ON display in the Museum is an eighteenth-century cabinet of geometrical solids; hidden in one of its drawers lie these boxwood spheres, illustrating Proposition 18 from Book XII of Euclid's *Elements of Geometry*.

At 25mm and 35mm in diameter, the spheres are two of about a hundred solids of varying shapes and sizes made to illustrate the Greek work of the 3rd century B.C. which was still the standard geometry text at the time these spheres were made. All the solids are made of boxwood, some, like the smaller sphere here, come in more than one piece and others have wire attachments.

Proposition 18 of Book XII of the *Elements *states that 'spheres are to one another in the triplicate ratio of their respective diameters' and is one of several applications of the so-called 'method of exhaustion' found there. Largely on evidence from Archimedes, the method of exhaustion is attributed to Eudoxus of Cnidus, who was a member of Plato's Academy from around 370 B.C. until his death some twenty or so years later.

The method is a *reductio ad absurdum* and was put to use by the Greeks as the method of proof of a wide range of different results. The smaller sphere, in coming apart, is actually two spheres, one inside the other, and as such points to a contradiction in Euclid's *reductio*: that the smaller sphere is actually larger than the sphere in which it sits.

The set of geometrical solids of which the spheres are part came from the workshop of George Adams senior (1709-1772) who kept a shop 'at the Sign of Tycho Brahe's Head' in Fleet Street, London. Adams's business was a large concern, selling a wide range of optical, mathematical and philosophical instruments. It is unlikely that he fashioned the spheres with his own hand, yet he did act as a sort of 'quality controller', insisting that he did 'always inspect and direct the several Pieces myself.'

Adams's reputation rested in no small measure on a number of royal appointments and many of the instruments now found in the King George III collection in the Science Museum were produced by Adams for the King.

The set dates from before 1757, since the address given on the ivory scrolls on the outside of the cabinet - 'the corner of Raquet Court in Fleet Street London' - was vacated by Adams after a fire in a neighbouring property in that year. Whilst the set may have been made much earlier than 1757, only two other examples are known to survive (in the Whipple Museum, Cambridge and a private collection) and these are dated between 1757 and 1760. Quite possibly all three sets were made within a relatively short space of time, in which case the Oxford set must have been made not long before Adams's move.

Things are made slightly less clear by the fact that the set of solids is mentioned in two of Adams's catalogues, in 1746 and 1753. However, in both cases these are long lists of unpriced items which seem to be intended to cover everything the customer could wish for, and do not necessarily imply that Adams stocked such a variety. Whether such a set of solids would have been included before Adams had made one is difficult to ascertain.

Other wooden geometrical solids can be found from both before and after the period of these sets. Earlier sets are mostly of the Platonic solids, which have their own important history and which proved inspirational to da Vinci and Kepler, amongst others. Later sets were often used as aids in crystallography teaching, as was probably the case with a box of 'Larkin's Geometrical Solids' also to be found in the Museum.

During the eighteenth century, geometrical solids might be used to demonstrate the conic sections or in centre of gravity experiments. The Adams set is somewhat unusual, as it is labelled for use with a specific text and is therefore possibly best viewed in the context of a whole range of presentations of Euclid's work being made at the same time.

As with the rest of the models in the set, the spheres present solid geometry in three dimensions, overcoming the difficulties a learner might face when attempting to come to grips with the two-dimensional representations found in text editions of Euclid. There were, however, a variety of schemes on offer which were promoted as methods of surmounting such problems.

As far back as the first English edition of the *Elements* in 1570, the editor Henry Billingsley used fold-up flaps of paper and cut-outs to add a three-dimensional element to the text, re assuring the reader that these cut-outs would solve any potential problems with which they might be faced.

Amongst the editions of Euclid brought out during the eighteenth century, several editors used methods similar to those of Billingsley, as well as more elaborate systems involving coloured threads and 'pop-up' solids. John Cowley, for instance, who later became Professor of Mathematics at the Royal Academy, Woolwich, published an appendix to Euclid's *Elements* in 1758 that contained a number of flat templates which the reader could then fold up to form various solids.

At the time, writers disagreed about the pros and cons of geometrical models. Whilst John Williamson (editor of a two volume *Elements* published in the 1780s) claimed that it was better for readers simply to get used to two-dimensional projections, Maria and Richard Edgeworth, in their *Practical Education*, recommended a set of models by Benjamin Donne, having 'no doubt of their entire utility'. Each opinion was aimed at promoting the particular advantages of the scheme of the writer in question. For instance, solid models were criticized for their hidden internal parts by those advocating paper and thread models.

Discussions of this sort might not have been held to be very important, were it not for the fact that, as Adams himself put it, mathematics had 'become a necessary part of almost every Gentleman's Education.' Mathematics also became fashionable in the eighteenth century, with popular journals such as the *Universal Magazine* and *Ladies' Diary* devoting space to mathematical puzzles.

It comes as no surprise therefore that close attention was paid to the ways in which geometry might be taught in a manner that was both productive and entertaining, qualities Adams would no doubt have been striving for when he designed these spheres.

Michael Rich

Selwyn College, Cambridge