Prepared in connection with his lectures as Professor of Perspective at the Royal Academy, Turner’s diagram is based on an illustration from Samuel Cunn’s

*Euclid’s Elements of Geometry* (London 1759, Plane Trigonometry). It illustrates Propositions 1, 2, 3 and 4 on Plane Trigonometry. The left figure represents Proposition 1, which is described as: ‘Theorem. The two sides of any right-angled triangle being given, the other side is also given’.

1 The right diagram illustrates three Propositions: Proposition 2, ‘Problem. The sine DE of the arc BD, and the radius CD, being given, to find the cosine DF’, Proposition 3, ‘Problem. The sine DE of any arc DB being given, to find DM or BM, the sine of half the arc’ and Proposition 4, ‘Problem. The sine BM of the arc BL being given, to find the sine of a double that arc’.

2^{ } There are sketches of the Propositions in Turner’s lecture notes.

3