## Extracts from W W Rouse Ball's papers

Below we present short extracts from six of Walter William Rouse Ball's papers. They are all published in

*The Mathematical Gazette*between 1910 and 1915. Although research into the history of mathematics has moved on a great deal in the 100 years since these papers were written, they are still well worth reading and we encourage those reading the following extracts to consult the original papers.**1. A School Course in Mathematics in the XVIIth Century, by W W Rouse Ball.**

*The Mathematical Gazette*

**5**(84) (1910), 202-205.

We have ... some information about the mathematical teaching at one English public school - Christ's Hospital, London - in the seventeenth century, which may be worth giving in detail, though it should be stated at once that the circumstances there were peculiar, and it is not typical of the work at a normal public school at that time. At that school there were special provisions, on the foundation of Charles II., for teaching mathematics and the elements of navigation to boys who intended to enter the royal navy. The King, after establishing this department, directed Sir Jonas Moore to compile a textbook for the boys. Moore died before he had finished it, but it was completed by his sons-in-law, W Hanway and J Potenger, with the aid of Mr Perkins, then the chief mathematical master in the school, Flamstead, and Halley. The whole was published in 1681, and contains sections on Arithmetic, Algebra, Practical Geometry, together with the substance of Euclid's Elements, i-vi, xi, xii, Plane and Spherical Trigonometry, Cosmography, Navigation, the Sphere, Astronomical and Logarithmic Tables, and Geography. The work had a considerable reputation, and is excellent of its kind, but we may infer from the correspondence described below that, in fact, it was not used in the school. This may have been due to the death of Mr Perkins in 1680 or 1681. A few years later the question of the school course in mathematics, admittedly insufficient, was raised, and it was determined at the same time, taking advantage of the opportunity, to connect or combine with the King's foundation another one in the school, due to the liberality of a Mr Stone. The new scheme, with a statement explaining what changes it introduced, was sent to Isaac Newton, and his criticisms thereon invited. A draft of his reply and some of his memoranda on the subject are preserved among the Portsmouth papers. The syllabus previously in use has not been preserved, but from the extant papers it would seem that at least it required the boys to study the earlier parts of the first book of Euclid's Elements; the 10th, 11th, and 12th propositions of the sixth book; and to learn arithmetic. The course lasted two years. Perhaps this represented all the theoretical mathematics then normally prescribed to the King's Scholars, for Newton notes the following omissions. There was no "symbolic arithmetic," i.e. no use was made of algebraical symbols and methods. There was no "taking of heights and distances and measuring of planes and solids." There was nothing "of spherical trigonometry, though this was requisite for many problems in astronomy, geography, and navigation." Nothing was taught of "Mercator's chart" or "of computing the way of a ship," or "of longitude, amplitude azimuths, and variations of the compass." And, lastly, there was no "word of reasoning about force and motion, though it be the very life and soul of mechanical skill and manual operations." He continues, "by these defects it's plain that the old scheme wants not only methodizing, but also an enlargement of the learning." Of course the specific criticisms were made with reference to the fact that the boys looked to a career in connection with the sea. ...

Of the final scheme there are various drafts, differing in minute details. I think we may take the following as being substantially that ultimately approved for the abler boys. The subjects of study were to be Arithmetic; Algebra; Plane and Solid Geometry, practical with rule and compass as well as theoretical; Plane Trigonometry; Drawing and Designing; Instruments and their use; Cosmography, including therein the rudiments of astronomy and the art of making maps and charts; the use of Spherical Triangles; and Mechanics. Also applications of the above to sea problems, laying down courses, and determining positions. Some schedules of the extent to which the subjects should be read are given, but perhaps the above description will suffice.

Of the final scheme there are various drafts, differing in minute details. I think we may take the following as being substantially that ultimately approved for the abler boys. The subjects of study were to be Arithmetic; Algebra; Plane and Solid Geometry, practical with rule and compass as well as theoretical; Plane Trigonometry; Drawing and Designing; Instruments and their use; Cosmography, including therein the rudiments of astronomy and the art of making maps and charts; the use of Spherical Triangles; and Mechanics. Also applications of the above to sea problems, laying down courses, and determining positions. Some schedules of the extent to which the subjects should be read are given, but perhaps the above description will suffice.

