The characteristic in the Clavis that is most striking to a modern reader is the total absence of indexes or exponents. There is much discussion in the leading treatises of the latter part of the sixteenth and the early part of the seventeenth century on the theory of indexes, but the modern exponential notation, aⁿ, is of later date. The modern notation, for positive integral exponents, first appears in Descartes’ Géométrie, 1637; fractional and negative exponents were first used in the modern form by Sir Isaac Newton, in his announcement of the binomial formula, in a letter written in 1676. This total absence of our modern exponential notation in Oughtred’s Clavis gives it a strange aspect. Like Vieta, Oughtred uses ordinarily the capital letters, A, B, C, . . . . to designate given numbers; A² is written Aq, A³ is written Ac; for A⁴, A⁵, A⁶ he has, respectively, Aqq, Aqc, Acc. Only on rare occasions, usually when some parallelism in notation is aimed at, does he use small letters[27] to represent numbers or magnitudes. Powers of binomials or polynomials are marked by prefixing the capital letters Q (for square), C (for cube), QQ (for the fourth power), QC (for the fifth power), etc.
Oughtred does not express aggregation by (). Parentheses had been used by Girard, and by Clavius as early as 1609,[28] but did not come into general use in mathematical language until the time of Leibniz and the Bernoullis. Oughtred indicates aggregation by writing a colon (:) at both ends. Thus, Q:A-E: means with him (A-E)². Similarly, √q:A+E: means √(A+E). The two dots at the end are frequently omitted when the part affected includes all the terms of the polynomial to the end. Thus, C:A+B-E=.. means (A+B-E)³=.. There are still further departures from this notation, but they occur so seldom that we incline to the interpretation that they are simply printer’s errors. For proportion Oughtred uses the symbol (::). The proportion a:b=c:d appears in his notation a·b::c·d. Apparently, a proportion was not fully recognized in this day as being the expression of an equality of ratios. That probably explains why he did not use = here as in the notation of ordinary equations. Yet Oughtred must have been very close to the interpretation of a proportion as an equality; for he says in his Elementi decimi Euclidis declaratio, “proportio, sive ratio aequalis ::” That he introduced this extra symbol when the one for equality was sufficient is a misfortune. Simplicity demands that no unnecessary symbols be introduced. However, Oughtred’s symbolism is certainly superior to those which preceded. Consider the notation of Clavius.[29] He wrote 20:60=4:x, x=12, thus: “20·60·4? fiunt 12.” The insufficiency of such a notation in the more involved expressions frequently arising in algebra is readily seen. Hence Oughtred’s notation (::) was early adopted by English mathematicians. It was used by John Wallis at Oxford, by Samuel Foster at Gresham College, by James Gregory of Edinburgh, by the translators into English of Rahn’s algebra, and by many other early writers. Oughtred has been credited generally with the introduction of St. Andrew’s cross × as the symbol for multiplication in the Clavis of 1631. We have discovered that this symbol, or rather the letter x which closely resembles it, occurs as the sign of multiplication thirteen years earlier in an anonymous “Appendix to the Logarithmes, shewing the practise of the Calculation of Triangles etc.” to Edward Wright’s translation of John Napier’s Descriptio, published in 1618.[30] Later we shall give our reasons for believing that Oughtred is the author of that “Appendix.” The × has survived as a symbol of multiplication.
Another symbol introduced by Oughtred and found in modern books is ~, expressing difference; thus C~D signifies the difference between C and D, even when D is the larger number.[31] This symbol was used by John Wallis in 1657.[32]
Oughtred represented in symbols also certain composite expressions, as for instance A+E=Z, A-E=X, where A is greater than E. He represented by a symbol also each of the following: A²+E², A³+E³, A²-E², A³-E³.
Oughtred practically translated the tenth book of Euclid from its ponderous rhetorical form into that of brief symbolism. An appeal to the eye was a passion with Oughtred. The present writer has collected the different mathematical symbols used by Oughtred and has found more than one hundred and fifty of them.
The differences between the seven different editions of the Clavis lie mainly in the special parts appended to some editions and dropped in the latest editions. The part which originally constituted the Clavis was not materially altered, except in two or three of the original twenty chapters. These changes were made in the editions of 1647 and 1648. After the first edition, great stress was laid upon the theory of indices upon the very first page, as also in passages farther on. Of course, Oughtred did not have our modern notation of indices or exponents, but their theory had been a part of algebra and arithmetic for some time. Oughtred incorporated this theory in his brief exposition of the Hindu-Arabic notation and in his explanation of logarithms. As previously pointed out, the last three chapters of the 1631 edition were considerably rearranged in the later editions and combined into two chapters, so that the Clavis proper had nineteen chapters instead of twenty in the additions after the first. These chapters consisted of applications of algebra to geometry and were so framed as to constitute a severe test of the student’s grip of the subject. The very last problem deals with the division of angles into equal parts. He derives the cubic equation upon which the trisection depends algebraically, also the equations of the fifth degree and seventh degree upon which the divisions of the angle into 5 and 7 equal parts depend, respectively. The exposition was severely brief, yet accurate. He did not believe in conducting the reader along level paths or along slight inclines. He was a guide for mountain-climbers, and woe unto him who lacked nerve.
Oughtred lays great stress upon expansions of powers of a binomial. He makes use of these expansions in the solution of numerical equations. To one who does not specialize in the history of mathematics such expansions may create surprise, for did not Newton invent the binomial theorem after the death of Oughtred? As a matter of fact, the expansions of positive integral powers of a binomial were known long before Newton, not only to seventeenth-century but even to eleventh-century mathematicians. Oughtred’s Clavis of 1631 gave the binomial coefficients for all powers up to and including the tenth. What Newton really accomplished was the generalization of the binomial expansion which makes it applicable to negative and fractional exponents and converts it into an infinite series.
As a specimen of Oughtred’s style of writing we quote his solution of quadratic equations, accompanied by a translation into English and into modern mathematical symbols.
As a preliminary step[33] he lets
Z=A+E and A>E;