MacTutor
JAMES BOOTH, LL.D., F.R.S., was born at Lava, County Leitrim, on August 25, 1806. He entered Trinity College, Dublin, in 1825 and was elected scholar in 1829. He graduated B.A. in 1832, M.A. in 1840, and LL.D. in 1842. In 1834 he received Bishop Berkeley's gold medal for Greek. He was a candidate for a fellowship and in 1835 was placed second among the unsuccessful candidates, in 1837 first, in 1838 fourth, in 1839 first, and in 1840 second, receiving a premium on each occasion and the Madden premium in 1837 and 1839, when he was first.
In 1840, the four candidates who received premiums were George Salmon, James Booth, William Roberts, and Michael Roberts, all well-known mathematicians. Dr. Salmon and Mr. William Roberts were elected Fellows in 1841, and Mr. Michael Roberts was elected in 1843, but Dr. Booth does not appear to have become a candidate again. Having thus narrowly failed to obtain a Fellowship, Dr. Booth left Ireland in 1840 and in the same year was appointed Principal of Bristol College, an office which he held until 1843. Among his colleagues at Bristol were Mr. F. W. Newman, who was professor of classics, and Dr. W. B. Carpenter, who was professor of natural philosophy and natural history. Dr. Booth was ordained deacon by the Bishop of Exeter in 1842 and priest in the same year by the Archbishop of Canterbury In 1843 he was appointed Vice-principal of the Collegiate Institution, Liverpool. He held this position until 1848, when he came to London, and was engaged in writing educational works and lecturing for the Society of Arts. In 1854 he was appointed to the sole charge of St. Anne's, Wandsworth, and in 1859 was presented to the vicarage of Stone, Buckinghamshire, by the Royal Astronomical Society. He held this position until his death, which took place on April 15, 1878. He was also chaplain to the Marquis of Lansdowne for twenty years. In 1864 he was appointed a Justice of the Peace for Buckinghamshire. He married Mary, daughter of Daniel Watney, Esq., of Wandsworth, who died in 1874. Dr. Booth leaves two sons and one daughter; and his sister Maria, who always lived with him, survives him.
Dr. Booth's writings were chiefly mathematical and educational. The former show him to have been an able and original mathematician, and the latter exercised considerable influence on the promotion of education among the middle and industrial classes. His earliest printed paper appeared in the Philosophical Magazine for 1840, under the title "On the Focal Properties of Surfaces of the Second Order"; but before this he must have devoted considerable time to his method of tangential coordinates, by which his name is best known to mathematicians. This method was published in 1843, in an octavo tract of 32 pages, entitled "On the Application of a New Analytic Method to the Theory of Curves and Curved Surfaces," the pages being headed "On Tangential Coordinates." The preface is dated March 25, 1840, but the date on the title page is 1843, and the author is described as "Professor of Mathematics in the Collegiate Institution, Liverpool." The method is not the same as that generally known by the name of tangential coordinates, viz. in which the position of a line is fixed by the perpendiculars let fall upon it from three fixed points, or, more conveniently, by the ratios of these perpendiculars, for in Dr. Booth's system the coordinates of a line are the reciprocals of the intercepts on two fixed axes. Thus, the ordinary equation of a straight line in Cartesian coordinates is : if now we fix the point (a, y) and let a, b vary, this point may be regarded as determined by the same equation. Putting and the equation becomes ; and ξ, ν being the variables, this is the tangential equation of the point (x, y) in Dr. Booth's form; while, if a, y are variables, the equation denotes the straight line whose intercepts on the axes are and . It is this duality of interpretation which is of capital importance; of more importance, indeed, than the invention of the new system of coordinates.
Professor Cayley, in the first chapter of the second edition (1873) of Salmon's Higher Plane Curves, writes in reference to line coordinates in general: "There is little occasion for any explicit use of line coordinates: but the theory is very important; it serves in fact to show that in demonstrating by point coordinates any descriptive theorem whatever, we demonstrate the correlative theorem deducible from it by the theory of reciprocal polars (or that of geometrical duality): viz., we do not demonstrate the first theorem and deduce from it the other, but we do at one and the same time demonstrate the two theorems: our (2, y, z) instead of meaning point coordinates may mean line coordinates, and the demonstration is in every step thereof a demonstration of the correlative theorem."
