GEORGE SALMON

HIS MATHEMATICAL WORK AND INFLUENCE

Ro d Gow

There are ve Ir i shb or n mathematicians who have b een deemed

worthy of inclus ion in the Biographical Dictionary of Mathem

aticians William Rowan Hamilton Henry J S Smith George

Gabriel Stokes Patrick DArcy and George Salmon G J Stoney

i s also include d but he should re ally b e cons idere d a physici st

James MacCullagh i s not include d although he was certainly a

mathematical physici st Of thes e it would generally b e agree d

that W R Hamilton whos e di scoverie s are routinely

us e d in s everal branches of mathematics i s the mo st illustr ious

Stokes and Smith are held in high e steem

internationally but their cre ative work was done outside Ireland

Stokes working at Cambridge and Smith at Oxford DArcy

i s probably not well known and he lived mo st of hi s adult life

in France

In this article we intend to di scus s the remaining Ir i sh math

ematician in this li st George Salmon who without making any

mathematical di scoverie s comparable with thos e of the three math

ematicians mentioned above exerted a gre at inuence on mathem

atical re s e arch and teaching in Europe and America in the s econd

half of the th century Thi s inuence was derived f rom four text

b o oks that Salmon published in Dublin b etween and

Each of thes e appe are d in at le ast three e ditions and photographic

repr ints of the nal e ditions of e ach b o ok made in the s by

the Chelse a Publishing Company of New York may still b e avail

able Salmons role as a di s s eminator and p opularizer of contem

p orary re s e arch in algebra and geometry has b een acknowledge d

by mo st hi storians of mathematics although there i s a tendency

George Salmon

for the shorter hi storie s to rep e at the same bas ic as s e s sment of

hi s imp ortance and little els e We would like here to outline some

details of hi s working life concentrating on hi s mathematical activ

ities It i s imp ortant to p oint out that Salmon b ecame Regius

Profe s sor of Divinity at Trinity College Dublin TCD in

and e s s entially ce as e d to b e involved in re s e arch level mathem

atics thereafter We will de scr ib e the contents of thos e b o oks

indicating novel fe atures which appe ale d to hi s re aders and di s

cus s their shortcomings s een f rom a later p oint of view We will

also provide contemporary evidence to illustrate the extent of the

inuence exerci s e d by Salmons textbo oks

Salmons life and career at Trinity College Dublin

There are s everal source s giving details of Salmons life One i s

an obituary by Rob ert Stawell Ball Ball had b een a pupil

of Salmons at TCD in Hi s obituary i s p erhaps a little

to o adulatory although it contains us eful information Ball was a

p opular lecturer and wr iter on astronomy as well as a profe s s ional

astronomer and he tende d to b e enthusiastic in hi s interests and

opinions Another obituary s igned C J J was wr itten by Charles

Jasp er Joly Joly was also an astronomer but he did or iginal

re s e arch in algebra e sp ecially quaternions and Cliord algebras

There i s another unsigned obituary in Nature p erhaps also

by Joly s ince it contains details rather s imilar to thos e in

The Dictionary of National Biography DNB has an article on

Salmon by John H Ber nard who was then Bi shop of Os sory

and later b ecame Archbishop of Dublin Although Ber nard studied

mathematics at TCD and hi s article shows appreciation and under

standing of Salmons mathematical work hi s account concentrates

on Salmons theological and cler ical work which we will largely

ignore here We found the article on Salmon by A J McConnell

in di sappointing for it s eemed to b e largely a paraphras e of

parts of Jolys obituary As a Provost of TCD was wr iting about

one of hi s pre dece s sors as Provost a go o d opportunity to give a

more detailed as s e s sment of Salmons achievements and inuence

to a worldwide audience was mi s s e d The hi story of TCD

has s everal reference s to Salmon and gives instances of hi s biting

IMS Bulletin

wit an interesting asp ect of Salmons p ersonality An e arlier hi s

tory of TCD that of Constantia Maxwell also gives extracts

f rom other appreciatory articles wr itten at the time of Salmons

death Since wr iting this pap er our attention has b een drawn to

an article by T D Sp e arman but we have b een unable to

make us e of the information contained therein

There are two comp eting birthplace s for Salmon Cork and

Dublin What i s certain i s that he was b or n on September

As f ar as we have b een able to di scover any article wr itten

up to that gives information on Salmons birthplace states

that he was b or n in Dublin Thus for example G T Stokes

wr iting on Salmon in The Review of the Churches for June

states

Dr Salmon was b or n in the city of Dublin Though b or n in Dublin

he came of a Cork f amily and circle which has made its mark on this age

of ours for he i s a cous in of the wellknown Profe s sor Edward Dowden

the cr itic

Edward Dowden was Profe s sor of English Literature

at TCD f rom until hi s death Salmon was rst cous in of

Dowdens mother

The obituary of Salmon in The Times of January

was wr itten by Ber nard but it do e s not mention Salmons birth

place The later obituary also by Ber nard in the Proceedings

of the British Academy for states that Salmon was b or n in

Dublin but Ber nards DNB article published in states that

he was b or n in Cork Ber nard corre sp onde d with Salmons relat

ives and f r iends when compiling hi s DNB article and he may have

found new information on the birthplace Nonetheles s the i s sue

remains p erplexing for while mo st recent articles on Salmon have

given Cork as hi s birthplace the standard source of information

on graduates of TCD gives Dublin and the current Ordnance

Survey of Ireland street guide to Dublin in its s ection on f amous

scientic p ersonages who have lived in Dublin do e s the same If

Cork i s the correct answer Salmon made no attempt to correct

erroneous articles published in hi s lifetime and p o s s ibly he did

not know the answer hims elf

Salmon matriculated at TCD in and graduated as rst

George Salmon

Senior Mo derator in Mathematics and Physics in ahead of

William Rob erts and William Bindon Blo o d The degree examin

ations were held in October at the b eginning of Michaelmas term

and thus Salmon was just nineteen years old when he graduated

As re aders may not b e f amiliar with the style of university examin

ations in mathematics in the nineteenth century we pre s ent b elow

some of the que stions that were p o s e d in Thes e are taken

f rom Six pap ers of ten que stions e ach were s et over three

days with one pap er in the mor ning and one in the evening The

examiners were Mr Malet Mr Sadle ir and Mr Luby There was a

strong bias towards mechanics in mo st of the que stions p o s e d with

calculus also prominent The fourth pap er s et by Mr Sadle ir con

tained s everal que stions on the geometry of lines and surf ace s and

this pap er may well have appe ale d to Salmon given hi s interest

in analytic geometry

Paper

If two r ight lines b e constantly p erp endicular to e ach other at a given

p oint and move in given planes their planes envelope a cone

Find the length and equations of the r ight line which i s the shortest

di stance b etween two given r ight lines in space

If three diameters of an ellip soid cut at r ight angle s the sum of the

square s of their recipro cals i s constant

What i s Meusniers theorem and how i s it proved

Find the general equation of surf ace s of revolution round any axi s

and de duce the partial dierential equation

If the ap ex of a cone which touches a surf ace of the s econd order

de scr ib e a surf ace of the same order the plane of contact will envelope

a surf ace of the s econd order

Find the equation of the surf ace formed by r ight lines joining the

p oints of intersection of two curves of double curvature made by a plane

whos e direction i s given and de duce the partial dierential equation

Salmon include d Meusniers theorem in hi s textbo ok GTD but

he mi s sp elle d the name as Meunier

Salmon sat the Fellowship Examination in and was elec

ted a Fellow of the College in He was ordained deacon in

and pr ie st in the Church of Ireland in On hi s b ecoming

a Fellow Salmon rece ived a college tutorship which he retained

until From he also s erved as a divinity lecturer The

IMS Bulletin

duties of a tutor had change d following the reforms in the tutorial

system introduced in the mids It was the cas e that every

student entering the college had to place hims elf under the tuition

of one of the Junior Fellows who was a tutor Up to tutors

had lectured only to their own pupils acro s s a wide range of sub

jects but following the reforms they admitted pupils of other

tutors to their lectures and they lectured on a more limited range

of sub jects Furthermore remuneration no longer depende d on

the number of pupils under the tutors guidance but only on the

tutors s enior ity Much of the stimulus for wr iting Salmons rst

two textbo oks must have come f rom the tutors lectures that he

gave during the s It i s interesting to note that Salmon s erved

as tutor to b oth of W R Hamiltons sons during the s they

did not di stinguish themselves academically

In the Erasmus Smiths Profe s sor of Mathematics at

TCD Charles Graves re s igned the profe s sorship He

had b een appointed in and published twenty e ight mathem

atical re s e arch pap ers in such are as as geometry quaternions and

the calculus of operations all p opular topics for re s e arch at TCD

during the th century In he published a translation of

two geometrical memoirs of Michel Chasles on cones of the s econd

degree and spherical conics together with notes and additions

and an appendix of hi s own work Gravess monograph was

probably inuential on Salmon and it i s quoted o ccas ionally in

Salmons textbo oks although Salmon mi stakenly wrote that its

date of publication was

It would have s eemed in view of Salmons eminence in the

mathematical world an eminence bas e d particularly on the repu

tation of hi s textbo oks that he was the worthiest succes sor to

Graves For whatever re ason however Salmon must have alre ady

decide d to concentrate hi s energies in theology instead of math

ematics and he never oere d hims elf as a candidate for the vacant

mathematical profe s sorship Instead a vacancy had o ccurre d in

the Archbishop Kings Lectureship in Divinity The oce of Arch

bi shop Kings Lecturer commande d a salary of p er annum

equal to that of the Profe s sorship of Mathematics a cons iderable

sum in the s According to Joly p Salmon had b een

George Salmon

le d to b elieve that a certain William Lee who had b een appointed

Fellow of the College in would not apply for the Lecture

ship If Lee were to apply for the p o s ition he was certain to b e

preferre d over Salmon by virtue of hi s s enior ity in the Fellowship

Unfortunately for Salmon Lee change d hi s mind after Michael

Rob erts had alre ady b een appointed Profe s sor of Mathematics

and obtained the lectureship Thus Salmon was f rustrated in this

promotion opportunity Rob erts retained the profe s sorship until

and Lee the lectureship until hi s death in Salmon

had to wait a further four years b efore achieving the s enior p o s

ition he de s erved albe it in hi s s econd sp eciality of divinity The

circumstances of hi s elevation to the divinity profe s sorship fol

lowing on f rom hi s decis ion not to oer hims elf as a candidate for

the mathematical profe s sorship are also de scr ib e d in note

p although there i s no mention of Lees p o s s ible role

Erasmus Smiths Profe s sorship of Mathematics fell vacant in Sal

mon for re asons which we have b een unable to di scover did not oer

hims elf as a candidate although he was s enior to the two Fellows who

did comp ete Michael Rob erts and Townsend and b etter qualie d than

e ither He can scarcely have b een re s erving hims elf for the Regius

Profe s sorship of Divinity as the o ccupant Butcher was only e ight

years older than Salmon and hi s re s ignation at the age of ftyseven to

b ecome Bi shop of Meath could not have b een pre dicted

In ppxxiiixxiv Ball de scr ib e s lectures by Salmon that he

attende d in

He was an admirable teacher I particularly remember the cours e on

conics It was pre sumed that we had re ad or were re ading hi s b o ok

so the lectures often to ok the line of showing us the improvements or

extensions which he was prepar ing for future e ditions He would kindly

encourage hi s pupils to make their comments and even honour them by

asking for their eorts to aid him in que stions which were o ccupying him

at the momemt He us e d to warn us that the go o d things were mo stly