**2. Four Fours. Some Arithmetical Puzzles, by W W Rouse Ball.**

*The Mathematical Gazette*

**6**(98) (1912), 289-290.

An arithmetical amusement, said to have been first propounded in 1881, is the expression in the ordinary arithmetic and algebraic notation of the consecutive integers from 1 upwards, as far as practicable, by the use of four "4"s. I have mentioned the problem in my Mathematical Recreations, but my friend Mr Oscar Eckenstein has now carried the solutions considerably further. I think a bare statement of our procedure may be of interest, and perhaps some readers of the Gazette may amuse themselves by filling in the details of the analysis or applying similar methods to higher numbers. The solutions will vary according to what we mean by ordinary arithmetic and algebraic notation. Here I will assume that we allow the use of brackets and the symbols for square roots, decimals (simple and repeating), factorials, and subfactorials [this is

The following numbers, forming what I will call the series

*n*!(1 -^{1}/

_{1!}+

^{1}/

_{2!}-

^{1}/

_{3!}+ ...

^{1}/

_{n!})], as well as those for addition, subtraction, multiplication, and division, and that we exclude indices (other than first powers) not expressible by a "4" or "4"s, and roots (other than square roots used a finite number of times). On this assumption we can express by four "4"s every number up to 873.

The following numbers, forming what I will call the series

*a*, are expressible by one "4": 1, 2, 3, 4, 6, 9, 24, 265, 720, ..., and we may conveniently use them in this form instead of writing them in the more cumbrous "4" notation. Moreover, since 1 is expressible by one "4," if we can express a number by less than four "4"s we can, by multiplying by 1 a sufficient number of times, express it by four "4"s. From this series it follows that if

*m*and

*n*are two numbers such that

*n*-

*m*is less than 10, then every number between

*m*and

*n*is expressible by m or n and one "4." The numbers 1 to 13, 15 to 18, 20 to 28, 30, 33, 36, forming the series b, are expressible by two "4"s. Hence, if

*m*and

*n*are two numbers such that

*n*-

*m*is less than 41 (and is not equal to 33), every number between

*m*and

*n*is expressible by

*m*or

*n*and two "4"s. If

*n*-

*m*is equal to 33, such expressions for the numbers

*m*+ 14 and

*m*+ 19, are not obtainable in this way. ...

**3. The Cambridge School of Mathematics, by W W Rouse Ball.**

*The Mathematical Gazette*

**6**(99) (1912), 311-323.

Cambridge is honoured this year by its selection for the first meeting held in England by the International Congress of Mathematicians. The Cambridge School has played a notable part in the development of mathematics in Britain, and it may be of interest if I briefly summarize the leading facts of its history, which is indeed closely connected with that of the University. The School is of more than respectable antiquity and inevitably it has been sometimes flourishing and sometimes the reverse. Here I desire rather to bring out the salient features in its history than to discuss details or individual achievements.

The University is among the oldest in Europe, having been founded about the close of the twelfth century. Its medieval curriculum and studies were much the same as those of other Universities of the time, and there is no need to describe them in this paper.

The modern development of the University begins with the Renaissance, which was warmly welcomed in Cambridge and was accompanied by a distinct development of mathematical teaching. I date the origin of its Mathematical School from this movement, and the first chapter of the history of the School may be said to cover the sixteenth and the early years of the seventeenth century. It is true that, during the greater part of this period, no notable advance in the theory of the subject was made at Cambridge, or indeed in Britain - the first important British discovery in mathematics being that of logarithms, published by Napier in 1614 - but it is worthy of remark that all the leading English mathematicians of the sixteenth century were educated at the University, and this fact even though their principal work was done elsewhere may justify my treating Cambridge as being then an important centre of mathematical teaching. ...

At the Restoration there was a general rearrangement of things academical as well as political. Just at that time, in 1663, a professorship in mathematics was founded at Cambridge, and this promoted a revival of interest in the subject. Isaac Barrow was the first occupant of the chair. His lectures - on the principles of the subject, geometrical optics, and properties of curves - are extant, but to his disappointment the attendance at them was small. He was however fortunate in having among his pupils Isaac Newton, in whose favour, in 1669, he resigned the chair, thus securing to Newton, when still under twenty-seven, the opportunity to prosecute and promulgate his discoveries. ...