The tangential coordinates invented by Dr. Booth have sometimes been referred to in England as Boothian coordinates; but in fact they had been introduced by Plücker as early as 1830, in a paper, "Ueber eine neue Art, in der analytischen Geometrie Puncte und Curven durch Gleichen dargestellt" (Crelle's Journal, vol. vi., pp. 107-146). Plücker considers a straight line, where A, B, C are connected by an equation , a, b, c being constants. The straight line then always passes through the point given by . Treating A, B, C as variables, and denoting them by u, v, w, he regards the equation as representing the point .
If the equation is written in the form , then this becomes identical with Dr. Booth's system on taking I. There is no question, therefore, that the Boothian tangential coordinates are really due to Plücker. Dr. Booth's discovery was a perfectly independent one; and his system differs from Plücker's in one detail, viz., the equations are not homogeneous. Dr. Booth seems to have been led to his system of coordinates by the "anomalous fact in the application of algebraic analysis to geometrical investigations, that while the locus of a point could be found from the simplest and most elementary considerations, the envelope of a right line or plane could be determined only by the aid of principles, artificial and obscure, derived from a higher department of analysis" (Tangential Coordinates, 1843, p. 1).
The Plückerian or Boothian coordinates may be said to stand in the same relation to three-point tangential coordinates (in which perpendiculars are let fall from three fixed points) as Cartesian coordinates stand to trilinear; for in the latter case, one side of the triangle of reference and in the former case, two of the points of reference move off to infinity. The connection is exhibited in the first chapter of Salmon's Higher Plane Curves (first edition, pp. 13, 14, and second edition, p. 9). Dr. Booth's original tract contained only 32 pages, but he continued to develop the subject and in his Treatise on some New Geometrical Methods (1873) more than 200 pages are devoted to this method of tangential coordinates, a good many examples being included. A number of questions have in recent years appeared in the Educational Times which have been worked out by this system of coordinates, and some of these are reprinted at the end of the second volume of New Geometrical Methods (1877), pp. 424-440. It has been thought desirable to enter thus fully into the subject of tangential coordinates because it is in connection with this field of research that Dr. Booth's name is most familiar to mathematicians, and also because, although he was anticipated by Plücker, the independent discovery by Dr. Booth of the analytical equivalent to the geometrical theory of reciprocal polars is not without historical interest.
Next to tangential coordinates and reciprocal polars, the subject on which Dr. Booth's writings are most considerable and important is that of elliptic integrals, especially in connection with the motion of a rigid body around a fixed point. His memoirs "On the Geometrical Properties of Elliptic Integrals" were published in the Philosophical Transactions for 1852 and 1854, and his papers on the motion of a rigid body around a fixed point appeared in the Philosophical Magazine and other journals. In these researches, the author confines himself to the elliptic integrals of Legendre and does not employ the elliptic functions—inverse to the integrals—introduced by Abel and Jacobi; and while he was thus perhaps hindered from carrying out his investigations as far as he might otherwise have done, the same reason has probably prevented his writings from being studied much. But however this may be, Dr. Booth's papers certainly represent a great amount of very valuable and original research.
In relation to this subject, his memoir "On the Trigonometry of the Parabola and the Geometrical Origin of Logarithms," which was printed in extenso in the British Association Report for 1856, should also be mentioned. The memoir relates to elliptic integrals in the case in which the modulus is unity. In elliptic functions, if the modulus k is set equal to zero, the theory becomes identical with circular trigonometry, and if k is set equal to unity, the theory becomes that of parabolic trigonometry. The length of an arc of a parabola, of latus rectum 4m, measured from the vertex, is ; and, in the case of k = 1, the first elliptic integral is
The theory thus belongs to the particular case 1 of elliptic functions, studied by Gudermann, and for which the amplitude becomes the function termed by Professor Cayley the Gudermannian, and written by him (Cayley's Elliptic Functions, p. 56). Dr. Booth's trigonometry of the parabola is thus a geometric illustration of the theory of the gudermannian.
