# Percival Frost's Books

Percival Frost has published three mathematics books, all of which have run to several editions. We give below the Prefaces of some of these editions. We have omitted some editions since, for example, the Preface to the Second Edition of the Curve Tracing book is almost identical to that of the first edition written twenty years earlier. A fifth edition of this book was published by the Chelsea Publishing Company in 1960, almost a hundred years after it was written.

**Click on a link below to go to that book****1. Newton's Principia Sections I, II, III (First edition)**

with Notes and Illustrations

also A collection of problems principally intended as examples of Newton's methods

by Percival Frost, M.A.

Late Fellow of St John's College;

Mathematical Lecturer of Jesus College.

In publishing the following work, the Author's principal intention has been to explain those difficulties which may be encountered on first reading the Principia, to illustrate the advantages of Newton's methods by shewing the extent to which they may be applied in the solution of problems, and to prepare the student for engaging in the study of the higher branches of mathematics by exhibiting in a geometrical form some of the processes employed in the Differential and Integral Calculus, and in the Analytical investigations in Dynamics.

The Author has endeavoured, in preparing the version of the first three Sections of the Principia, to adhere closely to the original form in the first Section and in the beginning of the second; and in the cases in which sentences have been interpolated or the form of the demonstration changed, such changes and interpolations have been marked by brackets.

In the second and third sections these indications of deviation from the original form have been discontinued.

In the first section in which the Lemmas are established, which are the basis of Newton's investigations, although it is generally advisable not to deviate from the original, yet in some cases his demonstrations are purposely expressed very concisely, and, in the fifth Lemma, he is contented with simply giving the enunciation, so that in these cases it is considered that no distinctness has been lost by the interpolations which have been made.

Throughout the Problems and Theorems which depend upon the sixth proposition, the variations have been replaced by equations. By this method of treating the subject, now commonly adopted in treatises professing to be versions of these Sections of the Principia, it is believed that clearer ideas of the meaning of each step of the demonstrations are obtained by the student.

The Author desires to make his acknowledgments of the great assistance which he has derived in the preparation of the Notes, from the study of Whewell's Method of Limits, from which the Articles 49 to 56 have been almost entirely taken : he has also made use of several editions of Newton, and especially of Carr's. He takes this opportunity of thanking several of his friends who have kindly given him their assistance in preparing his papers for the press, and in verifying the results of the Problems.

The principal portion of the Problems have been selected from the papers set in the examinations for the Mathematical Tripos and in the course of the College examination, especially in those of St John's College.

The Author has been influenced in this selection by his desire practically to bring before the student the great advantages which undoubtedly arise from a judicious use of geometrical methods.

It is only necessary to add, that care has been taken that no problems be introduced which are not capable of solution by methods given in the work.

Cambridge

25 October 1854

also A collection of problems principally intended as examples of Newton's methods

by Percival Frost, M.A.

Late Fellow of St John's College;

Mathematical Lecturer of Jesus College.

**Preface**In publishing the following work, the Author's principal intention has been to explain those difficulties which may be encountered on first reading the Principia, to illustrate the advantages of Newton's methods by shewing the extent to which they may be applied in the solution of problems, and to prepare the student for engaging in the study of the higher branches of mathematics by exhibiting in a geometrical form some of the processes employed in the Differential and Integral Calculus, and in the Analytical investigations in Dynamics.

The Author has endeavoured, in preparing the version of the first three Sections of the Principia, to adhere closely to the original form in the first Section and in the beginning of the second; and in the cases in which sentences have been interpolated or the form of the demonstration changed, such changes and interpolations have been marked by brackets.

In the second and third sections these indications of deviation from the original form have been discontinued.

In the first section in which the Lemmas are established, which are the basis of Newton's investigations, although it is generally advisable not to deviate from the original, yet in some cases his demonstrations are purposely expressed very concisely, and, in the fifth Lemma, he is contented with simply giving the enunciation, so that in these cases it is considered that no distinctness has been lost by the interpolations which have been made.

Throughout the Problems and Theorems which depend upon the sixth proposition, the variations have been replaced by equations. By this method of treating the subject, now commonly adopted in treatises professing to be versions of these Sections of the Principia, it is believed that clearer ideas of the meaning of each step of the demonstrations are obtained by the student.

The Author desires to make his acknowledgments of the great assistance which he has derived in the preparation of the Notes, from the study of Whewell's Method of Limits, from which the Articles 49 to 56 have been almost entirely taken : he has also made use of several editions of Newton, and especially of Carr's. He takes this opportunity of thanking several of his friends who have kindly given him their assistance in preparing his papers for the press, and in verifying the results of the Problems.

The principal portion of the Problems have been selected from the papers set in the examinations for the Mathematical Tripos and in the course of the College examination, especially in those of St John's College.

The Author has been influenced in this selection by his desire practically to bring before the student the great advantages which undoubtedly arise from a judicious use of geometrical methods.

It is only necessary to add, that care has been taken that no problems be introduced which are not capable of solution by methods given in the work.

