose two of the roots to be imaginary, namely, 
1 + -0001 V — \. All these roots are approximate values of 
the roots of the equation, which different processes, whether 
tentative or direct, may determine : and it is obvious that when 
two roots are equal, or nearly so, an inaccuracy of the approxi- 
mation to those roots which are employed in the depression of 
the primitive equation may convert real roots into imaginary, 
or conversely. Such consequences will never follow when the 
limits and nature of the roots are previously ascertained, and 
every root is determined independently of the rest ; but it is 
not very easy to prevent their occurrence when methods of ap- 
proximation are applied without any previous inquiries into the 
nature and limits of the roots, though the resulting conversion 
of imaginary roots into real, and of real roots into imaginary, 
may not deprive them of the character of true approximations 
to the values of the roots which are required to be determined. 
If the limits of the roots of an equation F ^c = be assigned, 
and if the Newtonian method of approximation be applied con- 
tinually to one of these limits a, we should obtain, for the value 
of the root, the series* 

«" ,^ .o «"' 

a-a'Ya+ ^ (F «)^ - ^--^ (F a)^ + &c. 
• Lagrange, li^sQlution des Equations Numeriques, Note xi. 

350 THIRD REPORT — 1833. 


«' =^, 

a" = - 


a! F" a F" 

(F' of ~ (F' a)3 

„, _ _ cH F" a 3 a' (F" of 

Wa 3 (F'^ af 

~ (F «)4 + (F a)^ • 

This series was first assigned by Euler, and the observations 
which we have had occasion to make in the preceding pages 
upon hnear approximations will at once explain the circum- 
stances under which it may be safely applied : it cannot be 
viewed, however, in any other light than as the analytical ex- 
pression for the result of the application of such linear approxi- 
mations, repeated as many times as there are terms of the series 
succeeding the first. 

The celebrated theorem of Lagrange, which is so extensively 
used in the solution of the transcendental equations which pre- 
sent themselves in physical and plane astronomy, will enable 
us to assign, likewise, a series for the least root, or for any 
function of the least root of an equation in terms of its coeffi- 
cients. Mr. Murphy, in a very able memoir in the Transac- 
tions of the Philosophical Society of Cambridge for 1831, has 
shown the mode in which such series may be determined, by 
means of a very simple rule, which admits of very rapid and 
very extensive application. The rule is as follows : 

" To find the series for the least root of the equation ip {x) — 0, 
divide the equation by x, and take the Napierian logarithm of 

the quotient which arises ; then the coefficient of — with its 

sign changed is the series which expresses the least root re- 

Thus, to find the series for the least root of the quadratic 

a;2 4. a a; + 6 = 0, 

find the coefficient, with its sign changed, of — in log ^^ 

1 + X \, which is 

\ b b^ 4 63 Q.5 b^ 8.7.6 ^ , o \ 


and therefoi-e identical with that which arises from the deve- 
lopement oi — -^ + V\-r ~ ^)- If^be greater than -^, the 

roots of the equation are impossible, and the series becomes 
divergent, and gives no result- 
Any function / {x) of the least root of an equation <p (a;) = 
may be found " by subtracting from / (0) the coefficient of 

— m f (x) log ?-LJ," This more general theorem evidently 

includes the former. 

" The sum of any assigned number {ni) of roots of the equa- 
tion <p {x) = is equal to the coefficient, with its sign changed, 

of — m log ^-^. 
X ^ x"" 

The expression for the sum of m roots of an equation which 
is thus obtained gives the arithmetical value of the sum of the 
m least roots. In estimating the order of magnitude of such 
roots no regard is paid to their signs of affection. 

Mr. Murphy has shown in what manner the same general 
proposition which is employed in the deduction of the results 
just given may be applied to the investigation of some of the 
most general theorems which have been employed in analysis 
for the developement and transformation of functions. Amongst 
many others the following very remarkable theorem seems to 
merit particular notice. 

If x^^, Xci, ^3, . . ^TO be the m least roots of the equation 

(x - a)"' - hF (x) = 0, 

= "^f^""^ + ^' l.2..{m-\)da--^ 

Jl_ d^"^-^ {f {a) {¥ {a)Y} 
*■ 1 .2' 1.2 {2m-\)da^'"-' "^ ^' 

If in this very general theorem we make w = 1, it becomes 
the theorem of Lagrange ; and if we make m equal to the di- 
mensions of the equation, or greater than ^y power of x in- 
volved in F {x), then it becomes the theorem which Cauchy 
has given, without demonstration, in the ninth volume of the 
new series of the Memoirs of the Institute, for the expression 
of the sum of the different values oi f{x), when x is succes- 
sively replaced by every root of the equation. 

The preceding conclusions, so very remarkable for their 
great generality, and for the very simple means employed in 


352 THIRD REPORT — 1833. 

their derivation, will be sufficient to direct the attention of the 
reader to the other contents of this very original and valuable 

There ai-e some other most important departments of the 
general theory of equations which it was my intention to have 
noticed, and without which no report upon the present state 
and recent progress of algebra can be said to be complete. 
Amongst these may be particularly mentioned the theory of 
elimination and the solution of simultaneous equations, and also 
the theory of the solution of literal and of implicit equations. 
The very undue length, however, to which this Report has 
already extended, and the arrangements which have been made 
connected with the publication of this volume, compel me, 
though most reluctantly, to omit them. I venture to indulge 
a hope, however, that I may be allowed upon some future oc- 
casion to add a short supplemental Report upon this extensive 
department of analysis, in which I may be enabled to supply 
some of the numerous deficiencies of the preceding sketch. 


Page 197, line 21, dele not 

— 215. — 3, /o»-r(r) r(l»-) reorfr(r) r(l — r) 

— 215, — 11 from the bottom,*. jfor a;^:^- 2 + a; 1 read a:^ + 2 a; + 1 

71 "' 71 

— 221, — 15, /or cos i — read cos- 1 — 

e e 

cos-i a , , a 

— 226, — 2 from the bottom, for / , , ,, read cos-i , „ , ,„ 

Va^ + o2 Va^ + 62 

— 234, — 7, for (0) read (0)» 

— 240, — 18, for In this last case (a — 6)" read If we suppose n to be 

an even whole number or a frac