Dr. Booth republished the whole of his mathematical writings, with considerable additions, in two volumes, entitled A Treatise on Some New Geometrical Methods. This work is dedicated to the President and Council of the Royal Astronomical Society. The first volume, which relates chiefly to tangent coordinates and reciprocal polars, was published in 1873; the second, which appeared in 1877, contains the papers on elliptic integrals, and there is also a treatise on the conic sections, each property being obtained directly from the cone, which is printed here for the first time, although the substance of it was communicated to the Royal Irish Academy as long ago as 1837. In the introduction to this work the author writes, "It has been to me a heavy drawback and deep discouragement that I have had no fellow workers to share in these researches. Neither have I entered into the labours of any. Without sympathy and without help I have worked upon those monographs presented to the public." The English mathematician who devotes himself to one or two special subjects can scarcely expect to have many fellow workers among his countrymen, but the theories to which Dr. Booth's writings have been enormously developed on the Continent in the last fifty years, and the comparative neglect to which he alludes may be traced to the fact that it is these foreign researches which, being expressed in more modern forms, have been generally referred to. For example, the investigations of Jacobi and those who have followed him on the motion of a rigid body round a fixed point, in which the theta functions are employed, are naturally better known than that of Dr. Booth, in which only the integrals appear.
In 1846, when at Liverpool, Dr. Booth published a tract of 108 pages on Education and Educational Institutions considered with reference to the Industrial Professions and the Present Aspect of Society, and also, in the same year, an Address to the Literary and Philosophical Society of Liverpool, of which he was Presi-dent. He also published in 1847 Examination the Province of the State; or, the Outlines of a Practical System for the Extension of National Education (pp. 73), which attracted a good deal of attention. Dr. Booth was treasurer and chairman of the Council of the Society of Arts from 1855 to 1857, and during this period he delivered many lectures relating to education, some of which were printed as pamphlets. Several of his addresses were published by the Society of Arts under the titles Systematic Instruction and Periodical Examination (1857) and How to Learn and What to Learn (1857), and the latter went through several editions very rapidly. Dr. Booth also edited, and wrote the introduction to, the volume of the Speeches and Addresses of His Royal Highness The Prince Albert which was published by the Society of Arts in 1857. By his lectures and writings, as well as by his influence as Chairman of the Council, Dr. Booth may almost be said to have founded the Society of Arts' system of examinations. It is true that examinations had been held in 1854 and 1855, but very few candidates had presented themselves. In 1856, the first year in which Dr. Booth's plan was adopted, there were 52 candidates, the examination being held only in London, and in 1857, when the examinations were held at London and Huddersfield, there were above 200 candidates. Dr. Booth retired from the Council of the Society in 1857 in consequence of a difference of opinion between himself and the Council, the main point in dispute being as to whether the examinations should be partly oral, as Dr. Booth desired, or should be conducted wholly by printed papers, as pro-posed by Mr. Harry Chester, of the Privy Council Office, and adopted by the Council. It should be remembered that at this time the Oxford and Cambridge local examinations had not been founded, and the only public examination open to the middle and industrial classes was that of the College of Preceptors.
As a preacher Dr. Booth was eloquent, and when minister of St. Anne's, Wandsworth, attracted large congregations. When in 1859 the vicarage of Stone, the advowson of which had been given to the Royal Astronomical Society by Dr. Lee in 1844, became vacant, it was felt that Dr. Booth's services to science and to education gave him a strong claim to the appointment, and although not then a Fellow of the Society, he was (with the full approbation of Dr. Lee, who was a member of Council at the time) presented to the living, in which he succeeded the late Rev. Joseph Bancroft Reade, F.R.S., well known in connexion with the early history of photography, microscopy, &c. Among Dr. Booth's theological writings may be mentioned The Bible and its Interpreters (1861), A Sermon on the Death of Admiral W. H. Smyth, D.C.L., F.R.S. (1865), and The Lord's Supper, a Feast after Sacrifice (1870). Dr. Booth was elected a Fellow of the Royal Society on January 22, 1846, and of the Royal Astronomical Society on June 10, 1859.
