are much too thinly distributed in space for their resistance to be sensible, while the Cartesian ether, which completely fills space, must under all circumstances give rise to a very great resistance."

Newton's views on the subjects which concern us at present are thus summarized by Rosenberger."

"All particles of matter accessible to our senses and all bodies move as if they attracted one another in the direct ratio of their masses and the inverse ratio of the squares of their distances from one another. This mutual action is not action at a distance but is brought about by some intermediary not yet certainly to be determined. Since, however, a matter suitable to play the part of this intermediary cannot be found, this agent is probably immaterial and in all likelihood God himself. It is doubtful whether God does this directly or indirectly, though probably the former. Hence follows that the mutual action in all matter is not an essential and necessary property of it like impenetrability and mobility, but a quality given to matter by the creator in perfect freedom and in the way which seemed proper to him."

It is not impossible that this was Newton's real view. But it seems impossible to doubt that Newton decidedly leaned toward the hypothesis of a very rare ether. Certainly he thought that he had refuted the Cartesian theory of a dense ether. And Newton himself has given some further indication of what his views were, and this indication seems, like the letters to Bentley, to support our conjecture. Leibniz had asked Newton what was his opinion of what Huygens had said in the appendix, Discours de la Cause de la Pesanteur, to his Traité de la Lumière of 1690. On October 26, 1693, Newton replied:48

"I cannot admit that a subtle matter fills the heavens,

Op. cit., p. 422.

"Mentioned in another connection by Rosenberger, op. cit., pp. 461-462.

for the celestial motions are too regular to arise from vortices, and vortices would only disturb the motion. But if any one should explain gravity and all its laws by the action of some subtle mediums, and should show that the motions of the planets and comets were not disturbed by this matter, I should by no means oppose it."

It would seem that Newton felt just as strongly as did Descartes or Huygens or Leibniz or Faraday or Maxwell a need for a medium by which to transmit force. A weapon sometimes used against Cartesian doctrines by those whose interests or emotions led them to condemn what they fancied might be construed as a slight on the omnipotence of God, was that Descartes's world was so planned as to leave God out of account altogether. But something like this is the ideal of every scientific man. However pious a man of science may be-and some even eminent men of science have been models of unthinking devotion to certain religious sects he will try to explain by natural causes phenomena which have hitherto appeared inexplicable except as miracles wrought by God. It is difficult to imagine that religious belief could ever interfere with this postulate of scientific investigation, and thus it can hardly be believed that Newton preferred to help himself out by supposing certain actions on the part of God, when there was a chance that gravitation might be explained by the hypothesis of a rarefied ether.

CAMBRIDGE, ENGLAND.

PHILIP E. B. JOURDAIN.

CRITICISMS AND DISCUSSIONS.

THE WORKS OF WILLIAM OUGHTRED.*

CLAVIS MATHEMATICAE.

William Oughtred (1574 (?)-1660), though by profession a clergyman, was one of the world's great teachers of mathematics and should still be honored as the inventor of that indispensable mechanical instrument, the slide-rule. It is noteworthy that he showed a marked disinclination to give his writings to the press. His first paper on sun-dials was written at the age of twenty-three, but we are not aware that more than one brief mathematical manuscript was printed before his fifty-seventh year. In every instance, publication in printed form seems to have been due to pressure exerted by one or more of his patrons, pupils or friends. Some of his manuscripts were lent out to his pupils who prepared copies for their own use. In some instances they urged upon him the desirability of publication and assisted in preparing copy for the printer. The earliest and best known book of Oughtred was his Clavis mathematicae. As he himself informs us, he was employed by the Earl of Arundel about 1628 to instruct the Earl's son, Lord William Howard (afterwards Viscount Stafford), in the mathematics. For the use of this young man Oughtred composed a treatise on algebra which was published in Latin in the year 1631 at the urgent request of a kinsman of the young man, Charles Cavendish, a patron of learning.

The Clavis mathematicae, in its first edition of 1631, was a booklet of only 88 small pages. Yet it contained in very condensed form the essentials of arithmetic and algebra as known at that time.

Aside from the addition of four tracts, the 1631 edition underwent some changes in the editions of 1647 and 1648 which two are much alike. The twenty chapters of 1631 are reduced to nineteen in 1647 and in all the later editions. Numerous minute alterations

For further details see the author's article on "The Life of Oughtred" in The Open Court, August, 1915, where fuller references are given to some of the books cited here.

1The full title of the Clavis of 1631 is as follows: Arithmeticae in numeris et speciebus institutio: Quae tum logisticae, tum analyticae, atque adeo totius

from the 1631 edition occur in all parts of the books of 1647 and 1648. The material of the last three chapters of the 1631 edition is re-arranged with some slight additions here and there. The 1648 edition has no preface. In the print of 1652 there are only slight alterations from the 1648 edition; after that the book underwent hardly any changes, except for the number of tracts appended, and brief explanatory notes added at the close of the chapters in the English edition of 1694 and 1702. The 1652 and 1667 editions were seen through the press by John Wallis; the 1698 impression contains on the title-page the words: Er Recognitione D. Johannis Wallis, S.T.D. Geometriae Professoris Saviliani.