threshed out but still there were some le avings I also attende d

Salmons lectures on Elementary Theoretical Dynamics and never

shall I forget the delight with which we rece ived hi s illustrations of the

us e of dierential equations in dynamics

IMS Bulletin

Dur ing the tenure of hi s tutorship Salmon had opportunities to

lecture honours students and introduce them to the new mathem

atical theorie s and techniques b e ing develope d at that time We

may form some idea of the content of thes e lectures by lo oking at

various examination pap ers that he wrote during the s In

he examined for Bi shop Laws Mathematical Premium and

include d the que stions

Find the condition that two conics should touch whos e equations are

given

Find the multiple p oints of the curve

A B C D

In the Degree Examination of he s et a pap er on Algebra

Dierential Calculus and Analytic Geometry which include d the

que stion

Solve the equation x

We wonder whether he intende d that the solution in terms of square

ro ots which ar i s e s in the theory of the construction of a regular

p olygon of s ide s should b e given

In the Degree Examination of he s et que stions on Phys

ical Astronomy and he also examined in the same year on the

Hi storical Bo oks of the Old Testament for the Regius Profe s sor

of Divinitys Premium In Salmon then Donegal Lecturer

in Mathematics examined on the Theory of Curves and Algebra

for the Mo deratorships in Mathematics and Mathematical Physics

Among hi s que stions were

Write down the equations of the tangents at the p oints of inexion

of the curve

x y z mxy z

What i s the value of the determinant

George Salmon

and nd the determinant which i s its square

Salmon marrie d Frances Anne Salvador in and the couple

initially lived at Carleton Terrace Rathmines Dublin They

had moved to Heytesbury Terrace by remaining there until

or They then moved to Wellington Road this house

i s probably still standing as the house s in Wellington Road form

a continuous terrace dating f rom the Victorian era with number

the s econd last house in the terrace Finally having b een

appointed Provost of Trinity College in Salmon to ok up

re s idence in the Provosts Hous e There were s ix children of the

marriage four b oys and two girls but only two of the children

survived Salmon who die d on January Hi s wife had

die d in An In Memoriam notice wr itten by E J Gwynn

and recalle d in p de scr ib e s Salmons funeral pro ce s s ion

Our Provost was the rst man in Ireland whos e preeminence all

acknowledge d the immense pro ce s s ion which on the th January

stretched halfway f rom Chapel to cemetery was the outward and

vi s ible s ign of a national gr ief

Salmon i s bur ie d with e ight f amily members in the f amily vault in

Mount Jerome Cemetery Dublin The vault i s lo cated on the south

s ide of the path known as the Long Walk The unprepare d vi s itor

i s unlikely to nd the vault by chance as there i s no headstone

and the lid of the vault i s covered in foliage The words FAMILY

VAULT OF GEORGE SALMON are di scer nible with diculty

Not more than yards away on the same s ide of the Long Walk

as the Salmon vault but s et back further f rom the path i s the

grave of Salmons f amous fellow mathematician William Rowan

Hamilton The headstone of Hamilton i s still p erfectly legible and

i s f ar e as ier to nd than Salmons vault

Thos e who are interested in Salmons career as Profe s sor of

Divinity the theological b o oks he published and in hi s fteen years

as Provost of TCD may re ad the articles and b o oks that we men

tioned at the b eginning of this s ection According to G T Stokes

Salmon enjoyed

the mo st r ichly endowed academical p o s ition if not in Europe at le ast

in the Br itish Islands For owing to the lib erality of a deceas e d b ene

IMS Bulletin

f actor of the last th century the Provost has an e state of hi s own

which makes hi s p o s ition worth more than a year

Thi s income may account for why Salmon was able to make gifts of

over to TCD a large sum of money in the th century

p There i s a s e ated marble statue of Salmon by John Hughes

unveile d on June which stands near the Campanile in the

main quadrangle of the Trinity College the statue or iginally sto o d

els ewhere

While we are not attempting to de scr ib e Salmons p ersonality

in this article two anecdotes about him told in pp

may b e ar rep etition here

Many storie s are told of the gre at mans abs entminde dnes s Standing

carele s sly dre s s e d on the kerbstone one day and staring blankly at the

sky with hi s long white hair quivering in the breeze he was mi staken

for a blind b eggar by a kind old lady who oere d to le ad him acro s s the

street

One Sunday he accidentally brought to St Patricks Cathedral the

same charity s ermon that he had pre ached there a year b efore However

he went through with it to the bitter end re asoning that half the con

gregation had not b een there that one quarter had heard and forgotten

the s ermon and that the remaining quarter would b e very glad to hear

it againa pro ce s s of exhaustion worthy of Euclid hims elf

Salmons mathematical re s e arch

In the bibliography of this pap er we have given a li st of Salmons

mathematical re s e arch pap ers Thes e are taken f rom the Royal

So cietys Catalogue of Scientic Papers vols V and VI I I and we

hope that the li st i s re asonably complete It will b e s een f rom

the titles that almo st all hi s pap ers are devoted to geometrical

sub jects Dur ing the th century many mathematicians connec

ted with TCD published textbo oks or re s e arch pap ers on geo

metry often of the synthetic rather than the analytic variety

in this re sp ect Salmon was exceptional as hi s bias was usu

ally to the analytic s ide of geometry Two textbo ok wr iters

active during Salmons pr ime were John Mulcahy whos e Prin

ciples of Modern Geometry acknowledge d the inuence of

Gravess translation of Chasles and Salmons Conic Sections and

George Salmon

Richard Townsend whos e Chapters on the Modern Geometry of

the Point Line and Circle two vols sought to

f amiliar ize re aders with the work of Michel Chasles and to pro

gre s s b eyond Mulcahys b o ok Thos e Fellows who wrote geo

metrical re s e arch pap ers include d James MacCullagh Charles

Graves William Rowan Hamilton often employing quaternionic

techniques Andrew Se arle Hart Salmon Richard Townsend

William Snow Bur ns ide John Kells Ingram John William Stubbs

It i s interesting to note that in a letter to W R Hamilton dated

February and published in p Augustus De Mor

gan wrote

How comes it that the geometrical extensions take such ro ot in Ireland

Yourself C Graves MacCullagh Salmon C are all up in them

Hardly a soul in England care s about them

In hi s reply dated February p Hamilton wrote

I think that there is a gre ater or at le ast a more general aptitude for

pure geometry in Ireland than in England The Fellows of T C D are

nearly all geometers and some of them are extremely go o d ones

Salmons pap ers are for the mo st part quite short often no more

than re s e arch notes or announcements Moreover he published

very little mathematical re s e arch after obtaining the profe s sorship

of divinity Salmon was cle arly inuence d by the climate of geo

metric re s e arch that exi sted at TCD but we cannot say for cer

tain that Salmon was a follower of any particular p erson active at

the College in the s or s except p o s s ibly MacCullagh

some of whos e work Salmon de scr ib e s in hi s b o oks He s eems to

have b een largely s elfmade inuence d by hi s re ading of Ponce

lets b o ok and the re s e arch pap ers of such contemporar ie s as

Cayley Hermite and Sylvester Regarding Salmons own re s e arch

Joly wrote in p

It would b e mo st unfair to Salmon to judge hi s contributions to math

ematics by hi s pap ers alone He had a gre at di slike to the physical

trouble of wr iting he mo de stly communicated hi s di scoverie s to f r iends

or re s erved them for incorp oration in hi s b o oks so that it i s a matter of

extreme diculty to say how much i s hi s Of Salmons or iginal

contributions to science the mo st worthy of notice are hi s solutions of

IMS Bulletin

the problem of the degree of a surf ace recipro cal to a given surf ace hi s

re s e arches in connection with surf ace s sub ject to given conditions ana

logous to thos e of Chasles in plane curves hi s clas s ication of curves of

double curvature hi s conditions for rep e ated ro ots of an equation and

hi s theorem of the constant anharmonic ratio of the four tangents f rom

a p oint on a cubic curve

Joly do e s not mention what i s in our opinion Salmons ma jor

achievement in mathematical re s e arch namely hi s work concer n

ing the twenty s even lines on a cubic surf ace In keeping with

Jolys remark above Salmon wrote very little in the re s e arch liter

ature on this sub ject To give some details we recall that a cubic

surf ace in the three dimensional complex pro jective space P C

3

i s dened by an equation of the form

f x x x x

0 1 2 3

where f i s a homogeneous p olynomial in the x of degree Thus

i

the socalle d Fermat surf ace

3 3 3 3

x x x x

0 1 2 3

i s an example of a cubic surf ace The surf ace i s said to b e non

singular if there i s no p oint P on the surf ace for which

f P

x

i

for all i We may s imilarly dene a surf ace of degree n us ing a

homogeneous p olynomial of degree n

It i s known that in general a surf ace of degree n in P C

3

contains no lines lying entirely within the surf ace if n See

for example p Salmon sketches a generic pro of of this

in GTD p A surf ace of degree or contains innitely

many lines The cas e of degree i s e sp ecially interesting as Cayley

showed in that a nons ingular cubic surf ace contains at mo st

a nite number of lines Furthermore with some eort one can

show that a nons ingular cubic surf ace always contains at le ast one

George Salmon

line s ee for example Propo s ition In corre sp ondence

with Cayley during Salmon proved that there are exactly

twenty s even lines on a nons ingular cubic surf ace

The rst ma jor article on the twenty s even lines i s that of

Cayley Cayley ende d hi s pap er in the following manner

I may mention in conclus ion that the whole sub ject of this memoir

was develope d in a corre sp ondence with Mr Salmon and in particular

that I am indebted to him for the determination of the number of lines

up on the surf ace and for the investigations connected with the repre s

entation of the twentyseven lines by means of the letters a c e b d

f as develope d above

The same volume of the Cambridge and Dublin Mathematical

Journal contained a pap er by Salmon which pre s ented supple

mentary notes to Cayleys pap er

Salmon devoted a few page s in the various e ditions of GTD

to the sub ject of the twenty s even lines and de scr ib e d hi s contri

bution in a fo otnote as follows

The theory of r ight lines on a cubical surf ace was rst studied in the

year in a corre sp ondence b etween Mr Cayley and me the re sults

of which were published Cambridge and Dublin Mathematical Journal

vol IV pp Mr Cayley rst ob s erved that a denite number

of r ight lines must lie on the surf ace the determination of that number

as above and the di scus s ions in Art were supplie d by me

It was largely left to other workers to elucidate the b e autiful com

binatorial and algebraic relations that are as so ciated with the con

guration of the twenty s even lines and an extensive literature

on the sub ject exi sts There s eems to b e no elementary approach

to wr iting down explicitly the algebraic form of the lines even in

the cas e of the Fermat surf ace mentioned above There i s a Galoi s

theoretic problem le ading to an equation of degree obtained by

elimination techniques that i s attached to the que stion of determ

ining the lines A re asonably elementary approach to an explicit

investigation may b e found in Section For a more abstract

approach one may consult Chapter V Section

Salmons last published re s e arch pap er that of i s on

the elementary theory of numbers and it i s a rather feeble eort

IMS Bulletin

Salmon b egan hi s pap er as follows

I wi sh to make a few remarks viz to nd the number of gure s in

the p er io d of a circulating decimal whos e recipro cal i s a pr ime number

Thi s i s evidently the same as to nd what p ower of i s divi s ible by

x

the pr ime with remainder or to solve the equation 

Salmon thus wi shed to nd the order of mo dulo a given pr ime

The sub ject was scarcely or iginal at the time Salmon wrote s ince

it was tre ated in such university textbo oks as that of Hymers

p but Salmon made no reference to any published source s

and hi s re sults are e s s entially tr ivial