J. W. L. G.
In 1840, the four candidates who received premiums were George Salmon, James Booth, William Roberts, and Michael Roberts, all well-known mathematicians. Dr. Salmon and Mr. William Roberts were elected Fellows in 1841, and Mr. Michael Roberts was elected in 1843, but Dr. Booth does not appear to have become a candidate again. Having thus narrowly failed to obtain a Fellowship, Dr. Booth left Ireland in 1840 and in the same year was appointed Principal of Bristol College, an office which he held until 1843. Among his colleagues at Bristol were Mr. F. W. Newman, who was professor of classics, and Dr. W. B. Carpenter, who was professor of natural philosophy and natural history. Dr. Booth was ordained deacon by the Bishop of Exeter in 1842 and priest in the same year by the Archbishop of Canterbury In 1843 he was appointed Vice-principal of the Collegiate Institution, Liverpool. He held this position until 1848, when he came to London, and was engaged in writing educational works and lecturing for the Society of Arts. In 1854 he was appointed to the sole charge of St. Anne's, Wandsworth, and in 1859 was presented to the vicarage of Stone, Buckinghamshire, by the Royal Astronomical Society. He held this position until his death, which took place on April 15, 1878. He was also chaplain to the Marquis of Lansdowne for twenty years. In 1864 he was appointed a Justice of the Peace for Buckinghamshire. He married Mary, daughter of Daniel Watney, Esq., of Wandsworth, who died in 1874. Dr. Booth leaves two sons and one daughter; and his sister Maria, who always lived with him, survives him.
Dr. Booth's writings were chiefly mathematical and educational. The former show him to have been an able and original mathematician, and the latter exercised considerable influence on the promotion of education among the middle and industrial classes. His earliest printed paper appeared in the Philosophical Magazine for 1840, under the title "On the Focal Properties of Surfaces of the Second Order"; but before this he must have devoted considerable time to his method of tangential coordinates, by which his name is best known to mathematicians. This method was published in 1843, in an octavo tract of 32 pages, entitled "On the Application of a New Analytic Method to the Theory of Curves and Curved Surfaces," the pages being headed "On Tangential Coordinates." The preface is dated March 25, 1840, but the date on the title page is 1843, and the author is described as "Professor of Mathematics in the Collegiate Institution, Liverpool." The method is not the same as that generally known by the name of tangential coordinates, viz. in which the position of a line is fixed by the perpendiculars let fall upon it from three fixed points, or, more conveniently, by the ratios of these perpendiculars, for in Dr. Booth's system the coordinates of a line are the reciprocals of the intercepts on two fixed axes. Thus, the ordinary equation of a straight line in Cartesian coordinates is : if now we fix the point (a, y) and let a, b vary, this point may be regarded as determined by the same equation. Putting and the equation becomes ; and ξ, ν being the variables, this is the tangential equation of the point (x, y) in Dr. Booth's form; while, if a, y are variables, the equation denotes the straight line whose intercepts on the axes are and . It is this duality of interpretation which is of capital importance; of more importance, indeed, than the invention of the new system of coordinates.
Professor Cayley, in the first chapter of the second edition (1873) of Salmon's Higher Plane Curves, writes in reference to line coordinates in general: "There is little occasion for any explicit use of line coordinates: but the theory is very important; it serves in fact to show that in demonstrating by point coordinates any descriptive theorem whatever, we demonstrate the correlative theorem deducible from it by the theory of reciprocal polars (or that of geometrical duality): viz., we do not demonstrate the first theorem and deduce from it the other, but we do at one and the same time demonstrate the two theorems: our (2, y, z) instead of meaning point coordinates may mean line coordinates, and the demonstration is in every step thereof a demonstration of the correlative theorem."
