MATHEMATICS, “A SCIENCE OF THE EYE”
Oughtred was a great admirer of the Greek mathematicians—Euclid, Archimedes, Apollonius of Perga, Diophantus. But in reading their works he experienced keenly what many modern readers have felt, namely, that the almost total absence of mathematical symbols renders their writings unnecessarily difficult to read. Statements that can be compressed into a few well-chosen symbols which the eye is able to survey as a whole are expressed in long-drawn-out sentences. A striking illustration of the importance of symbolism is afforded by the history of the formula
ix=log(cos x+i sin x).
It was given in Roger Cotes’ Harmonia mensurarum, 1722, not in symbols, but expressed in rhetorical form, destitute of special aids to the eye. The result was that the theorem remained in the book undetected for 185 years and was meanwhile rediscovered by others. Owing to the prominence of Cotes as a mathematician it is very improbable that such a thing could have happened had the theorem been thrust into view by the aid of mathematical symbols.
In studying the ancient authors Oughtred is reported to have written down on the margin of the printed page some of the theorems and their proofs, expressed in the symbolic language of algebra.
In the preface of his Clavis of 1631 and of 1647 he says:
Wherefore, that I might more clearly behold the things themselves, I uncasing the Propositions and Demonstrations out of their covert of words, designed them in notes and species appearing to the very eye. After that by comparing the divers affections of Theorems, inequality, proportion, affinity, and dependence, I tryed to educe new out of them.
It was this motive which led him to introduce the many abbreviations in algebra and trigonometry to which reference has been made in previous pages. The pedagogical experience of recent centuries has indorsed Oughtred’s view, provided of course that the pupil is carefully taught the exact meaning of the symbols. There have been and there still are those who oppose the intensive use of symbolism. In our day the new symbolism for all mathematics, suggested by the school of Peano in Italy, can hardly be said to be received with enthusiasm. In Oughtred’s day symbolism was not yet the fashion. To be convinced of this fact one need only open a book of Edmund Gunter, with whom Oughtred came in contact in his youth, or consult the Principia of Sir Isaac Newton, who flourished after Oughtred. The mathematical works of Gunter and Newton, particularly the former, are surprisingly destitute of mathematical symbols. The philosopher Hobbes, in a controversy with John Wallis, criticized the latter for that “Scab of Symbols,” whereupon Wallis replied:
I wonder how you durst touch M. Oughtred for fear of catching the Scab. For, doubtlesse, his book is as much covered over with the Scab of Symbols, as any of mine. . . . . As for my Treatise of Conick Sections, you say, it is covered over with the Scab of Symbols, that you had not the patience to examine whether it is well or ill demonstrated.[142]
Oughtred maintained his view of the importance of symbols on many different occasions. Thus, in his Circles of Proportion, 1632, p. 20:
This manner of setting downe Theoremes, whether they be Proportions, or Equations, by Symboles or notes of words, is most excellent, artificiall, and doctrinall. Wherefore I earnestly exhort every one, that desireth though but to looke into these noble Sciences Mathematicall, to accustome themselves unto it: and indeede it is easie, being most agreeable to reason, yea even to sence. And out of this working may many singular consectaries be drawne: which without this would, it may be, for ever lye hid.