Euclid is well known as the author of The Elements of Geometry, a text that has been at the centre of mathematical teaching for 2000 years, during which time it has been repeatedly edited and translated into many languages.

Searching the Whipple Library collection for Euclid, I came across several edition of the  Elements with various corrections and supplements.  But what absorbed my attention most were the volvelles, inserts and pyramids in some of these books. As I’ve read more about Euclid’s geometry it has become more obvious that the visualisation of three-dimensional objects from two-dimensional figures on the page might be difficult for some readers, especially novices (such as me).  So one of the solutions was to past cut-out figures on additional slips of paper in order to make ‘pop-up’ figures such as pyramids and cubes.  These different cut-outs and projections which appeared on the pages in woodcuts in two dimensions, could be folded upwards to demonstrate the three-dimensional figures described in the text. Here there are some examples:

‘Euclid’s elements of geometry, from the latin translation of Commandine : to which is added a treatise of the nature of arithmetic of logarithms : likewise another of the elements of plain and spherical trigonometry : with a preface, shewing the usefulness and excellency of this work’  (London:  Printed by Tho. Woodward … and sold by J. Osborn …, 1733), by John Keil … The whole revised … Also many faults … are shewn … An ample account of which may be seen in the preface by Samuel Cunn. (STORE CR 1:51)

This book contains Book 1-6 and 11-12 of Euclid’s Elements in the Latin translation by Federico Commandino. It includes 16 folded leaves of plates of drawings and three small volvelles (paper wheel charts) which are fastened on an end-paper with some up-folded parts.

‘An appendix to the Elements of Euclid, in seven books:  Containing forty-two moveable schemes for forming the various kinds of solids, and their sections, by which the doctrine of solids in the eleventh, twelfth, and fifteenth books of Euclid is illustrated, and rendered more easy to learners than heretofore’… (London: Sold by T. Cadell, [1765?]), by John Lodge Cowley… (STORE 86:25)

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The idea of the ‘pop-up’ shapes is that once the readers have learnt how to visualize these types of three-dimensional objects with the aid of the figures, they will be able to imagine all other types of shapes themselves.

Being amused by the imaginative power of pictures and the manual use of the movable parts, I came up with the idea to make up the ‘pop-ups’ by myself. And I couldn’t pick a better time for doing this, as my 11-years-old son Jan was preparing for his SATS! So we both decided to travel back in time to the maths of Ancient Greece and revise geometry through these reproductions: