154 



NA TURE 



[Dec: i6, li 



briefly to the range of subjects wliich were included in the 

 examination. Of the nature of the questions proposed prior 

 to 1S28, the first year in which all the papers were printed, 

 very little can be known except what can be gathered from 

 the problem papers and the specimens of the other 

 papers that have been preserved ; ' but there can be no 

 doubt that their character was determined by the ordinary 

 Cambridge treatises then in use, which, it is well known, 

 were far behind the corresponding treatises published on 

 the Continent. Woodhouse's " Principles of Analytical 

 Calculation" (1S03), and " Plane and Spherical Trigono- 

 metry" (1809) are the earliest indications of the introduc- 

 tion of the analytical element into the mathematics of the 

 University ; a more decided impulse in this direction 

 was given by the translation of Lacroix's " Differential 

 and Integral Calculus " by Herschel, Peacock, and Bab- 

 bage (1817), followed by Peacock's "Examples on the 

 Differential and Integral Calculus," and Herschel's " Kx- 

 amples on the Calculus of Finite Differences " (1820). 



The reform in the mathematical studies of the Univer- 

 sity which w-as effected by Herschel, Peacock, and Babbage, 

 is well known. It is to them that we mainly owe the 

 revival of mathematics in this country, and the restora- 

 tion of intercourse with the rest of Europe after three- 

 quarters of a century of isolation. Peacock was Mode- 

 rator in 1S17, and he ventured to introduce the symbol 

 of differentiation into the examination, his colleague, 

 however, retaining the old fluxional notation. The old 

 system made its appearance once more in 181 8, but in 

 l8ig Peacock was Moderator again, with a colleague who 

 shared his views, and the change was fully accomplished. 



The introduction of the notation and language of the 

 differential calculus into the Senate House examina- 

 tion forms an important landmark in the history of 

 Cambridge mathematics. From that time onward the 

 University began to make up slowly but surely the 

 ground she had lost ; step by step the analytical pro- 

 cesses and methods superseded the older geometrical 

 modes of treatment ; and each year saw a substantial in- 

 crease in the range of subjects included in the course of 

 study. 



Only second in importance to the revolution effected 

 by the substitution of the differential for the fluxional 

 calculus was the rise of analytical geometry in the first 

 thirty years of the century ; and, considering the amount 

 of attention that this subject has received at Cambridge 

 in the last fifty years, and the accessions that have been 

 made in this country to the analytical theory of curves 

 and surfaces, a peculiar interest attaches to the introduc- 

 tion into the University of the algebraic treatinent of geo- 

 metry and the early stages of its development. The first 

 edition of Wood's "Algebra," which appeared in 1795, 

 contained, as Part IV., a chapter of thirty pages "Cn the 

 Application of Algebra to Geometry," in which are given 

 the equations of the straight line, ellipse, cissoid, conchoid, 

 and other curves, the construction of equations, &c. This 

 chapter remained unchanged in the ninth edition (1830), 

 and seems to have formed the only introduction to ana- 

 lytical geometry existing in the University until 1826, 

 when Hamilton - published his " Principles of Analytical 

 Geometry, designed for the use of Stuclents in the Uni- 

 versity." This was not the first English treatise on ana- 

 lytical geometry, as Lardner's " Algebraic Geometry " 

 was published, three years earlier, in 1823 ; but it was 

 the first Cambridge book, and the first which included 

 solid geometry. The problem papers from iSoo to 1820 

 show that at Uie beginning of the century analytical geo- 

 metry was always represented to some extent, though 

 scarcely as an independent subject, most of the questions 

 relating to areas, loci, &c., in which but little more than 



* The problem papers were printed from 1779; but only those of the 

 present century are accessible in the Cambridge University Calendars and 

 other publications. 



2 Late Dean of Salisbury ; born April 3, 1794 ; died February 7, 1880. 



the mode of representation by means of ordinates and 

 abscissae was involved. Hymers published his " Ana- 

 lytical Geometry of Three Dimensions" in 1830, and his 

 " Conic Sections '' in 1837. The latter at once superseded 

 Hamilton's treatise, and remained the standard work on 

 the subject for many years. 



In applied mathematics the character of the questions 

 proposed was largely influenced by the publication of 

 Whewell's " Mechanics " (1819), Whewell's " Dynamics " 

 (1823), Coddington's "Optics" (1823), Woodhouse's 

 "Plane Astronomy" (1S21-23), and Airy's "Tracts" 

 (1S26). A second edition of this last work, which appeared 

 in 1831, contained a tract on the " Undulatory Theory of 

 Light,'' a subject which was freely represented in the 

 examination for many years. Not only were the ques- 

 tions modified, in character and range, by the publication 

 of new mathematical treatises in the University, but they 

 were also affected to a certain extent by some of the pro- 

 fessorial lectures. At this time, too, the Smith's Prize 

 examination exerted a beneficial eflect upon the Senate 

 House examination, certain classes of questions which 

 were originally introduced into the former having shortly 

 afterwards been admitted into the latter. Between 1830 

 and 1840, questions in definite integrals, Laplace's coeffi- 

 cients, electricity, magnetism, and heat were also intro- 

 duced. There were no regulations of any kind, and the 

 responsibility of introducing innovations and alterations 

 rested solely with the Moderators and Examiners. The 

 uncertainty as to the subjects that the examination would 

 embrace, and the want of any due notice of any extension 

 of them, were found to be serious inconveniences to the 

 higher class of students, although, as has been already 

 stated, the introduction of a new subject had been gene- 

 rally preceded by the publication of a work by a Cam- 

 bridge mathematician, in which it was treated in a 

 manner adapted to the examination. 



The Board of Matheinatical Studies was created by the 

 Senate on October 31, 1848, and in May of the following 

 year they issued a report to the Senate in which, after 

 giving a short review of the past and existing state of 

 mathematical studies in the Lhiiversity, they recom- 

 mended that, considering the great number of subjects 

 occupying the attention of the candidates and the doubt 

 existing as to the range of subjects from which questions 

 might be proposed, the mathematical theories of elec- 

 tricity, magnetism, and heat should not be admitted as 

 subjects of examination. In the following year they issued 

 a second report in which they recommended the omis- 

 sion of elliptic integrals, Laplace's coefficient?, capillary 

 attraction, the figure of the earth considered as hetero- 

 geneous, &c., besides certain limitations of the questions 

 in lunar and planetary theory, &c. In making these 

 recommendations the Board expressed their opinion that 

 they were only giving definite form to what had become 

 the practice in the examination, and were only putting 

 before the candidates such results as they might them- 

 selves have deduced by the study of the Senate House 

 papers of the last few years. The Board also recom- 

 mended that the papers containing book-work and riders 

 should be shortened. 



From 1823 onwards, the examination was conducted ia 

 each year by four examiners — the two Moderators and 

 the two Examiners, the Moderators of one year be- 

 coming as a matter of course the Examiners of the next. 

 Thus of the four examiners in each year two had taken, 

 part in the examination of the previous year. The con- 

 tinuity of the examination was w-ell kept up by this 

 arrangement ; but perhaps it had the effect of causing its 

 traditions to be rather too punctiliously observed, the 

 papers of each year being, as regards the subjects in- 

 cluded, exact counterparts of the corresponding papers 

 of the previous year. The resolutions of the Board in 

 1849 50 were not binding on the successive Moderators 

 and Examiners up to 1872, but each year they seem to 