Cambridge

25 October 1854

**2.Newton's Principia Sections I, II, III (Second Edition)**

**Preface.**

In publishing the following work, my principal intention is to explain difficulties, which may be encountered by the student on first reading the Principia, and to illustrate the advantages of a careful study of the methods employed by Newton, by showing the extent to which they may be applied in the solution of problems. I have also endeavoured to give assistance to the student who is engaged in the study of the higher branches of Mathematics, by representing in a geometrical form several of the processes employed in the Differential and Integral Calculus, and in the analytical investigations of Dynamics.

In my version of the first section and the beginning of the second I have adhered as closely as I could to the original form; and, in the cases in which sections have been interpolated, or the form of demonstration changed, I have indicated such changes and interpolations by brackets.

Although it is generally advisable not to deviate from Newton's words in the demonstrations of the Lemmas, yet in many cases, I suppose, purposely, he expressed himself very concisely, as in Lemmas IV and X; and he was contented with simply giving the enunciation of Lemma V; in these cases, therefore, interpolations are made which, I believe, are in accordance with Newton's plan of demonstration.

Throughout the Problems and Theorem which depend upon the sixth proposition, the variations are replaced by equations; by this method of treating the subject, I conceive that clearer ideas of the meaning of each step are obtained by the student.

I take this opportunity to acknowledge the great assistance which I have derived in the preparation of my notes, from the study of Whewell' Method of Limits, from which the Articles 55-60 have been almost entirely taken; I have also made use of several editions of Newton, and especially of Carr's.

The Problems are principally selected from the papers set in the examinations for the Mathematical Tripos, and in the course of the College examinations; the results of these problems are given either in the statements or at the end of the work, but I have not thought it advisable to supply hints for the solution, because I imagine that the student would have been deprived thereby of the advantage which it is the object of a problem to secure. It is only necessary to add that I have been careful to introduce no problems which are not capable of solution by methods given in the work.

I desire to express my thanks to Mr Hadley of St John's College for several valuable suggestions, and also to Mr Cockshott of Trinity College, and to Mr King of Jesus College, for their kindness in correcting the errors of the press, and in testing the accuracy of the problems, which, I believe, are nearly free from mistakes.

Cambridge

13 November 1863

**3. A Treatise of Solid Geometry**

by the Rev Percival Frost, M.A.

Late Fellow of St John's College

Mathematical Lecturer of King's College

and the Rev Joseph Wolstenholme, M.A.

Fellow and Assistant Tutor of Christ's College.

The Authors of the following Treatise have endeavoured to present before students as comprehensive a view of the subject, as certain limitations have allowed them to do. The necessity of these limitations has developed itself in the course of preparing the work during a period of four years. The study of innumerable papers, by the most celebrated mathematicians of all countries, has convinced the authors that the subject is almost inexhaustible, and that, to have treated all parts of it with anything approaching to the fulness with which they have treated the first portion, would have swelled their work in a fearful proportion to what it has already attained.

Intending, as they have done, to make the subject accessible, at least in the earlier portions, to all classes of students, they have endeavoured to explain completely all the processes which are most useful in dealing with ordinary theorems and problems connected with the straight line, and plane, and particular surfaces of the second degree, and, in doing so, their object has been to direct the student to the selection of the methods which are best adapted to the exigencies of each problem. In the more difficult portions of the subject, they have considered themselves to be addressing a higher class of students, and here they have tried to lay a good foundation on which to build, if any of their readers should wish to pursue their studies in any department of the science, beyond the limits to which the work extends.

The authors would willingly have given references to all the writers from whom they have derived information in the course of their work, but they have found this to be impossible, and they regret it the less, because it will not be supposed that they lay claim to every thing in which they have made no reference. They have, however, in a very large number of cases mentioned the names of eminent men, who have advanced the boundaries of the subject, and they hope it will be apparent, that they have appreciated the labours of such men as Cayley, Salmon, McCullagh, Roberts and Townsend; at all events they are sensible that, in many departments, the treatise lately published by Salmon on the same subject proves how far their own work is from being perfect.

They cannot conclude this work without making acknowledgments to Mr Ferrers of Caius College, and Mr Horne of St John's College for their kindness in examining and commenting upon the proof sheets of the earlier parts of their work, and at the same time without expressing their regret that they have not escaped a large number of errors, which it will be punishment enough to them to see tabulated in an adjoining page.

1863.

Late Fellow of St John's College

Mathematical Lecturer of King's College

and the Rev Joseph Wolstenholme, M.A.

Fellow and Assistant Tutor of Christ's College.

**Preface.**The Authors of the following Treatise have endeavoured to present before students as comprehensive a view of the subject, as certain limitations have allowed them to do. The necessity of these limitations has developed itself in the course of preparing the work during a period of four years. The study of innumerable papers, by the most celebrated mathematicians of all countries, has convinced the authors that the subject is almost inexhaustible, and that, to have treated all parts of it with anything approaching to the fulness with which they have treated the first portion, would have swelled their work in a fearful proportion to what it has already attained.