During the century following the death of Newton, the work produced at Cambridge was unimportant. There were already two professorships in mathematics: additional chairs were founded, one in 1749 by Thomas Lowndes in Astronomy and Geometry, and another in 1783 by Richard Jackson in Natural and Experimental Philosophy and Chemistry. Most of the professors were however undistinguished, and indeed but little interested in the subject. Then, and well into the following century, a mathematical chair was often regarded only as a prize or a means of securing leisure, and at best, merely as offering a position where a man could pursue his own researches undisturbed by other duties. Notwithstanding this, Cambridge remained the centre of mathematical studies in Britain. Teaching in the subject was excellently organized, and the number of students in it steadily increased. This was due partly to the immense influence exerted by the Colleges and by "Pupil-mongers," but in the latter half of the century is mainly attributable to the development of a rigorous system of examination in mathematics which for a long time formed the chief avenue to University distinctions. ...

The prominent features of the history of the Cambridge School of Mathematics during the nineteenth century are the further development of the system of coaching and the confinement of the subjects studied to those scheduled in the Tripos regulations, accompanied by a striking revival of interest in the subject, and the appearance of a remarkable group of mathematical physicists....

**The Medieval Period.**The University is among the oldest in Europe, having been founded about the close of the twelfth century. Its medieval curriculum and studies were much the same as those of other Universities of the time, and there is no need to describe them in this paper.

**The Renaissance Period.**The modern development of the University begins with the Renaissance, which was warmly welcomed in Cambridge and was accompanied by a distinct development of mathematical teaching. I date the origin of its Mathematical School from this movement, and the first chapter of the history of the School may be said to cover the sixteenth and the early years of the seventeenth century. It is true that, during the greater part of this period, no notable advance in the theory of the subject was made at Cambridge, or indeed in Britain - the first important British discovery in mathematics being that of logarithms, published by Napier in 1614 - but it is worthy of remark that all the leading English mathematicians of the sixteenth century were educated at the University, and this fact even though their principal work was done elsewhere may justify my treating Cambridge as being then an important centre of mathematical teaching. ...

**The Newtonian Period.**At the Restoration there was a general rearrangement of things academical as well as political. Just at that time, in 1663, a professorship in mathematics was founded at Cambridge, and this promoted a revival of interest in the subject. Isaac Barrow was the first occupant of the chair. His lectures - on the principles of the subject, geometrical optics, and properties of curves - are extant, but to his disappointment the attendance at them was small. He was however fortunate in having among his pupils Isaac Newton, in whose favour, in 1669, he resigned the chair, thus securing to Newton, when still under twenty-seven, the opportunity to prosecute and promulgate his discoveries. ...

**The Eighteenth Century.**During the century following the death of Newton, the work produced at Cambridge was unimportant. There were already two professorships in mathematics: additional chairs were founded, one in 1749 by Thomas Lowndes in Astronomy and Geometry, and another in 1783 by Richard Jackson in Natural and Experimental Philosophy and Chemistry. Most of the professors were however undistinguished, and indeed but little interested in the subject. Then, and well into the following century, a mathematical chair was often regarded only as a prize or a means of securing leisure, and at best, merely as offering a position where a man could pursue his own researches undisturbed by other duties. Notwithstanding this, Cambridge remained the centre of mathematical studies in Britain. Teaching in the subject was excellently organized, and the number of students in it steadily increased. This was due partly to the immense influence exerted by the Colleges and by "Pupil-mongers," but in the latter half of the century is mainly attributable to the development of a rigorous system of examination in mathematics which for a long time formed the chief avenue to University distinctions. ...

**The Nineteenth Century.**The prominent features of the history of the Cambridge School of Mathematics during the nineteenth century are the further development of the system of coaching and the confinement of the subjects studied to those scheduled in the Tripos regulations, accompanied by a striking revival of interest in the subject, and the appearance of a remarkable group of mathematical physicists....

**4. Newton, by W W Rouse Ball.**

*The Mathematical Gazette*

**7**(112) (1914), 349-360.

Isaac Newton's investigations will always rank among the chief achievements of our race. It is intended in this sketch to give an outline of his life and what he did. ...

