In this very imperfect sketch I have confined myself almost exclusively to the Universities; I have done so chiefly for the lack of material that could be presented in a sufficiently definite form. De Morgan has made the remark *(Arithmetical Books,* p. vi)

I cordially agree with De Morgan in this attitude. The historian of mathematics should know not merely the work of the pioneers but the extent to which mathematics permeates the community, and the history is shorn of much of its value if it pay little or no attention to the humbler exponents through whom the subject is brought to bear upon the general culture of the citizen. I would suggest therefore as a research that would be of distinct value an enquiry into the curricula and textbooks of the schools during the period I have been dealing with. So far as I am aware there is no satisfactory account of such matters in existence. There is one textbook of Arithmetic whose fortunes have been traced with some success and that is Gray'sIt is essential to true history that the minor and secondary phenomena of the progress of mind should be more carefully examined than they have been. ... Copernicus and Newton would fill a large space, though the history of knowledge were written down to that of every individual who ever opened a book: but it seems to me that they and their peers are made to fill all the space. Nor will it be otherwise until the historian has at his command a readier access to second and third rate works in large numbers; so that he may write upon effects as well as causes.

*Arithmetic,*on which Mr Tweedie contributed a most interesting article to the

*Proceedings*of our Society (vol. 43, pp. 70-80, 1924). Research on the lines indicated by Mr Tweedie seems to me to be called for, and is surely within the compass of our members. It would not demand a wide knowledge of modern developments but it would call for patience and industry; above all it would need to be based on actual inspection of books and not on second-hand statements about books. Probably no one University would provide specimens of all books reported to exist; many even may have completely disappeared - the special fate of the early editions of popular school books; but in the four University Libraries and in Public Libraries there are probably many textbooks whose existence is apparently unknown.

The following list of books that I have seen and turned over, though not thoroughly examined, except in a few cases, may be noted.

**On Arithmetic or Arithmetic and Algebra.**

*Tyrocinia Mathematica*. By George Sinclair. Glasgow: 1661.

*Arithmetica Infinita*. By Rev. George Brown. No place. 1717-18.

A set of tables of decimals of a £, multiples of the farthing.

*A new System of Arithmetic, both theoretical and practical*. By Alexander Malcolm. London 1730.

Malcolm was a graduate of Marischal College, Aberdeen, and this book is of quite striking merit. He was a teacher and writing master in Aberdeen; eventually he made his way to America where he died in 1763.

*Arithmetic, Rational and Practical*. By John Mair. The edition I have seen is the 5th. Edinburgh: 1794.

Mair was rector of Perth Academy, and the author of several textbooks for schools; that on Bookkeeping ran through several editions.

*Introduction to Arithmetick*. By John Wilson, A.L.M. Teacher of Mathematics and consequently Navigation. Edinburgh, 1741.

*A Short System of Arithmetic and Bookkeeping*. By Robert Hamilton. London: 1788.

Hamilton was Professor of Mathematics in Marischal College. A much more important book is his:

*Introduction to Merchandize*.

It ran through several editions, the first (which I have not seen) being published at Edinburgh in 1777. The copy I possess is a later edition, "new-modelled and adapted to the improved methods and information of the present time." By Elias Johnston, Teacher of Mathematics in Edinburgh. Hamilton is much better known through his work in Political Economy. His *Inquiry concerning the Rise and Progress, the Redemption and Present State of the National Debt of Great Britain,* published in 1813, made a very considerable name for him in political circles.

*The Practical Figurer or an Improved System of Arithmetic*. By William Halbert. Paisley: 1789.

Halbert was schoolmaster at Auchinleck; the book has a portentously long title. Of no special merit; pretentious.

*The Young Ladies Arithmetic*. By John Greig. 2nd Ed. London: 1800.

Greig was a graduate of Marischal College; the book is stated to have gone through many editions.

The school books on Arithmetic by Gray and Melrose went through many editions; Melrose's was edited, in the later issues, by Ingram and Trotter. A study of Melrose on the lines on which Mr Tweedie treated Gray would, I think, show a remarkable longevity.

There are probably many books besides these which I have not seen. The schoolbooks on Geometry are almost limited to editions of Euclid, based on Simson, such as Playfair's and Ingram's - both excellent books. To the texts of Euclid was frequently appended a short textbook of *Trigonometry, both Plane and Spherical*; I possess issues of Simson, Playfair and Ingram which contain such Appendices. No doubt many of these would be for University use, but certainly not all.

Translations of Simson's *Conic Sections*, chiefly the first three books, seem to have been frequently used; an edition I possess is "a new and revised, edition," issued at Glasgow in 1817. The *Elements of Conic Sections,* by Richard Jack, Teacher of Mathematicks in Edinburgh (Edinburgh: 1742) owes a good deal to Simson's *Conics.*

A separate book on Trigonometry, with the title *Elements of Trigonometry, plane and spherical; with the Principles of Perspective and Projection of the Sphere,* by John Wright, was published at Edinburgh in 1772.

Perhaps the most characteristic type of book on mathematics is that represented by the well-known works of John Davidson, Alexander Ingram, and, though less known, Alexander Ewing. Of the personal history of these men I know little. Davidson was a teacher in Burntisland, Ingram and Ewing in Edinburgh; I think Davidson was a schoolmaster, but Ingram and Ewing "private teachers," of whom there seem to have been a considerable number in the University towns. As is well known these books range over the whole field of elementary mathematics and many of its applications. I am old fashioned enough to hold that they were an excellent introduction to mathematics; they were prized in many a village school, were used for the training of the sons of artisans and farmers at a time when the parish was almost self contained, and secured for the schoolmaster a status as a scholar that is not nowadays accorded to an Honours graduate.

This short list could probably be extended very considerably, and I hope that we shall not have long to wait for a thorough investigation of the whole subject.