Both before and after the time of Oughtred much was written on sun-dials. Such instruments were set up against the walls of prominent buildings, much as the faces of clocks in our time. The inscriptions that were put upon sun-dials are often very clever: "I count only the hours of sunshine," "Alas, how fleeting." A sun-dial on the grounds of Merchiston Castle, in Edinburgh, where the inventor of logarithms, John Napier, lived for many years, bears the inscription, "Ere time be tint, tak tent of time" (Ere time be lost, take heed of time).

Portable sun-dials were sometimes carried in pockets, as we carry watches. Thus Shakespeare, in As You Like It, Act II, Sc. 7:

1658.

"And he drew a diall from his poke."

Watches were first made for carrying in the pocket about

Because of this literary, scientific and practical interest in methods of indicating time it is not surprising that Oughtred devoted himself to the mastery and the advancement of methods of time-measurement.

Besides the accounts previously noted, there came from his pen: The Description and Use of the double Horizontall Dyall: Whereby not onely the hower of the day is shewne; but also the Meridian Line is found: And most Astronomical Questions, which may be done by the Globe, are resolved. Invented and written by W. O., London, 1636.

The "Horizontall Dyall" and "Horologicall Ring" appeared again as appendices to Oughtred's translation from the French of a book on mathematical recreations.

The fourth French edition of that work appeared in 1627 at Paris, under the title of Recreations mathematique, written by "Henry van Etten," a pseudonym for the French Jesuit Jean Leurechon (1591-1690). English editions appeared in 1633, 1653 and 1674. The full title of the 1653 edition conveys an idea of the contents of the text:

Mathematicall Recreations, or, A Collection of many Problemes, extracted out of the Ancient and Modern Philosophers, as Secrets and Experiments in Arithmetick, Geometry, Cosmographie, Musick, Opticks, Architecture, Statick, Mechanicks, Chemistry, Waterworks, Fire-works, &c. Not vulgarly manifest till now. Written first in Greek and Latin, lately compil'd in French, by Henry Van Etten, and now in English, with the Examinations and Augmenta

tions of divers Modern Mathematicians. Whereunto is added the Description and Use of the Generall Horologicall Ring. And The Double Horizontall Diall. Invented and written by William Oughtred. London, Printed for William Leake, at the Signe of the Crown in Fleet-street, between the two Temple-Gates. MDCLIII. The graphic solution of spherical triangles by the accurate drawing of the triangles on a sphere and the measurement of the unknown parts in the drawing, was explained by Oughtred in a short tract which was published by his son-in-law, Christopher Brookes, under the following title:

The Solution of all Sphaerical Triangles both right and oblique By the Planisphaere: Whereby two of the Sphaerical partes sought, are at one position most easily found out. Published with consent of the Author, By Christopher Brookes, Mathematique Instrumentmaker, and Manciple of Wadham Colledge, in Oxford.

Brookes says in the preface: "I have oftentimes seen my Reverend friend Mr. W. O. in his resolution of all sphaericall triangles both right and oblique, to use a planisphaere, without the tedious labour of Trigonometry by the ordinary Canons: which planisphaere he had delineated with his own hands, and used in his calculations more than Forty years before."

Interesting as one of the sources from which Oughtred obtained his knowledge of the conic sections is his study of Mydorge. A tract which he wrote thereon was published by Jonas Moore, in his Arithmetick in two books.... [containing also] the two first books of Mydorgius his conical sections analyzed by that reverend devine Mr. W. Oughtred, Englished and completed with cuts. London, 1660. Another edition bears the date 1688.

To be noted among the minor works of Oughtred are his posthumous papers. He left a considerable number of mathematical papers which his friend Sir Charles Scarborough had revised under his direction and published at Oxford in 1676 in one volume under the title, Gulielmi Oughtredi, Etonensis, quondam Collegii Regalis in Cantabrigia Socii, Opuscula Mathematica hactenus inedita. Its nine tracts are of little interest to a modern reader.

Here we wish to give our reasons for our belief that Oughtred is the author of an anonymous tract on the use of logarithms and on a method of logarithmic interpolation which, as previously noted, appeared as an "Appendix" to Edward Wright's translation into English of John Napier's Descriptio, under the title, A Description of the Admirable Table of Logarithmes, London, 1618.

The "Appendix" bears the title, "An Appendix to the Logarithmes, showing the practise of the Calculation of Triangles, and also a new and ready way for the exact finding out of such lines and Logarithmes as are not precisely to be found in the Canons." It is an able tract. A natural guess is that the editor of the book, Samuel Wright, a son of Edward Wright, composed this "Appendix." More probable is the conjecture which (Dr. J. W. L. Glaisher informs me) was made by Augustus De Morgan, attributing the authorship to Oughtred. Two reasons in support of this are advanced by Dr. Glaisher, the use of x in the "Appendix" as the sign of multiplication (to Oughtred is generally attributed the introduction of the cross x for multiplication in 1631), and the then unusual designation "cathetus" for the vertical leg of a right triangle, a term appearing in Oughtred's books. We are able to advance a third argument, namely the occurrence in the "Appendix" of (S*) as the notation for sine complement (cosine), while Seth Ward, an early pupil of Oughtred, in his Idea trigonometriae demonstratae, Oxford, 1654, used a similar notation (S'). It has been stated elsewhere that Oughtred claimed Seth Ward's exposition of trigonometry as virtually his own. Attention should be called also to the fact that, in his Trigonometria, page 2, Oughtred uses (') to designate 180°-angle.

