In England a modified notation soon sprang up in which ratio was indicated by two dots instead of a single dot, thus A:B::C:D. The reason for the change lies probably in the inclination to use the single dot to designate decimal fractions. W. W. Beman pointed out that this modified symbolism (:) for ratio is found as early as 1657 in the end of the trigonometric and logarithmic tables that were bound with Oughtred’s Trigonometria.[92] It is not probable, however, that this notation was used by Oughtred himself. The Trigonometria proper has Oughtred’s A.B::C.D throughout. Moreover, in the English edition of this trigonometry, which appeared the same year, 1657, but subsequent to the Latin edition, the passages which contained the colon as the symbol for ratio, when not omitted, are recast, and the regular Oughtredian notation is introduced. In Oughtred’s posthumous work, Opuscula mathematica hactenus inedita, 1677, the colon appears quite often but is most likely due to the editor of the book.
We have noticed that the notation A:B::C:D antedates the year 1657. Vincent Wing, the astronomer, published in 1651 in London the Harmonicon coeleste, in which is found not only Oughtred’s notation A.B::C.D but also the modified form of it given above. The two are used interchangeably. His later works, the Logistica astronomica (1656), Doctrina spherica (1655), and Doctrina theorica, published in one volume in London, all use the symbols A:B::C:D exclusively. The author of a book entitled, An Idea of Arithmetick at first designed for the use of the Free Schoole at Thurlow in Suffolk . . . . by R. B., Schoolmaster there, London, 1655, writes A:a::C:c, though part of the time he uses Oughtred’s unmodified notation.
We can best indicate the trend in England by indicating the authors of the seventeenth century whom we have found using the notation A:B::C:D and the authors of the eighteenth century whom we have found using A.B::C.D. The former notation was the less common during the seventeenth but the more common during the eighteenth century. We have observed the symbols A:B::C:D (besides the authors already named) in John Collins (1659),[93] James Gregory (1663),[94] Christopher Wren (1668-69),[95] William Leybourn (1673),[96] William Sanders (1686),[97] John Hawkins (1684),[98] Joseph Raphson (1697),[99] E. Wells (1698),[100] and John Ward (1698).[101]
Of English eighteenth-century authors the following still clung to the notation A.B::C.D: John Harris’ translation of F. Ignatius Gaston Pardies (1701),[102] George Shelley (1704),[103] Sam Cobb (1709),[104] J. Collins in Commercium Epistolicum (1712), John Craig (1718),[105] Jo. Wilson (1724).[106] The latest use of A.B::C.D which has come to our notice is in the translation of the Analytical Institutions of Maria G. Agnesi, made by John Colson sometime before 1760, but which was not published until 1801. During the seventeenth century the notation A:B::C:D acquired almost complete ascendancy in England.
In France Oughtred’s unmodified notation A.B::C.D, having been adopted later, was also discarded later than in England. An approximate idea of the situation appears from the following data. The notation A.B::C.D was used by M. Carré (1700),[107] M. Guisnée (1705),[108] M. de Fontenelle (1727),[109] M. Varignon (1725),[110] M. Robillard (1753),[111] M. Sebastien le Clerc (1764),[112] Clairaut (1731),[113] M. L’Hospital (1781).[114]
In Italy Oughtred’s modified notation a, b::c, d was used by Maria G. Agnesi in her Instituzioni analitiche, Milano, 1748. The notation a:b::c:d found entrance the latter part of the eighteenth century. In Germany the symbolism a:b=c:d, suggested by Leibniz, found wider acceptance.[115]
It is evident from the data presented that Oughtred proposed his notation for ratio and proportion at a time when the need of a specific notation began to be generally felt, that his symbol for ratio a.b was temporarily adopted in England and France but gave way in the eighteenth century to the symbol a:b, that Oughtred’s symbol for proportion :: found almost universal adoption in England and France and was widely used in Italy, the Netherlands, the United States, and to some extent in Germany; it has survived to the present time but is now being gradually displaced by the sign of equality =.
Oughtred’s notation to express aggregation of terms has received little attention from historians but is nevertheless interesting. His books, as well as those of John Wallis, are full of parentheses but they are not used as symbols of aggregation in algebra; they are simply marks of punctuation for parenthetical clauses. We have seen that Oughtred writes (a+b)² and √a+b thus, Q:a+b:, √:a+b:, or Q:a+b, √:a+b, using on rarer occasions a single dot in place of the colon. This notation did not originate with Oughtred, but, in slightly modified form, occurs in writings from the Netherlands. In 1603 C. Dibvadii in geometriam Evclidis demonstratio numeralis, Leyden, contains many expressions of this sort, √·136+√2048, signifying √(136+√2048). The dot is used to indicate that the root of the binomial (not of 136 alone) is called for. This notation is used extensively in Ludolphi à Cevlen de circulo, Leyden, 1619, and in Willebrordi Snellii De circuli dimensione, Leyden, 1621. In place of the single dot Oughtred used the colon (:), probably to avoid confusion with his notation for ratio. To avoid further possibility of uncertainty he usually placed the colon both before and after the algebraic expression under aggregation. This notation was adopted by John Wallis and Isaac Barrow. It is found in the writings of Descartes. Together with Vieta’s horizontal bar, placed over two or more terms, it constituted the means used almost universally for denoting aggregation of terms in algebra. Before Oughtred the use of parentheses had been suggested by Clavius[116] and Girard.[117] The latter wrote, for instance, √(2+√3). While parentheses never became popular in algebra before the time of Leibniz and the Bernoullis they were by no means lost sight of. We are able to point to the following authors who made use of them: I. Errard de Bar-le-Duc (1619),[118] Jacobo de Billy (1643),[119] one of whose books containing this notation was translated into English, and also the posthumous works of Samuel Foster.[120] J. W. L. Glaisher points out that parentheses were used by Norwood in his Trigonometrie (1631), p. 30.[121]
The symbol for the arithmetical difference between two numbers, ~, is usually attributed to John Wallis, but it occurs in Oughtred’s Clavis mathematicae of 1652, in the tract on Elementi decimi Euclidis declaratio, at an earlier date than in any of Wallis’ books. As Wallis assisted in putting this edition through the press it is possible, though not probable, that the symbol was inserted by him. Were the symbol Wallis’, Oughtred would doubtless have referred to its origin in the preface. During the eighteenth century the symbol found its way into foreign texts even in far-off Italy.[122] It is one of three symbols presumably invented by Oughtred and which are still used at the present time. The others are × and ::.
The curious and ill-chosen symbols,