Newton took his B.A. degree in the Lent Term, 1665 N.S. In that spring the plague appeared, and for a couple of years he lived mostly at home, though with occasional residence at Cambridge. Probably at this time his creative powers were at their highest. His use of fluxions may be traced back to May, 1665; his theory of gravitation originated in 1666; and the foundation of his optical discoveries would seem to be only a little later. In an unpublished memorandum made some years later (cancelled, but believed to be correct in the part here quoted), he thus described his work of this time: "In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions. And the same year I began to think of gravity extending to the orb of the Moon, and . .. from Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centres of their orbs I deduced that the forces which keep the Planets in their orbs must [be] reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematics and Philosophy more than at any time since." ...

Newton definitely rejected the wave theory [of light], but he never fully accepted the corpuscular theory which is commonly associated with his name. In fact, the assumption of the existence of material corpuscles, for which no other evidence exists, expelled in immense numbers by a source of light, was repugnant to his idea of legitimate scientific conjecture, though he regarded it as a possible scheme. As his views have been often misunderstood, I describe them at length. His opinion was that all space is permeated by an elastic ether capable of transmitting vibrations, that this ether pervades all bodies and is not necessarily uniform, that possibly electricity and gravitation may be due to it, though in what way we do not know, that it may be essential to the production of light, but that light cannot be due to its vibrations since light rays travel in straight lines. Light, he went on to say, is "something propagated by lucid bodies." It may be "an aggregate of peripatetic qualities," or it may arise from "multitudes of unimaginable small and swift corpuscles," or it may be "any other corporeal emanation or any impulse or motion of any other medium ... diffused through the main body of ether," or anything else which they that like not these views "can imagine proper for the purpose.... Let every man here take his fancy: only whatever light be, I suppose it consists of rays differing from one another in contingent circumstances, as bigness, form, or vigour." Of these vague hypotheses, that referring to corpuscles was the simplest: it was generally adopted by Newton's followers, and commonly attributed to him, though, in fact, his object seems to have been to present a theory free from speculation as to the mechanism that produced the phenomena.

Newton took his B.A. degree in the Lent Term, 1665 N.S. In that spring the plague appeared, and for a couple of years he lived mostly at home, though with occasional residence at Cambridge. Probably at this time his creative powers were at their highest. His use of fluxions may be traced back to May, 1665; his theory of gravitation originated in 1666; and the foundation of his optical discoveries would seem to be only a little later. In an unpublished memorandum made some years later (cancelled, but believed to be correct in the part here quoted), he thus described his work of this time: "In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions. And the same year I began to think of gravity extending to the orb of the Moon, and . .. from Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centres of their orbs I deduced that the forces which keep the Planets in their orbs must [be] reciprocally as the squares of their distances from the centres about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematics and Philosophy more than at any time since." ...

Newton definitely rejected the wave theory [of light], but he never fully accepted the corpuscular theory which is commonly associated with his name. In fact, the assumption of the existence of material corpuscles, for which no other evidence exists, expelled in immense numbers by a source of light, was repugnant to his idea of legitimate scientific conjecture, though he regarded it as a possible scheme. As his views have been often misunderstood, I describe them at length. His opinion was that all space is permeated by an elastic ether capable of transmitting vibrations, that this ether pervades all bodies and is not necessarily uniform, that possibly electricity and gravitation may be due to it, though in what way we do not know, that it may be essential to the production of light, but that light cannot be due to its vibrations since light rays travel in straight lines. Light, he went on to say, is "something propagated by lucid bodies." It may be "an aggregate of peripatetic qualities," or it may arise from "multitudes of unimaginable small and swift corpuscles," or it may be "any other corporeal emanation or any impulse or motion of any other medium ... diffused through the main body of ether," or anything else which they that like not these views "can imagine proper for the purpose.... Let every man here take his fancy: only whatever light be, I suppose it consists of rays differing from one another in contingent circumstances, as bigness, form, or vigour." Of these vague hypotheses, that referring to corpuscles was the simplest: it was generally adopted by Newton's followers, and commonly attributed to him, though, in fact, his object seems to have been to present a theory free from speculation as to the mechanism that produced the phenomena.

