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6 Laws, 81 
8 lb. 819 a 
18 INTRODUCTORY 
to the plane of the circle, all its diameters will look equal 
(Prop. 34), but if the joining line is neither perpendicular to 
the plane of the circle nor equal to its radius, diameters with 
which it makes unequal angles will appear unequal (Prop. 35); 
if a visible object remains stationary, there exists a locus such 
that, if the eye is placed at any point on it, the object appears 
to be of the same size for every position of the eye (Prop. 38). 
(2) The second branch is Catoptric, or the theory of mirrors, 
exemplified by the Gatoptrica of Heron, which contains, 
e. g., the theorem that the angles of incidence and reflexion 
are equal, based on the assumption that the broken line 
connecting the eye and the object reflected is a minimum. 
(3) The third branch is a-KgvoypacfHKrj or, as we might say, 
scene-painting, i.e. applied perspective. 
Under the general term of mechanics Geminus 1 dis 
tinguishes (1) opyavoirouK-q, the art of making engines of war 
(cf. Archimedes’s reputed feats at the siege of Syracuse and 
Heron’s (SeXoTrouKd), (2) OavpaTonouKri, the art of making 
wonderful machines, such as those described in Heron’s 
Pneumatica and Automatic Theatre, (3) Mechanics proper, 
the theory of centres of gravity, equilibrium, the mechanical 
powers, &c., (4) Sphere-making, the imitation of the move 
ments of the heavenly bodies; Archimedes is said to have 
made such a sphere or orrery. Last of all, 2 astronomy 
is divided into (1) yvcopoviKy, the art of the gnomon, or the 
measurement of time by means of the various forms of 
sun-dials, such as those enumerated by Vitruvius, 3 (2) perecopo- 
(tkottlkt), which seems to have included, among other things, 
the measurement of the heights at which different stars cross 
the meridian, (3) SLorrrpLKri, the use of the dioptra for the 
purpose of determining the relative positions of the sun, 
moon, and stars. 
Mathematics in Greek education. 4 
The elementary or primary stage in Greek education lasted 
till the age of fourteen. The main subjects were letters 
(reading and writing followed by dictation and the study of 
1 Proclus on End. I, p. 41. 3-18. 2 lb., pp. 41. 19-42. 6. 
3 Vitruvius, De architectura, ix, 8. 
4 Cf. Freeman, Schools of Hellas, especially pp. 100-7, 159.