Although Salmon ce as e d to b e involved in re s e arch level

mathematics so on after obtaining hi s profe s sorship he still main

tained an interest in mathematical que stions at the recre ational

level s ince hi s name appe ars in virtually every li st of con

tr ibutors to the publication Mathematical Questions with Their

Solutions From the Educational Times Thi s s er ial was

published b etween and and it contained over the

years statements and solutions of more than mathematical

problems We have not found many example s of problems or

solutions contributed by Salmon that were actually published

but the following problem of hi s shows him working in an

uncharacteri stic are a

A teetotum of n s ide s i s spun an indenite number of times and the

numbers turning up are adde d together what i s the chance that a given

number s will b e actually arr ived at

A teetotum i s a small top inscr ib e d with letters or numbers

Salmons mathematical textbo oks

Salmons name would scarcely attract any attention among math

ematicians so many years after hi s death if hi s reputation was

bas e d only on hi s re s e arch pap ers As we stated in the introduc

tion Salmons lasting f ame lie s in the inuence exerted by four

textbo oks he wrote Thes e were A Treatise on Conic Sections A

Treatise on the Higher Plane Curves Lessons Introductory to the

Modern Higher Algebra A Treatise on the Analytic Geometry of

Three Dimensions We have given details of the various e ditions of

thes e b o oks in the bibliography of this article The information i s

George Salmon

mainly extracted f rom the Br itish Mus eums General Catalogue of

Printed Books to and The National Union Catalogue pre

impr ints although the source s do not include all e ditions

of Salmons textbo oks We will cons ider e ach b o ok in a s eparate

s ection of this article It will b e s een that e ach of the four titles

was translated into b oth French and German and thes e trans

lations often went through s everal e ditions which we have not

attempted to itemize The German translations of the four main

b o oks e ach by the German mathematician Otto Wilhelm Fie dler

who worked mainly in Z urich were e sp ecially highly

regarde d The hi storian of science J T Merz wrote as follows

of thes e translations p and their eect on the teaching of

geometry in Germany in the s econd half of the th century

Germany indee d had not b een wanting in or iginal re s e arch but the new

ideas of Mobius Steiner Staudt Pl ucker and Gras smann in geometry

found no adherents till mainly through the translations of Salmons

textbo oks by Fie dler a new spir it came over geometrical teaching

Just b efore Salmons death Merz de scr ib e d Salmons inuence in

pp in thes e words

The merit however of having brought together the new ideas which

emanated f rom the school of Poncelet and Chasles in France of Cayley

and Sylvester in England into a connected do ctr ine and of having

given the imp etus to the fundamental remo delling of the textbo oks and

schoolb o oks of algebra and geometry in this country Br itain and in

Germany b elongs undeniably to Dr Salmon of Dublin

Felix Kle in was also a gre at enthusiast for Salmons b o oks He

de scr ib e d them aectionately as follows

They are like ref re shing and instructive walks through woo d and eld

and cultivated gardens where the guide draws attention now to this

b e auty now to that strange appe arance without forcing everything

into a r igid system of f aultles s p erfection and without digging out

individual us eful plants and transferr ing them to cultivated soil accord

ing to pr inciple s of intensive cultivation In this ower garden we have

all grown up here we have gathered the foundation of knowledge on

which we have to build

The quote i s taken f rom an English language version of part of

IMS Bulletin

Kle ins p o sthumously published lectures vol I p found

in p The reviewer of Kle ins lectures in notes that

Salmon remarked when Kle in enquire d why at Dublin there were

only three f aculties theology clas s ics and mathematics whereas

at Cambridge all the newest sub jects were taughtYes but Cam

br idge i s so ambitious

Kle in was probably sympathetic to Salmons style s ince Kle in

was known not to b e to o interested in the intricate details of

theorie s and preferre d broad pictures He s imilarly prai s e d the

SalmonFie dler s er ie s of textbo oks in hi s Elementarmathematik

von hoheren Standpunkte aus

All that i s neede d in the geometry of invariant theory e sp ecially i s to

b e found in the textbo oks of G Salmon which have contributed mo st

to spre ad the ideas which ar i s e here The German e dition of Salmons

b o ok by W Fie dler has always enjoyed an unusually wide us e

The German version of Kle ins b o ok rst appe are d in The

quote above i s f rom the English translation of the third e dition

p

Fie dlers versions of Salmons b o oks were more than just

translations The rst e ditions were said to b e adapted into Ger

man Deutsch b e arb e itet and later ones were f reely adapted into

German f re i b e arb e itet Fie dler incorp orated material of hi s own

into the German versions

Salmon was an e arly member of the London Mathematical

So ciety LMS which he joined in In Henry Smith

then Pre s ident of the LMS gave an addre s s to the So ciety in which

he noted how young Br itish mathematicians often chos e to study

algebra and geometry through the inuence of Salmons textbo oks

p

Can we doubt that much of the preference for geometrical and algeb

raical sp eculation which we notice among our young mathematicians i s

due to the admirable works of Dr Salmon and can we doubt that if

other parts of mathematical science had b een equally fortunate in nd

ing an exp o s itor we should ob s erve a wider interest in and a juster

appreciation of the progre s s which has b een achieved

The General Index to e ach volume of the Proceedings of the Lon

George Salmon

don Mathematical Society b etween and s erves as a

way of measur ing how much practis ing mathematicians us e d Sal

mons textbo oks e ither for general reference or to motivate their

own re s e arch s ince citations given in the pap ers published in the

Proceedings during this time are li sted in this index A rough

count reveale d pap ers that cited Salmon one or more times

In particular e ight pap ers in vol XI I I and ten pap ers

in vol XIV cited textbo oks by Salmon Authors cit

ing Salmon include d F Br io schi A Cayley P A MacMahon

A R Forsyth J Hammond W Sp ottiswoo de M J M Hill

H Lamb E J Routh W K Cliord H J S Smith J Larmor

L J Rogers W W Rous e Ball F S Macaulay A N White

head A B Kempe A G Greenhill H G Zeuthen and A Man

nheim Thi s i s an impre s s ive li st which fully indicates the extent

of Salmons inuence The reference s b ecame fewer in the s

p erhaps as Salmons contemporar ie s die d or ce as e d to b e active

While we have catalogued f avourable opinions of Salmons

textbo oks it s eems to us that on o ccas ions Salmons wr iting

was denitely unclear and badly expre s s e d In p Bro ck

rai s e d the que stion of whether Salmons textbo oks were re ally as

cle ar and as p opular with students as hi s obituari sts claimed He

de scr ib e s an annotated copy of CS in Le ice ster University

Library which b e ars such comments as Half the Bo ok unt for

students but go o d for a Maths teacher of Analytical Geometry

not e asy in f act very dicult badly expre s s e d and explained

little or no us esheer rubbish I never remembere d to have re ad

a Chapter which require d so much elucidation and rece ived so

little In a copy of HPC that we have us e d we found the

comment I nd it imp o s s ible to make head or tail of this atro

ciously wr itten ungrammatical nons ens e I do not know what i s

as s erted in this attempt at a s entence While students sometimes

lay the blame for their own incomprehension at the feet of their

lecturers or with their textbo oks thes e example s sugge st that Sal

mons wr iting was not always appreciated by students more inter

e sted in pas s ing examinations than dwelling on the ner p oints of

mo dern geometry

IMS Bulletin

A Treatis e on Conic Sections

The e arly publishing hi story of Salmons mo st f amous and endur

ing textbo ok A Treatise on Conic Sections i s a little complicated

and do e s not s eem to have b een commented up on b efore by anyone

wr iting about Salmon We will try to unravel some of the mystery

that surrounds the rst appe arance of this b o ok

In the s Salmon had planned to wr ite a tre atis e in two

volumes on analytic geometry intende d for the us e of undergradu

ates at TCD The rst volume was to b e divide d into two parts

Part I cons i sting of elementary analytic geometry and Part II of

the geometry of conic s ections with sp ecial emphasi s on mo dern

developments in the sub ject The s econd volume Part III of the

tre atis e was to b e devoted to higher plane curves Thi s plan was

e s s entially fullle d with Conic Sections of forming the rst

volume and Higher Plane Curves of the s econd What i s

surpr i s ing to us i s that in Salmon decide d to i s sue e ach of

Parts I and II s eparately entitled A Treatise on Analytic Geo

metry Part I The Right Line and Circle which appe are d late

in and A Treatise on Analytic Geometry and on the Theory

of Plane Curves Part II Conic Sections which was to appe ar

in January Copies of e ach s eparate part are very rare and

neither i s li sted in the Br itish Mus eums General Catalogue of

Printed Books or The National Union Catalogue The Trinity

College Library has copies of Parts I and II b ound together in a

s ingle volume entitled on the spine Salmons Geometry III Thi s

copy was pre s ented to the Library by Salmon hims elf in

There i s also a copy of Part I only in the National Library of

Ireland

The pref ace of AGI which we include b elow makes it cle ar

that Salmon brought out the rst part of the tre atis e as a s eparate

volume to provide an elementary introduction to analytic geometry

suitable for rst and s econd year undergraduates at TCD

It was not my or iginal intention to publish an independent tre atis e on

Analytic Geometry but rather a supplement to the ordinary element

ary works on that sub ject in which I meant to give an account of the

pr incipal additions made by mo dern geometric and algebraic methods

to the theory of Curves and Surf ace s In attempting to execute this

George Salmon

de s ign I found some inconvenience f rom the f act that there was no

s ingle work in general us e among the students of this College to which

I could refer for elementary information The late Dr Lloyds tre atis e

had b een for some years out of pr int and no other work has taken its

place as a recogni s e d textbo ok for the us e of our junior students Thi s

deciency I have attempted to supply in the following page s which have

b een drawn up with an e sp ecial reference to the cours e of study pur

sue d in this college As the clas s now commencing the study of this

science are not require d during the approaching term to pro cee d b ey

ond the properties of the Right Line and Circle it has b een thought

advi sable to publish without delay that p ortion of the pre s ent work

which i s require d for their immediate us e I have for the same re ason

di scus s e d thes e properties at gre ater length I fe ar than mo st re aders

will approve The additions which I have made to the matter usually

contained in elementary tre atis e s cons i st chiey in a tolerably copious

collection of example s Chaps I I I and VI I and in an account of thos e

methods of abridge d notation Chaps IV and VI I I which have lately

b ecome so much us e d and which the Tutors Lectures have during the

last few years made generally known among the candidates for honors

The Second Part of this Treatis e containing the theory of Conic

Sections i s intende d to b e published at the b eginning of next Hilary

Term and the remainder of the work as so on after as i s found practic

able

I have to return thanks to Mr Townsend for many us eful sugge s

tions and for much kind help in sup er intending the progre s s of thes e

page s through the Pre s s and to the b oard of Trinity College for the

lib eral as s i stance which they have aorde d to the pre s ent publication

GEORGE SALMON

TRINITY COLLEGE

October th

The advertis ements at the b eginning of the Dublin University Cal

endar for announce d the appe arance of Salmons A

Treatise on Analytic Geometry Part I at a co st of s and stated

that Part II was to b e published immediately The back cover of

the copy of AGI in the TCD Library advertis e d that Part I I con

taining the theory of conic s ections was to b e published in Janu

ary At le ast one copy of Part II was pr inted and b ound