Intending, as they have done, to make the subject accessible, at least in the earlier portions, to all classes of students, they have endeavoured to explain completely all the processes which are most useful in dealing with ordinary theorems and problems connected with the straight line, and plane, and particular surfaces of the second degree, and, in doing so, their object has been to direct the student to the selection of the methods which are best adapted to the exigencies of each problem. In the more difficult portions of the subject, they have considered themselves to be addressing a higher class of students, and here they have tried to lay a good foundation on which to build, if any of their readers should wish to pursue their studies in any department of the science, beyond the limits to which the work extends.

The authors would willingly have given references to all the writers from whom they have derived information in the course of their work, but they have found this to be impossible, and they regret it the less, because it will not be supposed that they lay claim to every thing in which they have made no reference. They have, however, in a very large number of cases mentioned the names of eminent men, who have advanced the boundaries of the subject, and they hope it will be apparent, that they have appreciated the labours of such men as Cayley, Salmon, McCullagh, Roberts and Townsend; at all events they are sensible that, in many departments, the treatise lately published by Salmon on the same subject proves how far their own work is from being perfect.

They cannot conclude this work without making acknowledgments to Mr Ferrers of Caius College, and Mr Horne of St John's College for their kindness in examining and commenting upon the proof sheets of the earlier parts of their work, and at the same time without expressing their regret that they have not escaped a large number of errors, which it will be punishment enough to them to see tabulated in an adjoining page.

1863.

**4. Solid Geometry**

by Percival Frost, M.A.,

Formerly Fellow of St John's College

Mathematical Lecturer of King's College

A New Edition,

Revised and Enlarged of the Treatise by Frost and Wolstenholme.

VOL. I

It was with a feeling of great discouragement that I began the preparation of another Edition of this work, deprived, as I was, of the valuable assistance of my friend Mr Wolstenholme, in working with whom I had had so much pleasure while writing the First Edition. Mr Wolstenholme, who is now Professor of Mathematics in the Royal Indian Engineering College at Cooper's Hill, thought that there would be great difficulty in carrying on this work satisfactorily by correspondence, even if the important duties in which he is engaged did not fully occupy his time; I was, therefore, reluctantly obliged to undertake the whole labour of remodelling our original work.

As we contemplated making additions, and many alterations both in form and substance, my friend desired that his name might not appear in the Second Edition, and I have been compelled to alter the title of the work, and to take the responsibility of the changes which have been introduced.

The problems which appeared in the former Edition were for the most part original, and a large proportion of them were due to Mr Wolstenholme; in this department, therefore, a most important one in my opinion, I have not lost the advantage of his valuable assistance.

The present Edition is intended, in its complete form, to occupy two volumes, but for the convenience of Students who may wish to have in one volume all those portions of Solid Geometry which would be useful to them in their studies of Physical subjects, I have endeavoured, as far as I could without material departure from the arrangement which I considered best for the proper treatment of the subject, to include in the first volume nearly all that will be required from their point of view; with this object, I have reserved for the second volume those parts which are chiefly interesting as Pure Geometry.

The Student who desires to confine his reading to the more practical portions of the subject should omit Chapters VI, VII, VIII, IX, Art 156-167, XV and XVII, the Three-Plane system of Coordinates being employed exclusively in the remaining chapters.

I feel bound to say a few words with respect to my persistence in retaining the word 'Conicoid' to represent the locus for the equation of the second degree. It was natural that the distinguished analyst, who has done so much towards the investigation of the properties of surfaces of higher degrees than the second, should seek a term for that of the second degree, which would connect it with those of higher degrees. But I cannot help thinking it unfortunate that the terms ^ quadric' should have been selected, which had already a different meaning. I quote the words of the author of the well-known treatise on Higher Algebra:

I consider that the surface of the second degree at present, whatever may be the case in some future development, stands on a platform of its own, on account of the services which it has rendered to all departments of Mathematical Science, and well deserves a distinctive name instead of being recognised only by its number, a mode of designation which, I am informed, a convict feels so acutely. Man might be always called a biped, because besides himself there exist a quadruped, an octopus, and a centipede, but, on account of his superiority, it is more complimentary to call him by some special name.

The useful word 'conic' being well-established, the term 'conicoid' seems to suggest all that can be required, when it is employed to designate the locus of the equation of the second degree in three dimensions, at least so long as the analogous words spheroid, ellipsoid, and hyperboloid are in use, at all events it is not open to the great objection of being equally applicable to plane curves, as is the term 'quadric;' cubics and quartics being actually so employed in Salmon's

To the many excellent mathematicians, whose talent is shewn in the composition of the yearly College papers and the papers set for the Mathematical Tripos examination, I am indebted in the highest degree both for the problems which I have added to the collection, and also for the hints derived from them in the treatment of the subject itself.

I have also to make thankful acknowledgments for the valuable assistance received from my friends. Mr Moulton, of Christ's College, has given me great help in parts of the subject which, except in the chapters on the general equation, do not appear in this volume. Mr H M Taylor, Fellow of Trinity College, was kind enough to look over many of the proof sheets; and I am indebted to Mr Ritchie and Mr Main, of Trinity College, and Mr Stearn, of King's College, for their kindness in testing a large number of the problems, as well as in looking over the proof sheets throughout the process of publication. But especially I wish to thank Mr Stearn for the great assistance which he has rendered in superintending the work during my frequent absence from Cambridge, and also for his many valuable criticism.