**5. Pythagoras, by W W Rouse Ball.**

*The Mathematical Gazette*

**8**(115) (1915), 5-12.

Primarily Pythagoras was a religious teacher and philosopher, inculcating a stern system of morality, greatly superior to any system then known to the Greeks. For reasons which I outline below he held that the ultimate explanation of things rested on a knowledge of numbers and form, that thus the study of philosophy must necessarily be associated with the study of pure and applied mathematics, and that these latter also provided the best general mental training and discipline. I am not here concerned with his ethical or philosophical opinions, of which indeed our knowledge is vague. Even on the teaching of his School in science we cannot speak with absolute assurance, while the further fact that his instruction was entirely oral introduces the possibility that commentators have failed to distinguish his conclusions from those made later by his followers. Thus the precise extent of his discoveries is still a matter of opinion, but I think we may reasonably attribute to him the results I proceed to describe. That he created mathematics as a science there is no doubt. We cannot say in what order Pythagoras made his discoveries, but I conjecture that he was led to the study of geometry and numbers through his researches in natural philosophy. Of these, his investigations on acoustics were the most important. Throughout his life, Pythagoras was profoundly moved by music, and in his School he taught that it was one of the chief means for exciting and calming emotions - it purged the soul, said his followers, as medicine purges the body. By a happy chance he constructed or came across an instrument consisting of a string stretched over a vibrating board with a movable bridge by which the string could be divided into different lengths. To his delight he found that other things being equal the note given by the vibrating string depended only on its length, and that the lengths that gave a note, its fourth, its fifth, and its octave were in the ratio 6, 8, 9, and 12. Thus suddenly the whole of an intangible and artistic world seemed reduced to a question of numbers. It was not unnatural that he should suspect that other physical phenomena were explicable by similar means. This suspicion was strengthened by his noticing that many facts - seasons, the months, the tides, day and night, sleep and waking, the pulse, breathing, etc. - are periodic. Hence he argued that whatever be their explanation, it must involve a consideration of numbers, and he is said to have wondered whether the regular rhythmic sequences of the world would not ultimately be found to be analogous to the regular breathing of animals.

**6. Augustus De Morgan, by W W Rouse Ball.**

*The Mathematical Gazette*

**8**(116) (1915), 42-45.

The biography of De Morgan by his widow is now more than thirty years old, but the recent republication of three essays by him on Newton, reviewed in the October number of the Gazette, serves to recall one of the most striking figures in the London mathematical world of some fifty years ago. Augustus De Morgan created no new branches of knowledge and discovered little of note, yet when the scientific history of England in the nineteenth century is written his name will occupy a prominent position, for he profoundly influenced the opinions of the ordinary man of science and mathematician of his time. ...

As illustrative of his ingenuity in applying the laws of probability to numerous problems, I will mention a test which he proposed for determining the authorship of books. He believed that if different books on similar subjects written by a particular author were examined it would be found that the average number of letters per word in each book would agree to (perhaps) one place of decimal. Hence, if the average number of letters per word in two books on the same subject differed by more than that percentage, it was probable that the books were by different authors. He thought this experiment might be well worth making in cases where authorship was in question, and in particular in the case of the Greek text of some of the books of the New Testament, but as far as I know the test has never been applied. ...

That De Morgan was obstinate and somewhat eccentric I readily admit, and I do not consider he was a genius, but he leaves on my mind the impression of a lovable man, with intense convictions, of marked originality, having many interests, and possessing exceptional powers of exposition. In those cases where his actions were criticized it would seem that the explanation is to be found in his determination always to take the highest standard of conduct without regard to consequences; he hated suggestions of compromise, expediency, or opportunism. Such men are rare, and we do well to honour them.

As illustrative of his ingenuity in applying the laws of probability to numerous problems, I will mention a test which he proposed for determining the authorship of books. He believed that if different books on similar subjects written by a particular author were examined it would be found that the average number of letters per word in each book would agree to (perhaps) one place of decimal. Hence, if the average number of letters per word in two books on the same subject differed by more than that percentage, it was probable that the books were by different authors. He thought this experiment might be well worth making in cases where authorship was in question, and in particular in the case of the Greek text of some of the books of the New Testament, but as far as I know the test has never been applied. ...

That De Morgan was obstinate and somewhat eccentric I readily admit, and I do not consider he was a genius, but he leaves on my mind the impression of a lovable man, with intense convictions, of marked originality, having many interests, and possessing exceptional powers of exposition. In those cases where his actions were criticized it would seem that the explanation is to be found in his determination always to take the highest standard of conduct without regard to consequences; he hated suggestions of compromise, expediency, or opportunism. Such men are rare, and we do well to honour them.