The tangential coordinates invented by Dr. Booth have sometimes been referred to in England as Boothian coordinates; but in fact they had been introduced by Plücker as early as 1830, in a paper, "Ueber eine neue Art, in der analytischen Geometrie Puncte und Curven durch Gleichen dargestellt" (Crelle's Journal, vol. vi., pp. 107-146). Plücker considers a straight line, where A, B, C are connected by an equation , a, b, c being constants. The straight line then always passes through the point given by . Treating A, B, C as variables, and denoting them by u, v, w, he regards the equation as representing the point .
If the equation is written in the form , then this becomes identical with Dr. Booth's system on taking I. There is no question, therefore, that the Boothian tangential coordinates are really due to Plücker. Dr. Booth's discovery was a perfectly independent one; and his system differs from Plücker's in one detail, viz., the equations are not homogeneous. Dr. Booth seems to have been led to his system of coordinates by the "anomalous fact in the application of algebraic analysis to geometrical investigations, that while the locus of a point could be found from the simplest and most elementary considerations, the envelope of a right line or plane could be determined only by the aid of principles, artificial and obscure, derived from a higher department of analysis" (Tangential Coordinates, 1843, p. 1).
The Plückerian or Boothian coordinates may be said to stand in the same relation to three-point tangential coordinates (in which perpendiculars are let fall from three fixed points) as Cartesian coordinates stand to trilinear; for in the latter case, one side of the triangle of reference and in the former case, two of the points of reference move off to infinity. The connection is exhibited in the first chapter of Salmon's Higher Plane Curves (first edition, pp. 13, 14, and second edition, p. 9). Dr. Booth's original tract contained only 32 pages, but he continued to develop the subject and in his Treatise on some New Geometrical Methods (1873) more than 200 pages are devoted to this method of tangential coordinates, a good many examples being included. A number of questions have in recent years appeared in the Educational Times which have been worked out by this system of coordinates, and some of these are reprinted at the end of the second volume of New Geometrical Methods (1877), pp. 424-440. It has been thought desirable to enter thus fully into the subject of tangential coordinates because it is in connection with this field of research that Dr. Booth's name is most familiar to mathematicians, and also because, although he was anticipated by Plücker, the independent discovery by Dr. Booth of the analytical equivalent to the geometrical theory of reciprocal polars is not without historical interest.
Next to tangential coordinates and reciprocal polars, the subject on which Dr. Booth's writings are most considerable and important is that of elliptic integrals, especially in connection with the motion of a rigid body around a fixed point. His memoirs "On the Geometrical Properties of Elliptic Integrals" were published in the Philosophical Transactions for 1852 and 1854, and his papers on the motion of a rigid body around a fixed point appeared in the Philosophical Magazine and other journals. In these researches, the author confines himself to the elliptic integrals of Legendre and does not employ the elliptic functions—inverse to the integrals—introduced by Abel and Jacobi; and while he was thus perhaps hindered from carrying out his investigations as far as he might otherwise have done, the same reason has probably prevented his writings from being studied much. But however this may be, Dr. Booth's papers certainly represent a great amount of very valuable and original research.
In relation to this subject, his memoir "On the Trigonometry of the Parabola and the Geometrical Origin of Logarithms," which was printed in extenso in the British Association Report for 1856, should also be mentioned. The memoir relates to elliptic integrals in the case in which the modulus is unity. In elliptic functions, if the modulus k is set equal to zero, the theory becomes identical with circular trigonometry, and if k is set equal to unity, the theory becomes that of parabolic trigonometry. The length of an arc of a parabola, of latus rectum 4m, measured from the vertex, is ; and, in the case of k = 1, the first elliptic integral is
Dr. Booth's notation is peculiar: he introduces the symbols and (which have some resemblance in their mode of application to + and – ), defined by the equations
,
,
In the case of we have in Elliptic Functions
,
;
so that if , then viz.