but we are not certain if any copies were ever oere d for sale as

we have s een no advertis ements relating to Part I I The London

IMS Bulletin

Catalogue of Books which li sts b o oks published in Eng

land b etween and has entrie s for b oth A Treatise on

Analytic Geometry Part I and Conic Sections which were b oth

published in London by Whittaker and Co but makes no mention

of A Treatise on Analytic Geometry and on the Theory of Plane

Curves Part II Conic Sections What i s certain i s that Parts I

and II were combined to form the rst e dition of Conic Sections

which was published in Salmons obituari sts Ber nard and

Joly give a publication date of for the rst e dition of Conic

Sections whereas the correct date i s They must have con

fus e d the incomplete AGI published in with the complete

CS published in The page s of Part I form the rst

e ight chapters and the page s of Part II the last s ix chapters of

Conic Sections Part II contained the following advertis ement

PART I I I will b e published as sp ee dily as the Authors other engage

ments will p ermit and will contain an account of the properties of the

higher Plane Curves together with a General Index and Notes on the

hi story of the more remarkable Methods and Theorems

The rst and s econd e ditions of Salmons Conic Sections contained

no authors pref ace which i s a pity in view of the confus ion cause d

by the e arlier publication of A Treatise on Analytic Geometry The

rst e dition did however include a f rontispiece advertis ement the

rst part of which we di splay b elow the s econd part of the advert

i s ement i s just the advertis ement which had alre ady appe are d in

AGII

The following page s form Parts I and I I of a Treatis e on ANALYTIC

GEOMETRY and on the THEORY OF PLANE CURVES intende d

for the us e of Undergraduate Students of the University of Dublin

We will devote the re st of this s ection to cons ideration of the vari

ous e ditions of the complete Conic Sections The fourteen chapters

of CS are as follows

Chapter I The p oint

Chapter I I The r ight line

Chapter I I I Example s on the r ight line

Chapter IV The r ight line continued

Chapter V Equations above the rst degree

George Salmon

repre s enting r ight lines

Chapter VI The circle

Chapter VI I Example s on the circle

Chapter VI I I The circle continued

Chapter IX Properties common to all curves of the s econd

degree de duced f rom the general equation

Chapter X Equations of the s econd degree referre d to the

centre as or igin

Chapter XI The parabola

Chapter XI I Example s and mi scellaneous properties

of the conic s ections

Chapter XI I I Methods of abridge d notation

Chapter XIV Geometrical methods

The rst twelve chapters which o ccupy page s constitute

a f airly complete introduction to the study of analytic geometry in

the plane up to and including conic s ections suitable for under

graduates In AGII Salmon indicated that the more advanced

Chapters XI I I and XIV might b e omitted by re aders commencing

the study of conic s ections The work was p opular for teaching

purp o s e s partly on account of the extensive tre atment with plen

tiful example s of this elementary material The thirteenth chapter

was on the methods of abridge d notation a topic continued in all

subs equent e ditions although it has f allen out of f avour nowadays

For profe s s ional mathematicians it was the nal chapter o ccupy

ing pp in the rst e dition on geometrical methods that

was of gre atest interest We will br iey de scr ib e its contents

The chapter b egins with the pr inciple of duality and the

method of recipro cal p olars Of the method of recipro cal p olars

which concer ns a duality b etween p oints and lines Salmon wrote

in a fo otnote to CS

Thi s b e autiful method was introduced by M Poncelet whos e account

will b e found at the commencement to the fourth volume of Crelles

Jour nal

There then follows a s ection on harmonic and anharmonic proper

ties of conics Thi s s ection was inuence d by the work of Michel

Chasles It i s succeede d by the method of innites imals The nal

IMS Bulletin

s ection i s the method of pro jections which Salmon de scr ib e d as

follows

The method i s the invention of M Poncelet See hi s Traite published

in the year I shall b e glad if the slight sketch here given induces

any re ader to study a work f rom which I have p erhaps derived more

information than f rom any other in the theory of curves

Thi s fo otnote was mo die d in the fourth e dition as follows

See hi s Traite des Proprietes Projectives published in the year

a work which I b elieve may b e regarde d as the foundation of Mo dern

Geometry In it were taught the pr inciple s that theorems concer ning

innitely di stant p oints may b e extende d to nite p oints on a r ight line

that theorems concer ning systems of circle s may b e extende d to conics

having two p oints common and that theorems concer ning imaginary

p oints and lines may b e extende d to re al p oints and lines

The nal s ection of the fo otnote given above concer ning imaginary

p oints and lines was referre d to as the principle of continuity It

was a contentious i s sue when introduced by Poncelet as it involved

the us e of complex methods in re al geometry and it was strongly

oppo s e d by Cauchy who s erved as a referee for one of Ponce

lets related pap ers The article by Rene Taton on Poncelet

provide s information on hi s work and the problems it encountered

We also note how Salmon refers to the sub ject of Mo dern Geo

metry in this fo otnote Mo dern geometry as develope d in the

late th and e arly th century by Monge Poncelet Chasles

and others repre s ented the rst ma jor extension of pure geometry

as oppo s e d to analytic geometry b eyond the state in which the

Greeks had left it The two geometry textbo oks published in Dub

lin that we referre d to in Section thos e of Mulcahy and Town

s end contained the phras e Mo dern Geometry in their titles

The fourth and later e ditions of CS contained s eparate

chapters on e ach of the topics of recipro cal p olars harmonic and

anharmonic properties of conics the method of pro jections and

the method of innites imals A further chapter on invariants and

covariants of systems of conics was adjoined Thi s was cle arly

inuence d by Salmons newer work on invariant theory

George Salmon

Title page of

A Treatise on Analytic Geometry Part I

IMS Bulletin

Title page of

A Treatise on Analytic Geometry Part II

George Salmon

A fe ature of CS i s the inclus ion of extensive fo otnotes

many of which did not survive into later e ditions Some of thes e

fo otnotes contain cur ious admi s s ions of forgetfulnes s or lack of

knowledge of the literature For example on p of CS

he wrote

I forget where I met with the following method of obtaining the prop

erties of angle s subtende d at the fo cus f rom thos e of small circle s on a

sphere

but in later e ditions he decide d that John Mulcahy had given this

method

CS was the last to contain any change s All subs equent

e ditions have b een re i s sue s of the s ixth e dition They have contin

ue d to appe ar as Chelse a Publishing Company repr ints to the end

of the twentieth century Conic Sections was Salmons mo st p opu

lar textbo ok in terms of sale s and probably the one for which he i s

b e st known Bo oks on conic s ections appe are d f airly regularly dur

ing the th and th centurie s and the sub ject formed a key part

of a university mathematics cours e Some textbo oks such as that

of Todhunter also ran to s everal e ditions but none s eems have

attracted the aection that was re s erved for Salmons b o ok It was

p erhaps its mixture of elementary methods and introduction to the

mo dern geometry of the th century which appe ale d to re aders

One may compare Salmons b o ok with that of Hymers which

was us e d at Cambridge b efore Salmons b o ok appe are d Hymerss

b o ok i s little more than the rst e ight chapters of Salmons with

none of the more exciting mo dern material Todhunters

tre atis e dealt with some more advanced topics such as the method

of abridge d notation but again it i s mainly devoted to elementary

material

In on the hundredth anniversary of the rst appe arance

of CS the Mathematical Gazette contained a short celeb

ratory note by E H Neville entitled Salmon together with

a f rontispiece photograph of the title page of CS The s en

timents expre s s e d in the article s eem to have b een felt by many

p eople interested in the study and teaching of geometry We quote

the article in full

The le ar ner of one generation i s the teacher of the next and as a rule

IMS Bulletin

it i s only for a decade or so that a textbo ok or tre atis e plays its part

in the development of its sub ject directly what i s or iginal in matter

or method i s absorb e d by re aders and p ermeates the b o oks which they

in their turn wr ite Chrystals Algebra and Hob sons Trigonometry are

fountainheads but the young mathematician do e s not go to them now

he imbibe s their wi sdom through channels f ashioned in hi s own lifetime

To the rule there i s one outrageous incre dible glor ious exception

Salmon The name stands not for an unknown provost and profe s sor

of divinity nor generically for the four mathematical tre atis e s of which

three have suere d the common f ate Salmon i s A Treatis e on Conic

Sections containing an account of the mo st imp ortant mo dern algebraic

and geometric methods and nothing els e And our f rontispiece shows

that this b o ok which schoolboys and undergraduates of to day are urge d

to re ad as a matter of cours e has in sob er f act re ached its centenary

It i s the moment for our salute

The third e dition of CS was published in London rather than

in Dublin and this may have ensure d that it enjoyed a wider cir

culation than the rst two e ditions Unlike the rst and s econd

e ditions it contained an authors pref ace which did not explicitly

link the b o ok to HPC and it was no longer stated that it was

intende d for the us e of students of the University of Dublin Its

sale s may also have b een improved by the appe arance in December

of a f avourable b o ok review in a widely re ad scientic jour nal

According to p this review was wr itten by Thomas

Archer Hirst Hirst was one of the rst Br itish math

ematicians to study for a Ph D in mathematics which he gained

in at the University of Marburg in Germany Hi s thes i s topic

was the geometry of ellip soids and hi s mathematical re s e arch was

in the are a of geometrical transformations Hirst b ecame Profe s sor

of Pure Mathematics at University College London in and

later was Director of Studies at the Royal Naval College Green

wich He thought highly of Salmon as he expre s s e d the following

opinion of him in hi s jour nal p

For my part the more I know him the more I re sp ect him by nature

he i s a s ingleminde d mo de st man but he i s endowed with an intellec

tual p ower and a capability of work such as we rarely meet

Hirst was an inuential admini strator in mathematical and sci

George Salmon

entic so cieties and he was instrumental in s ecur ing Salmons

election as a Fellow of the Royal So ciety of London in

p Hi s review i s quite detailed and p erspicuous and it pin

p oints s everal re asons why Salmons b o ok was so p opular We

pre s ent the whole of the review here as it gives a go o d contem

p orary as s e s sment of Salmons masterpiece

It i s a source of cons iderable satisf action to nd amongst the crowd

of very imp erfect e ducational b o oks which are daily i s sue d f rom our

pre s s a tre atis e so truly valuable as the pre s ent and it i s also cheering

to le ar n that the public has so f ar recognize d its merits as to demand a

third e dition The b o ok i s now suciently well known otherwis e its title

might mi sle ad many for although it i s true that the gre ater part of it i s

devoted to an examination of the properties of the conic s ections yet this

b e ing done f rom an analytical p oint of view chiey it was neces sary to

prepare the re ader by a s imilar investigation of the properties of the line

and circle so that in f act the whole constitutes a very ecient tre atis e

on the elements of analytic geometry such as may with advantage b e

place d in the hands of every student who has mastered the elements

of Euclid plane tr igonometry and algebra We venture to as s ert that

amongst students alre ady acquainted with its merits this i s one of their

f avourite textbo oks for the tre atment throughout i s admirably cle ar

str ict and elegantin f act such as can b e achieved only by one who

b e s ide s that p erfect mastery of the sub ject which can only b e acquire d

by or iginal re s e arch p o s s e s s e s also that unacquirable talent of lucid