COLORADO COLLEGE.

FLORIAN CAJORI.

BERGSON'S THEORY OF INTUITION.

Probably the best example of Bergson's application of the intuitive method is to be found in his account of the ideal genesis of the intelligence in the third chapter of Creative Evolution. This gives us the gist of his whole philosophy, and serves to illustrate the difficulties of Bergson's view not only of the nature of intellect, but also of intuition itself. What Bergson proposes to do is "to engender intelligence, by setting out from the consciousness which envelopes it"; that is to say, he proposes that we should actually experience in our own selves the process by which duration, which is pure heterogeneity and pure activity, is degraded into the spatializing intellect and spatialized matter. The intellect left to itself, Bergson argues, naturally tends to the homogeneous and the extended and the static. That is to say, the impression we get of the intellect is as of something unmaking itself. "Extension appears only as a tension which is interrupted." But this suggests to us a

reality of which the intellect is merely the degradation and suppression. "The vision we have of the material world is that of a weight which falls; no image drawn from matter, properly so called, will ever give us the idea of the weight rising." But in the case of life we see "an effort to mount the incline that matter descends." Living things "reveal to us the possibility, the necessity even, of a process the inverse of materiality, creative of matter by its interruption alone" (p. 259), a reality which is purely active, a cosmic impulse which makes itself incessantly.

Now if by means of a powerful effort of the mind we succeed in attaining to this reality, if, as Bergson expresses it, "we put back our being into our will, and our will itself into the impulsion it prolongs, we understand, we feel, that reality is a perpetual growth, a creation pursued without end" (p. 252). But if then we relax the tension which this effort demands, we shall ourselves see, or rather be, the reverse movement by which the cosmic impetus is degraded, by a kind of process of solidification or chilling or crystalization, into matter and intellect. Reality is pure creative activity, but apparently this creative activity is interrupted or diverted, and in this interruption of the creative current the material world and the spatializing intellect arise. But the creative current is not degraded utterly nor all at once. It still retains even in its degradation some of the force of the main cosmic stream from which it has been diverted. And so the material world and the materialized and materializing intellect, short apparently of pure mathematics and the mathematical intellect, always exhibit two contrary movements. Matter tends naturally toward homogeneous space and necessary determination, just as the intellect left to itself tends toward geometry. But nevertheless this movement is always counteracted by some form of life the function of which is always to convert determination into indetermination and liberty.

By means of this theory, Bergson thinks, it is possible to avoid the difficulty which confronts the Kantian philosophy as to how it has come about that the categories are adapted to work upon the manifold of sensibility at all. Kant had supposed "that there are three alternatives, and three only, among which to choose a theory of knowledge: either the mind is determined by things, or things are determined by the mind, or between mind and things we must suppose a mysterious agreement. But the truth is that there is a fourth alternative which consists first of all in regarding the intellect as a special function of the mind, essentially turned toward

inert matter; then in saying that neither does matter determine the form of the intellect, nor does the intellect impose its form on matter, nor have matter and intellect been regulated in regard to one another by we know not what pre-established harmony, but that intellect and matter have progressively adapted themselves one to the other in order to attain at last a common form. This adaptation has, moreover, been brought about quite naturally, because it is the same inversion of the same movement which creates at once the intellectuality of mind and the materiality of things" (p. 217).

This theory seems to raise far more difficulties than it solves. In the first place it is difficult to understand how the cosmic impulse ever can become degraded at all. Is it because the cosmic impulse, which is God, unceasing life, action and freedom, becomes weary? If so, what becomes of the argument that the cosmic impulse is pure creative activity?—an argument which alone, according to Bergson, can save us from the difficulties and deadlocks of the intellect. The metaphor of the stream of life which becomes diverted by matter only to get a better grip on matter does not help in the least, because this theory was put forward as explaining the genesis of matter. Instead of pure duration explaining matter, matter has to be appealed to in order to explain duration.

Moreover there is a further difficulty in this account of the ideal genesis of matter in connection with Bergson's view of the nature and validity of mathematics. Matter is constituted by the reversal of the cosmic impetus, but this movement of matter toward externality and spatiality is never complete. "Matter is extended without being absolutely extended," because in every actual material system there is always a certain amount of interaction between the parts, whereas in a purely extended system every part would be utterly indifferent to every other part. But although the reversal of the cosmic impetus has originated at once "the intellectuality of mind and the materiality of things," yet the intellect outruns the spatiality of things, and so we get pure mathematics. If this is so then it is untrue to say, as Bergson does, that "intellect and matter have progressively adapted themselves one to another to attain at last a common form" (Creative Evolution, p. 217), and we have not bridged over the Kantian antithesis of matter and form.

"Our perception," Bergson says, "whose rôle it is to hold up a light to our actions, works a dividing up of matter that is always too sharply defined, always subordinate to practical needs, consequently always requiring revision. Our science, which aspires to