Dr. Booth republished the whole of his mathematical writings, with considerable additions, in two volumes, entitled A Treatise on Some New Geometrical Methods. This work is dedicated to the President and Council of the Royal Astronomical Society. The first volume, which relates chiefly to tangent coordinates and reciprocal polars, was published in 1873; the second, which appeared in 1877, contains the papers on elliptic integrals, and there is also a treatise on the conic sections, each property being obtained directly from the cone, which is printed here for the first time, although the substance of it was communicated to the Royal Irish Academy as long ago as 1837. In the introduction to this work the author writes, "It has been to me a heavy drawback and deep discouragement that I have had no fellow workers to share in these researches. Neither have I entered into the labours of any. Without sympathy and without help I have worked upon those monographs presented to the public." The English mathematician who devotes himself to one or two special subjects can scarcely expect to have many fellow workers among his countrymen, but the theories to which Dr. Booth's writings have been enormously developed on the Continent in the last fifty years, and the comparative neglect to which he alludes may be traced to the fact that it is these foreign researches which, being expressed in more modern forms, have been generally referred to. For example, the investigations of Jacobi and those who have followed him on the motion of a rigid body round a fixed point, in which the theta functions are employed, are naturally better known than that of Dr. Booth, in which only the integrals appear.
In 1846, when at Liverpool, Dr. Booth published a tract of 108 pages on Education and Educational Institutions considered with reference to the Industrial Professions and the Present Aspect of Society, and also, in the same year, an Address to the Literary and Philosophical Society of Liverpool, of which he was Presi-dent. He also published in 1847 Examination the Province of the State; or, the Outlines of a Practical System for the Extension of National Education (pp. 73), which attracted a good deal of attention. Dr. Booth was treasurer and chairman of the Council of the Society of Arts from 1855 to 1857, and during this period he delivered many lectures relating to education, some of which were printed as pamphlets. Several of his addresses were published by the Society of Arts under the titles Systematic Instruction and Periodical Examination (1857) and How to Learn and What to Learn (1857), and the latter went through several editions very rapidly. Dr. Booth also edited, and wrote the introduction to, the volume of the Speeches and Addresses of His Royal Highness The Prince Albert which was published by the Society of Arts in 1857. By his lectures and writings, as well as by his influence as Chairman of the Council, Dr. Booth may almost be said to have founded the Society of Arts' system of examinations. It is true that examinations had been held in 1854 and 1855, but very few candidates had presented themselves. In 1856, the first year in which Dr. Booth's plan was adopted, there were 52 candidates, the examination being held only in London, and in 1857, when the examinations were held at London and Huddersfield, there were above 200 candidates. Dr. Booth retired from the Council of the Society in 1857 in consequence of a difference of opinion between himself and the Council, the main point in dispute being as to whether the examinations should be partly oral, as Dr. Booth desired, or should be conducted wholly by printed papers, as pro-posed by Mr. Harry Chester, of the Privy Council Office, and adopted by the Council. It should be remembered that at this time the Oxford and Cambridge local examinations had not been founded, and the only public examination open to the middle and industrial classes was that of the College of Preceptors.
As a preacher Dr. Booth was eloquent, and when minister of St. Anne's, Wandsworth, attracted large congregations. When in 1859 the vicarage of Stone, the advowson of which had been given to the Royal Astronomical Society by Dr. Lee in 1844, became vacant, it was felt that Dr. Booth's services to science and to education gave him a strong claim to the appointment, and although not then a Fellow of the Society, he was (with the full approbation of Dr. Lee, who was a member of Council at the time) presented to the living, in which he succeeded the late Rev. Joseph Bancroft Reade, F.R.S., well known in connexion with the early history of photography, microscopy, &c. Among Dr. Booth's theological writings may be mentioned The Bible and its Interpreters (1861), A Sermon on the Death of Admiral W. H. Smyth, D.C.L., F.R.S. (1865), and The Lord's Supper, a Feast after Sacrifice (1870). Dr. Booth was elected a Fellow of the Royal Society on January 22, 1846, and of the Royal Astronomical Society on June 10, 1859.
J. W. L. G.
James Booth's obituary appeared in Journal of the Royal Astronomical Society 39:4 (1879), 219-225.