exp o s ition and i s guide d by that knowledge of the diculties usually

encountered by students which exp er ience only can give

Thi s third e dition i s revi s e d and enlarge d and contains many

improvements up on the former two with re sp ect to its type arrange

ment and its wellchos en and numerous s election of example s The

student will do well to work all thes e conscientiously and to pay par

ticular attention to thos e interesting chapters on abridge d notation

The last chapter contains a short but cle ar account of the mo st b e auti

ful mo dern geometric methods amongst which i s that b e autiful method

of recipro cal p olars rst introduced by Poncelet an acquaintance with

which may b e said to form an ep o ch in the hi story of every young

mathematician

The imp ortance of thes e methods may b e e stimated f rom the f act

that in the hands of Steiner with scarcely any help f rom algebra they

have b ecome the mo st p owerful instruments of di scovery and have given

a wonderful ins ight into the nature and properties of curves an ins ight

IMS Bulletin

which i s p erhaps more thorough and direct than any attainable by

one who i s accustomed to call in the more mechanical aid of algebraic

calculation

We do not wi sh here to revive the old and us ele s s di scus s ion on

the comparative merits of the algebraic and geometric methods b oth

have undoubtedly their advantages and b oth are indisp ensable In the

gre ater part of the pre s ent tre atis e however the former of thes e meth

o ds i s adopted and in drawing attention to the imp ortance of the last

chapter we would merely remark that if hi s ob ject b e to obtain a thor

ough knowlege of the properties of conics the student will do well to

combine b oth methods to a gre ater extent than i s done here for the

f act cannot b e di sputed that the very f acility with which re sults can b e

obtained algebraically may indirectly prevent that intimate acquaint

ance with the properties of curves which a r igid geometrical investiga

tion also s ecure s

In view of Salmons admitted indebtednes s to Poncelets pioneer

ing work e sp ecially that contained in it may b e informat

ive to take into account some of Poncelets published opinions on

Salmons work in particular CS Je anVictor Poncelet

was the inventor of many of the techniques of pro ject

ive geometry that were so p opular in the th century including

the methods of duality recipro cal p olars pro jection and complex

methods Hi s mathematical career was blighted by pr ior ity di s

putes and cr iticism of hi s pr inciple of continuity Hi s comments

in hi s later works and show that he was particularly

s ens itive to the que stion of pr ior ity of di scovery in mathematical

re s e arch and he to ok such mathematicians as Hamilton and Cayley

to task for f ailing to admit hi s Poncelets pr ior ity in such mat

ters as the problem of inscr ibing or circumscribing p olygons in or

around conic s ections For more on the work of Poncelet Cayley

and Salmon on this sub ject s ee the article by J A Todd

Recalling Jolys remark include d in Section that it i s matter of

extreme diculty to say how much of the material in hi s textbo oks

was or iginal to Salmon Poncelets comments at the end of the

quotation b elow taken f rom pp are understandable

Nonetheles s as the quotation indicates Poncelet acknowledge d

that Salmons Conic Sections helpe d to re scue the methods of hi s

George Salmon

own Traite des Proprietes projectives des gures f rom relat

ive oblivion

Ce s diverse s propo s itions sur le s p olygones dordre pair et impair

inscr its ou circonscr its aux s ections conique s s e retrouvent demontrees

ains i que b e aucoup dautre s par la voie geometrique dans le Chap I I

Sect IV de mon Traite des Proprietes projectives des gures

Elle s ont depuis quelque s annees s eulement attire lattention de

plusieurs eminents geometres etrangers MM Mo ebius et Gopel

Journal de Crelle s ir William Hamilton Lectures on qua

ternions Rev George Salmon Treatise on conic sections

p

There follows a de scr iption of the work of Mo ebius Gopel and

Hamilton on inscr ibing and circumscribing p olygons in or around p oly

gons

Ce s theoremes thos e of Mo ebius Gopel and Hamilton ont une

o s

analogie evidente avec ceux de s n a de ce Traite Proprietes

projectives dont ils sont la s imple extension au cas de le space comme

le remarque luimeme M Hamilton qui ne connais sait dailleurs mes

propres recherches que par le s citations de lexcellent Traite des sec

r

tions coniques du Rev D Salmon p o sterieur au mien de plus de trente

annees

A legard de ce dernier et savant ouvrage la publication de

jouit en France dune juste celebrite et na pas p eu contribue a relever

le s methode s de demonstration et de recherches geometrique s du Traite

de s Proprietes pro jectives de le spece de di scredit o u elle s etaient

tombees depuis par suite de facheuse s di scus s ions de pr ior ite et de

preventions auss i p eu justiees que mal deguisees Neanmoins on doit

o

regretter que dans le n de son remarquable Traite elementaire

M Salmon nait p oint accorde plus dattention a limp ortante et

delicate theorie de linscr iption de s p olygones aux conique s theorie

que a lexemple de M Townsend il rattache a la cons ideration plutot

synthetique quanalytique bien que symbolique et abreviative de s

faisceaux projectifs de droites convergentes dont M Gopel setait

egalement s ervi dans son Memoire allemand de Peutetre meme

s erait on en droit de lui repro cher comme a tant dautre s de navoir

pas toujours tenu un compte susament exact de la dierence a mon

s ens capitale entre decouvrir et demontrerSir Hamilton me parat

avoir mieux sai s i le spr it et la p ortee de la que stion dont il sagit

IMS Bulletin

De spite the cr iticism Poncelet had a high opinion of Salmon s ince

in p he de scr ib e d him as

le loyale et do cte Salmon cet interprete bienveillant et lucide de s nou

velles do ctr ines algebricogeometrique s

Another French geometer whos e work inuence d Salmon was

Michel Chasles Chasles hims elf wrote a tre atis e on

conic s ections but only the rst part ever appe are d We

found only one br ief reference to Salmon in p which we

include here

e

M Salmon dans son excellent Traite des Sections coniques edition

page e st parvenu a une equation fort s imple du lieu cherche

A Treatis e on the Higher Plane Curves

Salmons s econd mathematical textbo ok was a A Treatise on the

Higher Plane Curves with the subtitle intended as a sequel to A

Treatise on Conic Sections published in As we mentioned in

the previous s ection this was the s econd volume of a tre atis e on the

analytic geometry of space of two dimensions It i s aimed at a more

advanced mathematical audience than CS and was suitable only

for s enior undergraduates or re s e archers In the pref ace Salmon

acknowledge s the inuence of Poncelet Chasles and

Pl ucker on hi s work

The b o ok has s even chapters dealing with the following

sub jects tangential co ordinates general properties of algebraic

curves curves of the third degree curves of the fourth degree

transcendental curves general methods applications of the integ

ral calculus There are also twenty s ix page s of notes The notes

are revealing as they deal in part with determinants and elimin

ation theory and indicate the b eginnings of Salmons interest in

invariant theory the sub ject of hi s next b o ok in There i s

also a short di scus s ion of the meaning of complex number methods

in geometry including a mention of quaternions then a sub ject of

much re s e arch in Dublin

The style of wr iting in HPC i s lo o s er than that in CS and

at times Salmon di splays a nonchalance with regard to hi s back

ground re ading a trait we drew attention to in the previous s ec

tion In the pref ace he wr ites

George Salmon

Cons idere d as a b o ok for advanced re aders I regret that I have not had

the le i sure for the re ading neces sary to make this work as complete as it

ought to b e My knowledge of the older wr iters on Geometry i s for the

mo st part e ither s econdhand or sup ercial and I cannot even claim

acquaintance with all the mo st remarkable works and memoirs which of

late years have b een published on the sub ject

On p he wr ites vaguely

I ascr ib e the method of transformation to M Chasles b ecause I know

that I was f amiliar with it and b elieved it to b e hi s long b efore the

publication of any other pap ers on the sub ject cited b elow The only

wr iting of M Chasles I can now refer to i s note xxi p of the

Apercu Hi storique which however only tre ats of the cas e where n

I cannot tell whether I was le d to the general method as an obvious

extension of this or whether some pap er on the sub ject may not b e

found among M Chasless many contributions to scientic p er io dicals

Then as a fo otnote on p he wr ites

I nd that I was mi staken in suppo s ing p that M Chasles had

published anything on this sub ject b eyond the note to the Apercu Hi s

torique alre ady referre d to

Thi s correction to hi s e arlier surmi sal i s p o s s ibly explained by

the usual method of pr inting a textbo ok in which s ections of

manuscript were s et into type by the University Pre s s while others

were still b e ing wr itten

On p he shows a di sregard for re s e arching hi s source s

as he wr ites

Mr J S Mill in a pas sage in hi s Logic on which I cannot lay my hand

erroneously repre s ents Galileos exp er iments to have b een succes sful

Salmons slack style of wr iting did not meet with the approval

of the more painstaking T A Hirst who wrote as follows in hi s

p ersonal jour nal in p

I have b een wr iting French all week prepar ing my memoir on derived

surf ace s for Tortolinis Journal It i s a labor ious task There are

paragraphs that I have wr itten three or four times over b efore I could

satisfy myself I que stion whether many mathematicians of this day take

equal pains to b e lucid in their demonstrations Salmon in hi s Higher

Plane Curves has devoted a few page s to the same sub ject I wi sh he

IMS Bulletin

had taken as much trouble to b e cle ar in hi s demonstrations as I have

done

Ball wrote the following of HPC in xxiv

I must however add though I am p erhaps expre s s ing my own opinion

when I say that the Higher Plane Curves i s the le ast interesting of

Salmons four gre at works sic It lacks the magic of the Conics It has

not the imp ortance in nature p o s s e s s e d by the Quadrics which form

so large part of the Geometry of Three Dimensions Nor do e s it le ad

to such splendid theorie s and conceptions as do the le s sons in Modern

Higher Algebra

HPC s eems to have b een the le ast succes sful in sale s terms of

hi s textbo oks s ince it only ran to three e ditions the s econd not

b e ing i s sue d until twenty one years after the rst We quote f rom

xxv on the circumstances of its wr iting

By the time a s econd e dition of Higher Plane Curves was calle d for

Salmon had b ecome a Profe s sor of Divinity and had thus made engage

ments which in hi s own words left me no le i sure to make acquaintance

with recent mathematical di scoverie s or even to keep up any memory of

what I previously had known In this emergency he applie d to Profe s sor

Cayley to obtain advice as to the choice of some younger mathematician

who would undertake the e diting of the work To hi s agree able surpr i s e

Profe s sor Cayley oere d to do this hims elf Thi s oer was gratefully

accepted and the s econd e dition was the joint work of Salmon and

Cayley

Cayley contributed the rst chapter ten page s long on co ordin

ates as well as s everal articles to the new e dition Salmon also

incorp orated into the text material f rom various unpublished

manuscripts by Cayley including one on the theory of envelope s

The main change s in the s econd e dition were the introduction of

two new chapters on envelope s and on metrical methods and the

omi s s ion of one on applications of integral calculus

We have the feeling that initially at le ast Salmon probably