The cost of publishing may be a matter of some interest. When arranging for the printing of the 1667 edition of the Clavis, Wallis wrote Collins: “I told you in my last what price she [Mrs. Lichfield] expects for it, as I have formerly understood from her, viz., 40 1. for the impression, which is about 91d. a book."

mathematicae, quasi clavis est.—Ad nobilissimum spectatissimumque invenem Dn. Guilelmum Howard, Ordinis qui dicitur, Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arundeliae & Surriae, Comitis Mareschalli Angliae, &c filium-Londini, Apud Thomam Harpervm. M.DC.XXXI.

In all there appeared five Latin editions, the second in 1648 at London, the third in 1652 at Oxford, the fourth in 1667 at Oxford, the fifth in 1693 and 1698 at Oxford. There were two independent English editions: the first in 1647 at London, translated in greater part by Robert Wood of Lincoln College, Oxford, as is stated in the preface to the 1652 Latin edition; the second in 1694 and 1702 is a new translation, the preface being written and the book recommended by the astronomer Edmund Halley. The 1694 and 1702 impressions labored under the defect of many sense-disturbing errors due to careless reading of the proofs. All the editions of the Clavis, after. the first edition, had one or more of the following tracts added on: Eq. De Aequationum affectarum resolutione in numeris.

=

Eu. Elementi decimi Euclidis declaratio.

=

So. De Solidis regularibus tractatus.

An.

Fa.

De Anatocismo, sive usura composita.
Regula falsae positionis.

Ar. Theorematum in libris Archimedis de Sphaera & cylindro declaratio.
Ho. Horologia scioterica in plano, geometricè delineandi modus.

The abbreviated titles given here are, of course, our own. The lists of tracts added to the Clavis mathematicae of 1631 in its later editions, given in the order in which the tracts appear in each edition, are as follows: Clavis of 1647, Eq., An., Fa., Ho.; Clavis of 1648, Eq., An., Fa., Eu., So.; Clavis of 1652, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1667, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1693 and 1698, Éq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1694 and 1702, Eq.

The title-page of the Clavis was considerably modified after the first edition. Thus, the 1652 Latin edition has this title-page: Guilelmi Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia Socii, Clavis mathematicae dengo limata, sive potius fabricata. Cum aliis quibusdam ejusdem commentationibus, quae in sequenti pagina recensentur. Editio tertia auctior & emendatior Oxoniae, Excudebat Leon. Lichfield, Veneunt apud Tho. Robinson. 1652.

* Rigaud, op. cit., Vol. II, p. 476.

As compared with other contemporary works on algebra, Oughtred's distinguishes itself for the amount of symbolism used, particularly in the treatment of geometric problems. Extraordinary emphasis was placed upon what he called in the Clavis the "analytical art." By that term he did not mean our modern analysis or analytical geometry, but the art "in which by taking the thing sought as knowne, we finde out that we seeke." He meant to express by it condensed processes of rigid, logical deduction expressed by appropriate symbols, as contrasted with mere description or elucidation by passages fraught with verbosity. In the preface to the first edition (1631) he says:

"In this little book I make known...the rules relating to fundamentals, collected together, just like a bundle, and adapted to the explanation of as many problems as possible."

As stated in this preface, one of his reasons for publishing the book, is"...that like Ariadne I might offer a thread to mathematical study by which the mysteries of this science might be revealed, and direction given to the best authors of antiquity, Euclid, Archimedes, the great geometrician Apollonius of Perga, and others, so as to be easily and thoroughly understood, their theorems being added, not only because to many they are the height and depth of mathematical science (I ignore the would-be mathematicians who occupy themselves only with the so-called practice, which is in reality mere juggler's tricks with instruments, the surface so to speak, pursued with a disregard of the great art, a contemptible picture), but also to show with what keenness they have penetrated, with what mass of equations, comparisons, reductions, conversions and disquisitions these heroes have ornamented, increased and invented this most beautiful science."

The Clavis opens with an explanation of the Hindu-Arabic notation of decimal fractions. Noteworthy is the absence of the words "million," "billion," etc. Although used on the continent by certain mathematical writers long before this, these words did not become current in English mathematical books until the eighteenth century. The author was a great admirer of decimal fractions, but failed to introduce the notation which in later centuries came to be

See, for instance, the Clavis mathematicae of 1652, where he expresses himself thus (p. 11): "Speciosa haec Arithmetica arti Analyticae (per quam ex sumptione quaesiti, tanquam noti, investigatur quaesitum) multo accommodatior est, quam illa numerosa."

*Oughtred, The Key of the Mathematicks, London, 1647, p. 4.