Cambridge

October 1875.

Formerly Fellow of St John's College

Mathematical Lecturer of King's College

A New Edition,

Revised and Enlarged of the Treatise by Frost and Wolstenholme.

VOL. I

**Preface.**It was with a feeling of great discouragement that I began the preparation of another Edition of this work, deprived, as I was, of the valuable assistance of my friend Mr Wolstenholme, in working with whom I had had so much pleasure while writing the First Edition. Mr Wolstenholme, who is now Professor of Mathematics in the Royal Indian Engineering College at Cooper's Hill, thought that there would be great difficulty in carrying on this work satisfactorily by correspondence, even if the important duties in which he is engaged did not fully occupy his time; I was, therefore, reluctantly obliged to undertake the whole labour of remodelling our original work.

As we contemplated making additions, and many alterations both in form and substance, my friend desired that his name might not appear in the Second Edition, and I have been compelled to alter the title of the work, and to take the responsibility of the changes which have been introduced.

The problems which appeared in the former Edition were for the most part original, and a large proportion of them were due to Mr Wolstenholme; in this department, therefore, a most important one in my opinion, I have not lost the advantage of his valuable assistance.

The present Edition is intended, in its complete form, to occupy two volumes, but for the convenience of Students who may wish to have in one volume all those portions of Solid Geometry which would be useful to them in their studies of Physical subjects, I have endeavoured, as far as I could without material departure from the arrangement which I considered best for the proper treatment of the subject, to include in the first volume nearly all that will be required from their point of view; with this object, I have reserved for the second volume those parts which are chiefly interesting as Pure Geometry.

The Student who desires to confine his reading to the more practical portions of the subject should omit Chapters VI, VII, VIII, IX, Art 156-167, XV and XVII, the Three-Plane system of Coordinates being employed exclusively in the remaining chapters.

I feel bound to say a few words with respect to my persistence in retaining the word 'Conicoid' to represent the locus for the equation of the second degree. It was natural that the distinguished analyst, who has done so much towards the investigation of the properties of surfaces of higher degrees than the second, should seek a term for that of the second degree, which would connect it with those of higher degrees. But I cannot help thinking it unfortunate that the terms ^ quadric' should have been selected, which had already a different meaning. I quote the words of the author of the well-known treatise on Higher Algebra:

It is convenient to have a word to denote the function itself without being obliged to speak of the equation got by putting the function = 0. The term 'quantic' denotes, after Mr Cayley, a homogeneous function in general, using the words 'quadric,' 'cubic,' 'quartic,' 'n-ic,' to "denote quantics of the 2nd, 3rd, 4th, nth, degrees.Now, 'quadric,' as used in the other sense, is not even the equation found, but it takes two steps and becomes the locus of the equation.

I consider that the surface of the second degree at present, whatever may be the case in some future development, stands on a platform of its own, on account of the services which it has rendered to all departments of Mathematical Science, and well deserves a distinctive name instead of being recognised only by its number, a mode of designation which, I am informed, a convict feels so acutely. Man might be always called a biped, because besides himself there exist a quadruped, an octopus, and a centipede, but, on account of his superiority, it is more complimentary to call him by some special name.

The useful word 'conic' being well-established, the term 'conicoid' seems to suggest all that can be required, when it is employed to designate the locus of the equation of the second degree in three dimensions, at least so long as the analogous words spheroid, ellipsoid, and hyperboloid are in use, at all events it is not open to the great objection of being equally applicable to plane curves, as is the term 'quadric;' cubics and quartics being actually so employed in Salmon's

*Higher Plane Curves*, Chapters V and VI.To the many excellent mathematicians, whose talent is shewn in the composition of the yearly College papers and the papers set for the Mathematical Tripos examination, I am indebted in the highest degree both for the problems which I have added to the collection, and also for the hints derived from them in the treatment of the subject itself.

I have also to make thankful acknowledgments for the valuable assistance received from my friends. Mr Moulton, of Christ's College, has given me great help in parts of the subject which, except in the chapters on the general equation, do not appear in this volume. Mr H M Taylor, Fellow of Trinity College, was kind enough to look over many of the proof sheets; and I am indebted to Mr Ritchie and Mr Main, of Trinity College, and Mr Stearn, of King's College, for their kindness in testing a large number of the problems, as well as in looking over the proof sheets throughout the process of publication. But especially I wish to thank Mr Stearn for the great assistance which he has rendered in superintending the work during my frequent absence from Cambridge, and also for his many valuable criticism.

Cambridge

October 1875.

**5. Solid Geometry (Third Edition)**

by Percival Frost, D.Sc, F.R.S.

Formerly Fellow of St John's College

Fellow of King's College

Mathematical Lecturer of King's College

I have reprinted the principal part of the Preface to the Second Edition because it contains the expression of my feelings of regret when I was deprived of the co-operation of Dr Wolstenholme in the preparation of that Edition.