did not put as much eort into hi s wr iting of HPC as he did into

hi s other textbo oks although the s econd e dition was a cons ider

able improvement on the rst While the b o ok certainly had its

adherents and was e s s entially the only English language textbo ok

George Salmon

on its sub ject for many years it had le s s impact on mathematical

teaching and re s e arch than hi s other b o oks

Le s sons Introductory to the Mo dern Higher Algebra

The sub ject matter of MHA i s largely invariant theory or as

Salmon calle d it the algebra of linear transformations The b o ok

grew out of what was or iginally intende d to b e an appendix for

the then forthcoming b o ok GTD We ob s erved in Section

that a s imilar short appendix had alre ady appe are d in HPC

Salmons aim in wr iting MHA was to introduce hi s re aders to the

study of invariant theory which had b een develope d in the s

and s The exact nature of invariant theory i s dicult to

explain but it may b e s een as a combination of linear algebra and

algebraic geometry Nowadays it i s partly subsumed in the theory

of the repre s entations of clas s ical group s on tensor algebras David

Lewi s has given a de scr iption of invariant theory and its methods

in a previous e dition of this Bul letin

Salmon trace d the or igin of invariant theory to a pap er of

Bo ole of and he acknowledge d this pap er for having

introduced him to the study of linear transformations Cayley had

extende d Bo oles work to a much gre ater clas s of functions than

thos e cons idere d by Bo ole As Salmon explained in the pref ace to

MHA

The geometrical imp ortance of this theory invariant theory i s now

manifest When we are given the equation of any curve or surf ace

the theory of linear transformations at once pre s ents us with equations

repre s enting other curves and surf ace s and p o s s e s s ing p ermanent rela

tions to the given one which will b e unaected by any change of the

axe s of coordinates And in like manner the same theory pre s ents us

with certain functions of the given equation the vanishing of which must

expre s s a property of the given curve or surf ace wholly independent of

the choice of axe s Be s ide s thes e geometrical applications the theory

has other imp ortant us e s which I shall not stop to enumerate

In rather s imilar vein Salmon wrote in a fo otnote of MHA

p

invariants then are functions of the co ecients expre s s ing certain

xe d properties of the curve or surf ace which are independent of our

IMS Bulletin

choice of axe s such as the condition that a curve or surf ace should

have a double p oint C Covariants repre s ent certain other curves or

surf ace s having a xe d relation to the given one independent of our

choice of axe s

There are s eventeen le s sons in MHA but we will not

attempt to de scr ib e them in any detail The topics dealt with

include determinants elimination theory di scr iminants quantics

invariants and covariants canonical forms Sturms theorem and

orthogonal transformations A pro of of Borchardts theorem that

the e igenvalues of a re al symmetric matrix are re al i s b egun on

p Thi s pro of us ing Sturm remainders i s cons iderably more

complicated than mo dern pro ofs which exploit e igenvectors in

conjunction with the as so ciated quadratic form

Like the third e dition of CS the rst e dition of MHA was

the sub ject of a review in the Philosophical Magazine

We quote part of the review which makes a p o or attempt

at b e ing humorous largely at the exp ens e of the exotic names

which had b een introduced to the sub ject by Sylvester Re ading

the whole review mo st of which cons i sts of quotations f rom the

pref ace of MHA we can only conclude that the reviewer scarcely

re ad b eyond the pref ace

Within the last e ighteen years the old and welltrodden eld of Algebra

has b een invade d by a host of new and strange intruders with the

o dd sounding names of Determinants Hyp erdeterminants Di scr im

inants Emanants Invariants Evectants Bezoutiants He s s ians

having no connexion however with e ither Bo ots or Crucible s

Canonizants of no religion Dialytics and Quantics Many a

re ader of the Cambridge Mathematical Jour nal the Philo sophical

Magazine Philo sophical Transactions c has wondere d what it all

meantwondere d sometimes indee d whether there was any meaning

at al l in thes e new expre s s ions and symbols Very few even of the

b e st mathematicians of the day have paid much attention to the sub ject

as yet but they are b eginning to do so nding that there i s re ally

something like a new branch growing out of their old treenay more

that the young oshoot i s alre ady b e ar ing f ruit

In view of Cayleys imp ortance to the development of invariant the

George Salmon

ory it may b e worthwhile to note an article entitled Recent termin

ology in mathematics that Cayley wrote in or The

sub ject matter i s algebra e sp ecially invariant theory It touches

up on group s including the symmetric group S determinants

3

matrice s linear transformations invariants covariants canonical

forms among other things One of the mo st pre scient of Cayleys

articles de scr ib e s matrice s which he had brought to the publics

attention in a pap er of Although not str ictly relevant to

Salmons work for Salmon sp e aks exclus ively of determinants we

cannot forb e ar f rom quoting a little of Cayleys wr iting

Moreover in a system of s imple equations if the co ecients in the nat

ural square order are cons idere d apart by themselves this le ads to the

theory of matrices a theory which indee d might have prece de d that of

determinants the matrix i s so to sp e ak the matter of a determinant

Here i s hi s de scr iption of linear transformations le ading to the

sub ject of invariance

LINEAR TRANSFORMATIONS In this theory the variable s of a

function are suppo s e d to b e re sp ectively linear functions of a new s et

of variable s so that the function i s transformed into a s imilar function

of thes e new variable s with of cours e altered values of the co ecients

and the que stion was to nd the relations which exi sted b etween the

or iginal and new co ecients and the co ecients of the linear equations

The determinant comp o s e d of the co ecients of the linear equations i s

said to b e the modulus of transformation and when this determinant i s

unity the transformation i s said to b e unimodular It was ob s erved that

a certain function of the co ecients namely the di scr iminant p o s s e s s e d

a remarkable property found afterwards to b elong to it as one of a clas s

of functions calle d or iginally hyperdeterminants but now invariants

and it was in this manner that the problem of linear transformation le d

to the general theory of covariants

Likewis e here i s hi s denition of invariant

INVARIANT An invariant i s a function of the co ecients of a rational

and integral homogeneous function or quantic the characteri stic prop

erty whereof i s as follows namely if a linear transformation i s eected

on the quantic then the new value of the invariant i s to a f actor pres

equal to the or iginal value the f actor in que stion or quotient of the two

IMS Bulletin

values b e ing a p ower of the mo dulus of transformation and the two

values b e ing thus equal when the transformation i s unimo dular

After li sting the pr incipal textbo oks on the sub jects of the art

icle by Sp ottiswoo de Br io schi Baltzer Faa di Bruno Cayley

conclude d hi s article by recommending Salmons new textbo ok

And extending to nearly all the sub jects Salmon Le s sons introductory

to the mo dern higher algebra vo Dublin

When the s econd e dition of MHA appe are d in it was double

the s ize of the rst e dition and Salmon ap ologize d in the pre

f ace that its title was no longer appropriate as the work consti

tuted more of a tre atis e than a s et of le s sons The new e dition

was noteworthy for an extensive number of explicit calculations

of invariants Le s son XVI I Applications to binary quantics con

tains various vast formulae e ach o ccupying s everal page s The

b o ok may b e said to b e notorious for one particular calculation

that of a skew invariant E of degree fteen attached to a binary

s extic The re sult of the calculation contains terms and it

o ccupie s thirteen page s Ball thus de scr ib e d Salmons work on E

in hi s obituary xxvii

It i s in this b o ok MHA that Salmon has undertaken some of the

mo st formidable calculations that any pure mathematician has f ace d

The p ortentous invariant E alone require s s everal page s for its expre s

s ion He has told us that when the work was done he pro cee de d to

check it by testing whether the sum of the co ecients was zero as must

b e the cas e for a skew invariant He was di smayed by nding a bal

ance of s everal thousands but divining a p o s s ible error he halved this

balance and nding this was exactly one of the two co ecients he

immediately surmi s e d a wrong s ign which proved to b e the cas e and

the verication was complete

The two subs equent e ditions of MHA omitted this sp ectacular

formula Salmon also allude s on p of MHA to a formula

of almo st terms which he had calculated and which appe ars

in a pap er of Cayley Phil Trans p He spare d hi s

re aders the details

The inclus ion of thes e calculations which was hardly vital

to the ob ject of introducing re aders to the latest re s e arches on

George Salmon

invariant theory indicates an asp ect of Salmons mathematical

p ersonalitya tendency to b e involved in calculation for calcula

tions sake Joly di sapproved of this tendency s ince he wrote

p

It i s however cur ious that the f ascination with ar ithmetical work should

have detained Salmon on calculations such as that of E as above at a

time when Bo oles gre at conception was pushing on the mathematical

world to feveri sh haste in new di scovery

T A Hirst also ob s erved the love of calculation p

He i s a gre at calculator fond of calculating for its own sake I do

clas s him among the high mathematicians however The more re ady

reckoning element i s to o prominent in him

Although a fourth e dition of MHA appe are d in Salmon

admitted in its pref ace that he had not contributed to it He

wrote

The pre s sure of other engagements having prevented me f rom taking

any part in the preparation of this new e dition of my Modern Higher

Algebra I have to expre s s my obligations to the go o d oce s of my

f r iend Mr Cathcart in revi s ing the work and sup er intending its pas sage

through the pre s s

The contribution to the spre ad of knowledge about invariant the

ory made by the various e ditions of MHA and Fie dlers German

translations i s highlighted in the following extract f rom a rep ort

made to the Deutsche MathematikerVereinigung in by

W F Meyer hims elf a sp eciali st in invariant the

ory We quote f rom the English translation given by Merz in

p

Recogni s ing how the sp ecial re sults in this domain invariant theory

gradually acquire d a gre at bulk we must the more gratefully acknow

le dge the work of Salmonwho had alre ady in the direction of algebra as

well as geometry fur ni shed valuable contributions of hi s ownin under

taking the labour of collecting the widelyscattered material in a conci s e

monograph For the promulgation in Germany we have to thank Fie dler

b oth for hi s e dition of Salmon and for having alre ady given an inde

p endent introduction to the sub ject

As David Lewi ss article makes cle ar interest in the sub

IMS Bulletin

ject of invariant theory waned by the start of the th century

partly b ecause Hilberts abstract methods had solved some of the

central nitenes s problems of the sub ject Invariant theory b egan

to b e re interpreted in the th century us ing the to ols of group

repre s entation theory One may compare Salmons tre atment of

Hermites law of recipro city for the invariants of binary quantics

MHA p with that of Sp e i s er in x Sp e i s er

makes us e of the ClebschGordan formula for the repre s entations

