Extracts from letters of W. Gascoigne to Oughtred, of the years 1640 and 1641, throw some light upon mathematical teaching of the time:
Amongst the mathematical rarities these times have afforded, there are none of that small number I (a late intruder into these studies) have yet viewed, which so fully demonstrates their authors’ great abilities as your Clavis, not richer in augmentations, than valuable for contraction; . . . .
Your belief that there is in all inventions aliquid divinum, an infusion beyond human cogitations, I am confident will appear notably strengthened, if you please to afford this truth belief, that I entered upon these studies accidentally after I betook myself to the country, having never had so much aid as to be taught addition, nor the discourse of an artist (having left both Oxford and London before I knew what any proposition in geometry meant) to inform me what were the best authors.[59]
The following extracts from two letters by W. Robinson, written before the appearance of the 1647 English edition of the Clavis, express the feeling of many readers of the Clavis on its extreme conciseness and brevity of explanation:
I shall long exceedingly till I see your Clavis turned into a pick-lock; and I beseech you enlarge it, and explain it what you can, for we shall not need to fear either tautology or superfluity; you are naturally concise, and your clear judgment makes you both methodical and pithy; and your analytical way is indeed the only way. . . . .
I will once again earnestly entreat you, that you be rather diffuse in the setting forth of your English mathematical Clavis, than concise, considering that the wisest of men noted of old, and said stultorum infinitus est numerus, these arts cannot be made too easy, they are so abstruse of themselves, and men either so lazy or dull, that their fastidious wits take a loathing at the very entrance of these studies, unless it be sweetened on with plainness and facility. Brevity may well argue a learned author, that without any excess or redundance, either of matter or words, can give the very substance and essence of the thing treated of; but it seldom makes a learned scholar; and if one be capable, twenty are not; and if the master sum up in brief the pith of his own long labours and travails, it is not easy to imagine that scholars can with less labour than it cost their masters dive into the depths thereof.[60]
Here is the judgment of another of Oughtred’s friends:
. . . . with the character I received from your and my noble friend Sir Charles Cavendish, then at Paris, of your second edition of the same piece, made me at my return into England speedily to get, and diligently peruse the same. Neither truly did I find my expectation deceived; having with admiration often considered how it was possible (even in the hardest things of geometry) to deliver so much matter in so few words, yet with such demonstrative clearness and perspicuity: and hath often put me in mind of learned Mersennus his judgment (since dead) of it, that there was more matter comprehended in that little book than in Diophantus, and all the ancients. . . . .[61]
Oughtred’s own feeling was against diffuseness in textbook writing. In his revisions of his Clavis the original character of that book was not altered. In his reply to W. Robinson, Oughtred said:
. . . . But my art for all such mathematical inventions I have set down in my Clavis Mathematica, which therefore in my title I say is tum logisticae cum analyticae adeoque totius mathematicae quasi clavis, which if any one of a mathematical genius will carefully study, (and indeed it must be carefully studied,) he will not admire others, but himself do wonders. But I (such is my tenuity) have enough fungi vice cotis, acutum reddere quae ferrum valet, exsors ipsa secandi, or like the touchstone, which being but a stone, base and little worth, can shew the excellence and riches of gold.[62]
John Wallis held Oughtred’s Clavis in high regard. When in correspondence with John Collins concerning plans for a new edition, Wallis wrote in 1666-67, six years after the death of Oughtred:
. . . . But for the goodness of the book in itself, it is that (I confess) which I look upon as a very good book, and which doth in as little room deliver as much of the fundamental and useful part of geometry (as well as of arithmetic and algebra) as any book I know; and why it should not be now acceptable I do not see. It is true, that as in other things so in mathematics, fashions will daily alter, and that which Mr. Oughtred designed by great letters may be now by others be designed by small; but a mathematician will, with the same ease and advantage, understand Ac, and a³ or aaa. . . . . And the like I judge of Mr. Oughtred’s Clavis, which I look upon (as those pieces of Vieta who first went in that way) as lasting books and classic authors in this kind; to which, notwithstanding, every day may make new additions. . . . .
But I confess, as to my own judgment, I am not for making the book bigger, because it is contrary to the design of it, being intended for a manual or contract; whereas comments, by enlarging it, do rather destroy it. . . . . But it was by him intended, in a small epitome, to give the substance of what is by others delivered in larger volumes. . . . .[63]