I have especially retained my plea for the use of the term 'conicoid,' and the statement of my objections to the term 'quadric,' which weigh so much with me, that they have overbalanced my desire to do honour to the distinguished sponsor, to whom I owe so much, by accepting the name which he selected as most suitable.

In the present Edition the problems placed at the end of each chapter have been selected and arranged with great care in groups, each of which illustrates most of the points of the chapter to which they are attached.

In an Appendix, now nearly ready for the Press, I shall give hints sufficient for the solution of the problems.

The use of the 'solidus,' described by Prof Stokes in the Preface to his

I have in the body of the work given references for many of the mathematical papers which I have had occasion to use; but I find that I have omitted to attach the names of the authors to so many theorems that I think it best to supply the omission by giving here a list of articles which contain theorems due to authors not mentioned in the text.

...

The arrangement of lines of a cubic surface, called a double sixer. Art. 541, was found out by Schlafli.

I am afraid that there still remain many omissions, but I should like especially to acknowledge how much I owe to Dr Salmon for whatever knowledge I possess of many departments of the subject on which I have been engaged.

This is the proper place to express my thanks for the great assistance which I have received from Mr Chree, Mr Berry, and Mr Richmond, of King's College, who have kindly not only corrected the proof sheets, which is a very tedious business, but helped me materially by testing the correctness of a great many of the problems. I wish also to thank Mr Stearn, of King's College, for his help in the earlier portions of the work, and for his kind superintendence of the printing during my absence from Cambridge.

Cambridge

March 1886

Formerly Fellow of St John's College

Fellow of King's College

Mathematical Lecturer of King's College

**Preface.**I have reprinted the principal part of the Preface to the Second Edition because it contains the expression of my feelings of regret when I was deprived of the co-operation of Dr Wolstenholme in the preparation of that Edition.

I have especially retained my plea for the use of the term 'conicoid,' and the statement of my objections to the term 'quadric,' which weigh so much with me, that they have overbalanced my desire to do honour to the distinguished sponsor, to whom I owe so much, by accepting the name which he selected as most suitable.

In the present Edition the problems placed at the end of each chapter have been selected and arranged with great care in groups, each of which illustrates most of the points of the chapter to which they are attached.

In an Appendix, now nearly ready for the Press, I shall give hints sufficient for the solution of the problems.

The use of the 'solidus,' described by Prof Stokes in the Preface to his

*Physical Papers*, has enabled me to introduce a great deal of matter not contained in the last Edition without increasing the bulk of the volume. It certainly will not have been employed in vain if it induce any student to go through the work himself, in order to use another notation, instead of only reading the book.I have in the body of the work given references for many of the mathematical papers which I have had occasion to use; but I find that I have omitted to attach the names of the authors to so many theorems that I think it best to supply the omission by giving here a list of articles which contain theorems due to authors not mentioned in the text.

...

The arrangement of lines of a cubic surface, called a double sixer. Art. 541, was found out by Schlafli.

I am afraid that there still remain many omissions, but I should like especially to acknowledge how much I owe to Dr Salmon for whatever knowledge I possess of many departments of the subject on which I have been engaged.

This is the proper place to express my thanks for the great assistance which I have received from Mr Chree, Mr Berry, and Mr Richmond, of King's College, who have kindly not only corrected the proof sheets, which is a very tedious business, but helped me materially by testing the correctness of a great many of the problems. I wish also to thank Mr Stearn, of King's College, for his help in the earlier portions of the work, and for his kind superintendence of the printing during my absence from Cambridge.

Cambridge

March 1886

**6. An Elementary Treatise on Curve Tracing**

by Percival Frost, M.A.,

Formerly Fellow of St John's College, Cambridge;

Mathematical Lecturer of King's College.

I do not much like the idea of writing a preface, but I feel myself obliged to say a few words on the publication of what I have called a treatise, the term being very likely a misnomer.

Although my subject is Curve-tracing and not Curves, I am aware that some complete branches of this art are not alluded to at all.

The student might expect, in a treatise upon this subject, to find methods of drawing Polar Curves, Rolling Curves, Loci of Equations in Trilinear Coordinates, and Intrinsic Equations; he might also expect to find interesting Geometrical Loci discussed; these, and many other things immediately connected with the tracing of curves, have been deliberately omitted, for reasons which I consider good.

A treatise, if I had ventured upon it, at all comparable in exhaustive qualities with the excellent one of Salmon on Curves of Higher Orders, would have demanded, on the part of the student, far more extensive reading than I suppose him to possess; such a treatise would have required an advanced knowledge of Differential and Integral Calculus, of Higher Algebraical processes which do not appear in elementary treatises on Algebra, and of the science of projections, to understand which involves a familiarity with Solid Geometry, beyond the standard to which I have supposed the student to have attained.

My readers must not be disappointed if they do not meet with an historical survey of the researches which have been made in old times on modes of generation and properties of particular curves, and in modern times on the singularities of curves; such a survey would have been irrelevant to the object which I have proposed to myself.

I acknowledge myself, nevertheless, indebted to many of those old mathematicians for ideas, and especially to Cramer, for many curves which I have employed in illustrating points on which I have been engaged.