of the group GL C The noted group theori st Philip Hall drew

2

attention to the eectivenes s of more mo dern repre s entation theory

in hi s review of Sp e i s ers b o ok in

The fundamental nature of the connection b etween the clas s ical theory of

forms and repre s entation theory i s even to day not so widely appreciated

as it de s erves to b e That after only s even page s Sp e i s er i s alre ady

completing hi s s econd pro of of Hermites recipro city theorem for binary

forms i s an indication b oth of the us efulnes s of grouptheoretic methods

and also the smo othnes s of the pre s ent tre atment

A Treatis e on the Analytic Geometry of Three Dimen

s ions

Salmons last mathematical b o ok was A Treatise on the Analytic

Geometry of Three Dimensions rst published in Thi s

was the large st of Salmons works page s in length and the

later e ditions were even longer The rst e dition contained f

teen chapters on the following sub jects the p oint interpretation

of equations the plane properties of quadr ics in general clas s i

cation of quadr ics central surf ace s methods of abridge d nota

tion confo cal surf ace s cones and spheroconics general theory

of surf ace s curves and developable s f amilie s of surf ace s sur

f ace s derived f rom quadr ics surf ace s of third degree general the

ory of surf ace s Chapter XI I I Surf ace s derived f rom quadr ics

b egins with a di scus s ion of the wave surf ace introduced by Fres

nel in hi s study of light The wave surf ace had b een studied by

W R Hamilton who had pre dicted the phenomenon of conical

ref raction on the bas i s of hi s analysi s of cusp s on the surf ace The

surf ace had also b een studied by the Trinity College mathematical

physici st James MacCullagh Chapter XIV Surf ace s of the third

George Salmon

degree contained the theory of the twenty s even lines on a cubic

surf ace which we have alre ady de scr ib e d in Section

An interesting asp ect of GTD i s an article in the

appendix on the calculus of quaternions applie d to geometrical

investigations Hamilton had invented the quaternions in

and b oth Salmon and Cayley had attende d hi s lectures in TCD on

quaternions in June Hamilton was anxious to s ee the accept

ance of quaternionic methods as standard re s e arch to ols e sp ecially

in geometry and R P Graves records a corre sp ondence in

b etween Hamilton and Salmon on the sub ject of quaternions in

pp While Salmon s eems to have maintained doubts as

to whether quaternions could b e us e d to prove anything more con

veniently than by traditional methods he nonetheles s admitted the

b e auty of quaternionic methods Hamilton entertained high hope s

that Salmon would b ecome as gre at a devotee of quaternions as he

hims elf was and he encourage d Salmon to apply quaternions with

hi s own geometric ins ight as the following p ortion of a letter

p f rom Hamilton to Salmon dated September makes

cle ar

As you s eem to have withdrawn for the pre s ent f rom the sub ject qua

ternions let me at le ast ask you to allow me to thank you for the kind

attention which you have given lately to it

And do not imagine it to b e attery if I add that I cons ider you to

b e very likely to excel myself even in this which of thes e late years

may b e regarde d as my own department if you shall ever b e induced to

pursue it You p o s s e s s a far larger stock of geometrical know ledge

although I have always had something of a geometrical taste and carr ie d

away all the honors of my divi s ion in geometry when I was very long

ago an undergraduate in our University If you shall ever s er iously take

up the Quaternions your geometry will enable you to go f ar b eyond me

in that sub ject

Salmons interest in quaternions s eems to have dimini shed f rom

this time and Hamiltons hope s for a highprole di sciple at TCD

were never fullle d By the time of the third e dition of GTD the

appendix on quaternions had di sappe are d as Salmon cons idere d

that the sub ject was now adequately handled in the textbo ok of

Kelland and Tait

IMS Bulletin

Title page of

Lessons introductory to the Modern Higher Algebra

George Salmon

Just as he had as s i sted Salmon in the preparation of a new

e dition of HPC so also did Cayley as s i st him in wr iting the third

e dition of GTD which appe are d in Cayleys contribution

was le s s than hi s previous one but he nonetheles s provide d s everal

new articles and inuence d Salmons wr iting of other articles as

Salmon explains in the pref ace to GTD When the fourth

e dition of GTD was b e ing prepare d Salmon to ok no part in the

revi s ion le aving this to hi s colleague Mr Cathcart who likewis e

revi s e d the fourth e dition of MHA in as de scr ib e d in Section

GTD was the only one of Salmons mathematical b o oks to

undergo a substantial revi s ion after hi s death for a fth e dition

appe are d in two volumes published in and The e ditor

of the new work was Reginald A P Rogers a fellow of TCD The

following review of the s econd volume by M Long show that

enthusiasm for Salmons wr iting style and hi s choice of content still

remained high ten years after hi s death

It i s interesting to note how little change it has b een neces sary to make

in Salmons own work A word here and there has b een change d but

otherwis e the or iginal text i s almo st intact and the style and character

of it have b een carefully followed by the e ditor and thos e who have

helpe d him It i s not the le ast of Salmons many virtues as a wr iter

of textbo oks that what he says may b e adde d to by later e ditors but

need rarely b e di splace d to make way for newer methods of tre atment

Salmon once said of Cayley that he b oth di scovered the diggins and

got out some of the bigge st nuggets It i s often dicult to sort out

Salmons own nuggets but there can b e no doubt of hi s genius for

cle aning up and arranging thos e alre ady found and for p ointing out

the diggins which would repay further work Certainly many of the

nuggets contained in this new volume are the direct pro duct of the

indications he gave of directions in which further progre s s might b e

made

The mining analogy which Salmon applie d to Cayleys work in

invariant theory was made in Salmons appreciatory article

Ball continued the analogy although in the context of stone quar

rying rather than gold mining when he wrote in pp

Dr Salmon wr ites about Conic Sectionsand hi s work has b een a quarry

IMS Bulletin

in which all other wr iters at home and abroad have ever s ince mined

without even exhausting its re source s

In a pap er published as late as W L Edge p ointed out

that all ve e ditions of GTD as well as its French and German

translations contained an error in the calculation of the di scr imin

ant of a cubic surf ace in three dimensional pro jective space Edge

provide d corrected formulae and de spite the error he also testied

to the clar ity of Salmons wr iting

Salmons legacy and continuing inuence

We have provide d do cumentation in this article f rom varie d

source s to sugge st that Salmon was one of the mo st inuential

textbo ok wr iters of the th century Hi s gift as others saw it

was to draw together sub jects of current re s e arch di splay them

to the re ader and illustrate the theory by wellchos en example s

Inevitably a mathematician of a later century who re ads Salmons

b o oks will nd them lacking in preci s ion and r igour but hi s

style of exege s i s was typical for the time when the denition

lemma theorem pro of mo de of pre s entation found in current

mathematical textbo oks did not exi st

The sub ject matter of two of Salmons b o oks HPC and

GTD would b e clas s ie d as algebraic geometry a sub ject well

known for the diculties it pre s ents Until the s mo st b o oks

on algebraic geometry were impreci s e and often appe ale d to intu

ition or generic arguments that exclude d degenerate or trouble some

cas e s One has only to examine textbo oks of say J L Co olidge

to ob s erve this tendency Salmon hims elf was not exempt f rom

cr iticism of the inadequacy of hi s tre atment of certain topics as

the following review by F Bath of H Hiltons Plane Algebraic

Curves shows

Unfortunately the theorem regarding the p o stulation of an r ic through

certain of the intersections of an nic and an N ic and its pro of have

b een left in this chapter as Salmon left them incomplete and mi sle ading

It i s to b e regretted that the true theorem embracing all cas e s has not

yet b een made available in an English textbo ok particularly as the

e s s ential re sult was known to Jacobi

As the arguments of algebraic geometry were clar ie d and made

George Salmon

r igorous by such mathematicians as O Zar i ski B van der Waer

den A Weil and others so the sub ject b ecame le s s intuitive

and p erhaps le s s acce s s ible to the outsider who did not wi sh

to b e burdened by notions of intersection multiplicities divi sor

clas s e s holomorphic dierentials transcendence degree and so

on However as Zar i ski said at the International Congre s s of

Mathematicians in Our aim i s not to prove that our f athers

were wrong but that they were r ight Salmons nonr igorous

arguments have b een replace d by r igorous if longer arguments

To s ee this in practice a more mo dern pro of of a theorem of

Salmon on cubic curves may b e found in p and an

up dated version of a f avourite topic of Salmon systems of conics

and quadr ics may b e studied in pp

While mo dern re s e arch require s a dierent s et of textbo oks

for its bas ic reference s compare d with thos e us e d in th century

re s e arch nonetheles s Salmons textbo oks continue to b e quoted in

the re s e arch literature as we have found by browsing through a

few jour nals To take a random example in a collection of copies

of Mathematical Proceedings of the Cambridge Philosophical Soci

ety in our oce dated b etween and we found three

reference s to Salmons tre atis e s and which i s a

go o d citation record for b o oks that were nearly years old at

the time they were cited At the time of this wr iting the mo st

recent citation to Salmons work that we have s een i s of

Salmons place in the hi story of mathematics appe ars to b e

as sure d as hi s textbo oks gained an unusually high degree of recog

nition for at le ast years and in the cas e of Conic Sections for

more than years Recognition o ccurre d b oth at the element

ary level and at the re s e arch level Of universitylevel mathematics

b o oks wr itten or published in Ireland only Cas eys Sequel to Euc

lid or Bur ns ide and Pantons Theory of Equations can have b een

as succes sful as Salmons works and as scientic publishing i s vir

tually nonexi stent now in Ireland it i s unlikely that any other

scientic b o ok published in Ireland will ever b e as succes sful as

Salmons

Further progre s s on Salmons career and hi s relationships with

contemporary mathematicians will require study of hi s corre sp ond

IMS Bulletin

ence According to there are f amily pap ers and letters of

Salmon in the TCD library and we have alre ady noted that the

Hamilton corre sp ondence contains s everal letters f rom Salmon to

Hamilton on the sub ject of quaternions dating f rom In addi

tion the library of University College London has letters f rom

Salmon to T A Hirst dated b etween and

It i s p erhaps dicult now for us to appreciate how highly Sal

mon an academic and theologian was regarde d during the latter

part of hi s life The many appreciations of Salmon that we have

re ad wr itten b oth b efore and after he die d all p ortray a man

admire d to an extraordinary extent As it states in p

More p erhaps than any member of the College throughout its hi story

he left hi s contemporar ie s the impre s s ion of a gre at man and p o sterity

has s een no re ason to reverse this verdict

English Language Editions of Salmons Bo oks

AGI A Treatis e on Analytic Geometry Part I The Right Line and Circle

vo iv p Ho dge s and Smith Dublin