In cutting off so many vital parts of a complete treatise, I have to shew that I do not fall to the ground by sawing on the wrong side the branch on which I am sitting; I shall therefore explain, in as few words as I can, the objects which I have had in view in my work as it stands.

In order to make any rapid progress, in after years, in all the difficult subjects to which mathematical analysis is applied, it is absolutely necessary that, by some means or other, a student should, as early as possible, make himself familiar with all the ordinary instruments of his trade, such as he handles when he studies Algebra, Trigonometry, and Algebraical Geometry; his tastes may carry him with greater impetus in one direction than another, but he should remember that it is necessary to be strong all round, and even against the grain he should use efforts to avoid having weak points.

He must practise himself while he is young and his mind flexible, in all sorts of analytical processes and geometrical artifices. The solution of a great number of equations may be looked upon as one of the best exercises of one sort of faculties, and familiarity with the Binomial theorem and cognate subjects as essential, such as approximation to roots, expansion of a variety of functions in Algebra and Trigonometry, reversion of series, &c, accurate numerical calculations not being avoided.

I have reason to think that this kind of preliminary preparation for the study of the higher branches of mathematics has been much neglected in later years, and I am fortified in this opinion by observations made by Examiners of the greatest experience, who complain both of a want of power of work and of a want of individuality in the manner in which particular problems are attacked; this they attribute to defective early training and the omission of that practice which I have described as necessary.

Whether this practice has been neglected principally in consequence of the temptation to push forward to certain physical subjects, which have been recommended for the use of schools, I cannot say; but I have no doubt that the feeling of dignity acquired by entering upon the field of the Physical Sciences has enticed many a student from a course which, if pursued, would have enabled him to do in a few weeks what it has taken him many months to puzzle over.

If there is time for a student not only to attend to the dry work of polishing, but also to make himself acquainted with a little Mechanics and Hydrostatics, he should by all means do so; but everyone who has examined in public schools knows how little time there is for the study of Mathematics, and how sensible the mathematical masters are of the insufficiency of this time. Looking, therefore, upon the total amount of energy as nearly constant, I should have no hesitation in reserving for some future time the study of the Physical Sciences, which will not eventually suffer; whereas, to attempt after a certain age to acquire ease in mathematical operations is like a grown man trying to learn the violin.

Having, then, a distinct feeling of the absolute necessity of developing skill and power - I will not add cunning - and, at the same time, being perfectly sensible in what dry places the poor spirit of a student has been condemned to wander in the performance of his duty, I have selected the subject of this work in order to relieve him in the dull work involved in his preparation for climbing heights, by taking him along a very pleasant path, on which he may exercise in an agreeable way all his mathematical limbs, and, if he keep his eyes open, may see a variety of things which it will be useful to have observed when his real work begins.

For the subject, which I have chosen with this object in view, presents so many faces, pointing in directions towards which the mind of the intended mathematician has to radiate, that it would be difficult to find another which, with a very limited extent of reading, combines, to the same extent, so many valuable hints of methods of calculations to be employed hereafter, with so much pleasure in its present use.

For example, the subject of Graphical Calculation is coming more into use every day, and is applied with success to many difficult problems in Statics, Engineering, and Crystallography; hints of this the student will find in the practical solution of divers equations and in the determination of the number of their real roots, which are obtained by graphical methods with great facility.

Again, the methods of successive approximations which are employed in Optics and Astronomy are illustrated in the process of finding asymptotes and approximations to the forms of curves at a finite distance.

The comparison of large and small quantities of different orders of magnitude contains the staple of many of the most important applications of Mathematical Analysis; the Lunar and Planetary Theories depending almost entirely upon such considerations of relative magnitude.

The habit of looking towards an infinite distance, and discussing what takes place there, will render less startling a multitude of conceptions having in them a tendency to produce a feeling of vagueness, such, for instance, as the treatment of the mechanical effect of a couple as synonymous with that of an infinitely small force acting at an infinitely great distance.

As an important point, I would mention the tentative character of the inverse problem in which the form of a curve being given, its equation is investigated; the kind of uncertainty which will remain on the mind on account of defective estimation of magnitudes; and the necessity of a selection of what may appear the best of many possible solutions; all this will prepare the student for the disappointment which, having perhaps a wrong notion of what is meant by calling mathematics an exact science, he will feel in the conflict of theories by which it is attempted to reconcile the results of experiment in such subjects as Heat, Light, Electricity, and Molecular action generally; for an instance of this I may refer to the battle of philosophers about the direction of vibration of the ether in Plane Polarization.

The very uncertainty which exists in these subjects, the necessary balancing of evidence, and the difficulty of making up the mind as to what is to be believed, place such subjects, in the opinion of one at least of our greatest philosophers, among the best for the training of the intellect.

Looked upon as a special preparation for a special subject, I hope that my treatise may be considered useful in having given clear ideas, when the student enters upon the systematic treatment of the properties of curves; especially since the classification of curves according to degrees, and the subdivision of curves of the same degree into species is now being taken in hand by some eminent mathematicians.