AGII A Treatis e on Analytic Geometry and the Theory of Plane Curves Part

I I Conic Sections vo i errata p Ho dge s and Smith Dublin

CS A Treatis e on Conic Sections vo ii errata p Ho dge s and

Smith Dublin

nd e d vo iv p Ho dge s and Smith Dublin

rd e d vo xv p Longman Brown Green and Longmans Lon

don

th e d vo xvi p Longman Brown Green Longman and

Rob erts London

th e d vo xv p Longmans Green Ryder and Dyer London

th e d vo xv p Longmans Green and Co London

HPC A Treatis e on the Higher Plane Curves vo xii p Ho dge s and

Smith Dublin

nd e d vo xix p Ho dge s Foster and Co Dublin

rd e d vo xix p Ho dge s Foster and Figgi s Dublin

MHA Le s sons Introductory to the Mo dern Higher Algebra ix p Ho dge s

Smith and Co Dublin

George Salmon

nd e d vo xv p Ho dge s Smith and Co Dublin

rd e d vo xix p Ho dge s Foster and Co Dublin

th e d vo xv p Ho dge s Figgi s and Co Dublin

GTD A Treatis e on the Analytic Geometry of Three Dimensions vo xv

errata p Ho dge s Smith and Co Dublin

nd e d vo xv p Ho dge s Smith and Co Dublin

rd e d vo xvii errata p Ho dge s Foster and Co Dublin

th e d vo xix errata p Ho dge s Figgi s and Co Dublin

th e d vols revi s e d by Reginald A P Rogers Vol vo xxii

p Vol vo xvi p Longmans Green and Co London

Translations of Salmons Bo oks

a Analytische Geometrie der Kegelschnitte mit b e sonderer Ber ucksicht

igung der neueren Methoden von G Salmon Unter Mitwirkung de s Ver

f as s ers Deutsch b e arb e itet von Dr Wilhelm Fie dler Teubner Le ipzig

b Analytische Geometrie de s Raumes von George Salmon Deutsch

b e arb e itet von Dr Wilhelm Fie dler vols Teubner Le ipzig

c Vorlesungen zur Einf uhrung in die Algebra der linearer Transforma

tionen von George Salmon Deutsch b e arb e itet von Dr Wilhelm Fie dler

viii p Teubner Le ipzig

d Analytische Geometrie der hoheren ebenen Curven von George Sal

mon Deutsch b e arb e itet von Dr Wilhelm Fie dler xvi p Teubner

Le ipzig

e Lecons dalgebre superieur Traduit de langlais par M Bazin Aug

mente de notes par M Hermite xii p GauthierVillars Pari s

f Traite de geometrie a deux dimensions s ections conique s Traduit

de langlais sur la cinquieme edition par H Re sal et V Vaucheret

GauthierVillars Pari s

g Traite de geometrie analytique a troi s dimensions Ouvrage traduit de

langlais sur la quatrieme edition par O Chemin Pari s

h Traite de geometrie analytique courb e s planes Ouvrage traduit de

langlais par O Chemin et suivi dune etude sur le s p oints s inguliers

par G Halphen xxiii p GauthierVillars Pari s

IMS Bulletin

i Trattato analitico delle s ezioni coniche Versione italiana sullultima

e d ingle s e par N Salvatore Dino Aggiuntovi gli elementi di geometria

analitica a tre co ordinate e stratti dal trattato del Salmon Nap oli

Papers of George Salmon

On the properties of surf ace s of the s econd degree which corre sp ond to

the theorems of Pascal and Br ianchon on conic s ections Phil Mag

Nouv Ann Math

On the degree of a surf ace recipro cal to a given one Camb and

Dubl Math J

Note on the parabolic p oints of surf ace s Camb and Dubl Math J

On the generation of surf ace s of the s econd degree Pro c RIA

On the condition that a plane should touch a surf ace along a curve line

Camb and Dubl Math J

On the number of normals which can b e drawn f rom a given p oint

to a given surf ace Camb and Dubl Math J

Nouv Ann Math

Note on a re sult of elimination Camb and Dubl Math J

th

On the cone circumscribing a surf ace of the m order Camb and

Dubl Math J

On the tr iple tangent planes to a surf ace of the third order Camb and

Dubl Math J

On the clas s ication of curves of double curvature Camb and Dubl

Math J

On the condition that an equation should have equal ro ots Camb and

Dubl Math J

Lettre de Mr G Salmon de Dublin a lediteur de ce jour nal Sur le s

p oints dinexion de s courb e s de troi s ieme degre J f ur die re ine und

angewandte Math

Theoremes sur le s courb e s de troi s ieme degre J f ur die re ine und ange

wandte Math Nouv Ann Math

Sur la formulation de la courb e reciproque a une courb e donnee J f ur

die re ine und angewandte Math

George Salmon

On a clas s of rule d surf ace s Camb and Dubl Math J

On recipro cal surf ace s Pro c RIA

Exerci s e s in the hyperdeterminant calculus Camb and Dubl Math J

On the problem of the inandcircumscrib e d tr iangle Phil Mag

On the problem of the inandcircumscrib e d tr iangle Phil Mag

Geometrical notes Quart J Math

On the order of certain systems of equations Quart J Math

On the contact of r ight lines with surf ace s Quart J Math

Sur la theorie de deux conique s Nouv Ann Math

On curves of the third order Phil Trans Royal So ciety

Pro c Royal So ciety

On the equation of the surf ace of centres of an ellip soid Quart J Math

Rectication dun theoreme de Mm Steiner et Dewulf Le lieu de s

sommets de s angle s droits circonscr its a une courb e de la clas s e n e st

une courb e de degre n Nouv Ann Math

On the degree of the surf ace recipro cal to a given one Trans RIA

On quaternary cubics Phil Trans Royal So ciety

Pro c Royal So ciety

Sur le theoreme Faure et courb e s paralleles Nouv Ann Math

On the relation which connects the mutual di stances of ve p oints in

space Quart J Math

On the determination of the p oints of contact of double tangents to an

algebraic curve Quart J Math

Geometrical theorems Quart J Math

Notes on quadr iplanar co ordinates Quart J Math

On the determination of the fo ci of a conic Quart J Math

IMS Bulletin

Geometrical theorems Quart J Math

On the circle which touches the four circle s which touch the s ide s of a

given spherical tr iangle Quart J Math

On some p oints in the theory of elimination Quart J Math

On the number of surf ace s of the s econd degree which can b e de scr ib e d

to satisfy nine conditions Quart J Math

On some sp ecial forms of conics Quart J Math

On the limiting cas e s of certain conics Me s s enger of Mathematics

On p er io ds in the recipro cals of pr imes Me s s enger of Mathematics New

Ser ie s

Reference s

R S Ball Obituary notice of George Salmon Pro c London Math So c

new s er ie s xxiixxviii

W V Ball e d Remini scence s and Letters of Sir Rob ert Ball Cas s ell

and Co London

F Bath Review of Plane Algebraic Curves by H Hilton Math Gazette

J H Ber nard Salmon George in Dictionary of National

Biography Second Supplement vol I I I e d by S Lee Smith

Elder Co London

Biographical Dictionary of Mathematicians vols Charles Scr ibners

Sons New York

G Bo ole Exp o s ition of a general theory of linear transformations Cam

br idge Math Jour nal and

R H Bro ck Reforming mathematics e ducation an AngloIri sh connec

tion in Science in Ireland Tradition and Reform e d by

J R Nudds et al Trinity College Dublin Dublin

G D Burtchaell and T U Sadle ir Alumni Dubliniense s

Alex Thom Co Ltd Dublin

A Cayley On the tr iple tangent planes of surf ace s of the third order

Camb and Dubl Math J Collected Mathematical

Papers vol I pp

George Salmon

A Cayley Recent Terminology in Mathematics in The English Cyc

lopaedia Arts and Science vol V pp Bradbury and Evans

London

M Chasles Apercu hi storique sur lor igine et le developpement de s

methode s en geometrie M Hayez Brus s els

M Chasles Traite de s s ections conique s premiere partie Gauthier

Villars Pari s

A M Cohen and D B Wales GLorbits in a dimensional mo dule

for characteri stic J Alg

Dr George Salmon F R S Nature

Dublin University Examinations MDCCCXXXIX William Curry

Jun and Co Dublin

W L Edge The scroll of tangents of an elliptic quartic curve Math

Pro c Camb Phil So c

W L Edge The di scr iminant of a cubic surf ace Pro c RIA A

W C Fletcher Premature abstraction Math Gazette

C Graves Translation of two Memoirs On the Properties of Cones of

the Second Degree and On the Spherical Conics by M Chasles With

notes and additions and an Appendix On the Application of Analysis

to Spherical Geometry Grant and Bolton Dublin

R P Graves Life of Sir William Rowan Hamilton vol I I I Ho dge s

Figgi s Co Dublin

P Hall Review of Die Theorie der Gruppen von end licher Ordnung by

A Sp e i s er Math Gazette

R Hartshorne Algebraic Geometry Spr ingerVerlag Berlin He idel

b erg New York

T A Hirst Review of A Treatise on Conic Sections by G Salmon

Phil Mag

J Hymers A Treatis e on Conic Sections nd e d T Stevenson Cam

br idge

J Hymers A Treatis e on the Theory of Algebraical Equations rd e d

De ighton Bell and Co Cambridge

C J Joly George Salmon Pro c Royal So ciety

IMS Bulletin

F Kle in Vorlesungen uber die Entwicklung der Mathematik im

Jahrhundert vols Spr inger Berlin

F Kle in Elementary Mathematics f rom an Advanced Standpoint Vol

I I Geometry Translated f rom the German by E R He dr ick and

C A Noble Dover New York

D W Lewi s David Hilbert and the theory of algebraic invariants Ir i sh

Math So c Bulletin

M Long Review of A Treatise on the Analytic Geometry of Three

Dimensions by G Salmon th e d vol revi s e d by R A P Rogers

Math Gazette

A J McConnell Salmon George in Dictionary of Scientic Biography

vol XI I e d by C G Gilli spie Charles Scr ibners Sons New

York

R B McDowell and D A Webb Trinity College Dublin

An Academic Hi story Cambridge University Pre s s Cambridge

C Maxwell A Hi story of Trinity College Dublin The Uni

versity Pre s s Dublin

J T Merz A Hi story of Europe an Thought in the Nineteenth Century

vol I William Blackwoo d and Sons Edinburgh and London

J T Merz A Hi story of Europe an Thought in the Nineteenth Century

vol I I William Blackwoo d and Sons Edinburgh and London

W F Meyer Ber icht uber den gegenwartigen Stand der Invariantenthe

or ie Jahre sb er icht der Deutschen Mathematikervereinigung

E H Neville Salmon Math Gazette

P E Newstead Invariants of p encils of binary cubics Math Pro c

Camb Phil So c

JV Poncelet Traite de s proprietes pro jectives de s gure s Bachelier

Pari s

JV Poncelet Traite de s proprietes pro jectives de s gure s deuxieme

edition tome s econd GauthierVillars Pari s

JV Poncelet Applications danalyse et de geometrie tome premier

MalletBachelier Pari s

M Re id Undergraduate Algebraic Geometry London Math So c Stu

dent Texts Cambridge University Pre s s Cambridge

George Salmon

G Salmon Que stion in Mathematical Que stions with their Solu

tions From the Educational Times vol V C F Ho dgson Son

London

G Salmon Science worthies xxiiArthur Cayley Nature

H J S Smith On the pre s ent state and pro sp ects of some branches of

pure mathematics Pro c London Math So c

A Sp e i s er Die Theorie der Gruppen von endlicher Ordnung third

e dition Spr inger Berlin

R Taton Poncelet Je an Victor in Dictionary of Scientic Biography

vol XI e d by C G Gilli spie Charles Scr ibners Sons New

York

The Dublin University Calendar MDCCCXLVII I Ho dge s and Smith

Dublin

The Manuscript Papers of Br itish Scientists Her Ma je stys

Stationery Oce London

J A Todd Poncelets p or i stic p olygons Math Gazette

J A Todd Pro jective and Analytical Geometry revi s e d e d Sir Isaac

Pitman Sons Ltd London

I Todhunter A Treatis e on Plane Coordinate Geometry as Applie d to

the Straight Line and the Conic Sections Macmillan and Co Cam

br idge

B L van der Waerden Einf uhrung in die algebraische Geometrie

s econd e d Spr ingerVerlag Berlin He idelberg New York

R J Walker Algebraic Curves Pr inceton University Pre s s Pr inceton

C T C Wall Note on the invariant of plane cubics Math Pro c Camb

Phil So c

F P White Review of Vorlesungen uber die Entwicklung der Math

ematik im Jahrhundert I by F Kle in Math Gazette

T D Sp e arman George Salmon A Trinity Monday Di s

cours e May Hermathena

Acknowledgements I am grateful to the National Library of

Ireland and the Library of Trinity College Dublin for providing

IMS Bulletin

me with the f acilities to re ad rare material relating to this article

I am also grateful to Vincent Kinane of the Department of Early

Pr inted Bo oks Trinity College Library for informing me about the

copy of the two parts of Salmons Treatise on Analytic Geometry

in the Trinity College Library

Ro d Gow

Department of Mathematics

University College Dublin

Beleld

Dublin