With regard to the rejection of methods supplied by the Differential Calculus, I may observe that since the equations whose loci are investigated are rational equations, and never rise to a high degree, little would have been gained by the employment of such methods, since the Binomial Theorem is sufficient for all my purposes, and as ready in its application; independently of the consideration that I suppose myself to be instructing a student whose reading has been confined to very narrow limits.

As to the last chapter on Inverse Methods I trust that it will be looked upon as only a sketch. I have no doubt that the subject of it is capable of considerable perfection, and I shall be glad to have commenced, however defectively, so instructive a study.

To save the student trouble I may observe that I have used, as sufficiently near approximations in estimating their values, $√3 = \large\frac{7}{4}\normalsize , √5 = \large\frac{9}{4}\normalsize , √7 = \large\frac{8}{3}\normalsize , √6 = \large\frac{5}{2}\normalsize , √10 = \large\frac{16}{5}\normalsize$; and, with a view to the graphical solution of equations, I should advise him to practice himself in drawing a good parabola and in tracing readily the hyperbola from the equation $xy = ax + by + c$ for a variety of values of $a, b, c.$

In concluding this preface, or apology, I desire to say that I have read, with much advantage, some notes on Newton's enunciation of Lines of the Third Order by C R M Talbot; and that I am indebted for some valuable hints to Mr Clifford; but, especially, I must acknowledge myself in the highest degree indebted to two gentlemen, Mr H G Seth Smith and Mr G L Rives, of Trinity College, for their extreme kindness in guarding me against errors. The nature of the subject renders it extremely difficult to avoid mistakes; and, although very great pains have been taken to give correct drawings of the large number of curves which have been discussed, I am aware that there is much that is open to censure, and principally in parts for which my two friends are not at all answerable, many portions having been written when it was not possible to send them the proof sheets in time for revision.

Cambridge

January 1872.

Formerly Fellow of St John's College, Cambridge;

Mathematical Lecturer of King's College.

**Preface.**I do not much like the idea of writing a preface, but I feel myself obliged to say a few words on the publication of what I have called a treatise, the term being very likely a misnomer.

Although my subject is Curve-tracing and not Curves, I am aware that some complete branches of this art are not alluded to at all.

The student might expect, in a treatise upon this subject, to find methods of drawing Polar Curves, Rolling Curves, Loci of Equations in Trilinear Coordinates, and Intrinsic Equations; he might also expect to find interesting Geometrical Loci discussed; these, and many other things immediately connected with the tracing of curves, have been deliberately omitted, for reasons which I consider good.

A treatise, if I had ventured upon it, at all comparable in exhaustive qualities with the excellent one of Salmon on Curves of Higher Orders, would have demanded, on the part of the student, far more extensive reading than I suppose him to possess; such a treatise would have required an advanced knowledge of Differential and Integral Calculus, of Higher Algebraical processes which do not appear in elementary treatises on Algebra, and of the science of projections, to understand which involves a familiarity with Solid Geometry, beyond the standard to which I have supposed the student to have attained.

My readers must not be disappointed if they do not meet with an historical survey of the researches which have been made in old times on modes of generation and properties of particular curves, and in modern times on the singularities of curves; such a survey would have been irrelevant to the object which I have proposed to myself.

I acknowledge myself, nevertheless, indebted to many of those old mathematicians for ideas, and especially to Cramer, for many curves which I have employed in illustrating points on which I have been engaged.

In cutting off so many vital parts of a complete treatise, I have to shew that I do not fall to the ground by sawing on the wrong side the branch on which I am sitting; I shall therefore explain, in as few words as I can, the objects which I have had in view in my work as it stands.

In order to make any rapid progress, in after years, in all the difficult subjects to which mathematical analysis is applied, it is absolutely necessary that, by some means or other, a student should, as early as possible, make himself familiar with all the ordinary instruments of his trade, such as he handles when he studies Algebra, Trigonometry, and Algebraical Geometry; his tastes may carry him with greater impetus in one direction than another, but he should remember that it is necessary to be strong all round, and even against the grain he should use efforts to avoid having weak points.

He must practise himself while he is young and his mind flexible, in all sorts of analytical processes and geometrical artifices. The solution of a great number of equations may be looked upon as one of the best exercises of one sort of faculties, and familiarity with the Binomial theorem and cognate subjects as essential, such as approximation to roots, expansion of a variety of functions in Algebra and Trigonometry, reversion of series, &c, accurate numerical calculations not being avoided.

I have reason to think that this kind of preliminary preparation for the study of the higher branches of mathematics has been much neglected in later years, and I am fortified in this opinion by observations made by Examiners of the greatest experience, who complain both of a want of power of work and of a want of individuality in the manner in which particular problems are attacked; this they attribute to defective early training and the omission of that practice which I have described as necessary.

Whether this practice has been neglected principally in consequence of the temptation to push forward to certain physical subjects, which have been recommended for the use of schools, I cannot say; but I have no doubt that the feeling of dignity acquired by entering upon the field of the Physical Sciences has enticed many a student from a course which, if pursued, would have enabled him to do in a few weeks what it has taken him many months to puzzle over.

If there is time for a student not only to attend to the dry work of polishing, but also to make himself acquainted with a little Mechanics and Hydrostatics, he should by all means do so; but everyone who has examined in public schools knows how little time there is for the study of Mathematics, and how sensible the mathematical masters are of the insufficiency of this time. Looking, therefore, upon the total amount of energy as nearly constant, I should have no hesitation in reserving for some future time the study of the Physical Sciences, which will not eventually suffer; whereas, to attempt after a certain age to acquire ease in mathematical operations is like a grown man trying to learn the violin.

Having, then, a distinct feeling of the absolute necessity of developing skill and power - I will not add cunning - and, at the same time, being perfectly sensible in what dry places the poor spirit of a student has been condemned to wander in the performance of his duty, I have selected the subject of this work in order to relieve him in the dull work involved in his preparation for climbing heights, by taking him along a very pleasant path, on which he may exercise in an agreeable way all his mathematical limbs, and, if he keep his eyes open, may see a variety of things which it will be useful to have observed when his real work begins.

For the subject, which I have chosen with this object in view, presents so many faces, pointing in directions towards which the mind of the intended mathematician has to radiate, that it would be difficult to find another which, with a very limited extent of reading, combines, to the same extent, so many valuable hints of methods of calculations to be employed hereafter, with so much pleasure in its present use.

For example, the subject of Graphical Calculation is coming more into use every day, and is applied with success to many difficult problems in Statics, Engineering, and Crystallography; hints of this the student will find in the practical solution of divers equations and in the determination of the number of their real roots, which are obtained by graphical methods with great facility.

Again, the methods of successive approximations which are employed in Optics and Astronomy are illustrated in the process of finding asymptotes and approximations to the forms of curves at a finite distance.

The comparison of large and small quantities of different orders of magnitude contains the staple of many of the most important applications of Mathematical Analysis; the Lunar and Planetary Theories depending almost entirely upon such considerations of relative magnitude.

The habit of looking towards an infinite distance, and discussing what takes place there, will render less startling a multitude of conceptions having in them a tendency to produce a feeling of vagueness, such, for instance, as the treatment of the mechanical effect of a couple as synonymous with that of an infinitely small force acting at an infinitely great distance.

As an important point, I would mention the tentative character of the inverse problem in which the form of a curve being given, its equation is investigated; the kind of uncertainty which will remain on the mind on account of defective estimation of magnitudes; and the necessity of a selection of what may appear the best of many possible solutions; all this will prepare the student for the disappointment which, having perhaps a wrong notion of what is meant by calling mathematics an exact science, he will feel in the conflict of theories by which it is attempted to reconcile the results of experiment in such subjects as Heat, Light, Electricity, and Molecular action generally; for an instance of this I may refer to the battle of philosophers about the direction of vibration of the ether in Plane Polarization.

The very uncertainty which exists in these subjects, the necessary balancing of evidence, and the difficulty of making up the mind as to what is to be believed, place such subjects, in the opinion of one at least of our greatest philosophers, among the best for the training of the intellect.

Looked upon as a special preparation for a special subject, I hope that my treatise may be considered useful in having given clear ideas, when the student enters upon the systematic treatment of the properties of curves; especially since the classification of curves according to degrees, and the subdivision of curves of the same degree into species is now being taken in hand by some eminent mathematicians.

With regard to the rejection of methods supplied by the Differential Calculus, I may observe that since the equations whose loci are investigated are rational equations, and never rise to a high degree, little would have been gained by the employment of such methods, since the Binomial Theorem is sufficient for all my purposes, and as ready in its application; independently of the consideration that I suppose myself to be instructing a student whose reading has been confined to very narrow limits.

As to the last chapter on Inverse Methods I trust that it will be looked upon as only a sketch. I have no doubt that the subject of it is capable of considerable perfection, and I shall be glad to have commenced, however defectively, so instructive a study.

To save the student trouble I may observe that I have used, as sufficiently near approximations in estimating their values, $√3 = \large\frac{7}{4}\normalsize , √5 = \large\frac{9}{4}\normalsize , √7 = \large\frac{8}{3}\normalsize , √6 = \large\frac{5}{2}\normalsize , √10 = \large\frac{16}{5}\normalsize$; and, with a view to the graphical solution of equations, I should advise him to practice himself in drawing a good parabola and in tracing readily the hyperbola from the equation $xy = ax + by + c$ for a variety of values of $a, b, c.$

In concluding this preface, or apology, I desire to say that I have read, with much advantage, some notes on Newton's enunciation of Lines of the Third Order by C R M Talbot; and that I am indebted for some valuable hints to Mr Clifford; but, especially, I must acknowledge myself in the highest degree indebted to two gentlemen, Mr H G Seth Smith and Mr G L Rives, of Trinity College, for their extreme kindness in guarding me against errors. The nature of the subject renders it extremely difficult to avoid mistakes; and, although very great pains have been taken to give correct drawings of the large number of curves which have been discussed, I am aware that there is much that is open to censure, and principally in parts for which my two friends are not at all answerable, many portions having been written when it was not possible to send them the proof sheets in time for revision.

Cambridge

January 1872.

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