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For in the case of rays which, for example, fall parallel upon a spherical surface AFE, intersecting one another, after refraction, at different points, as this figure represents; what can the waves of light be, in this transparent body, which are cut at right angles by the converging rays? For they can not be spherical.
All the reflexions of those rays which fall upon the quarter-circle AB will touch a curved line AFE, of which line the end E is at the focus of the hemisphere, that is to say, at the point which divides the semi-diameter BD into two equal parts.
If any one tripped when repeating it he had to pay a forfeit. Another might be in rhyme and run as follows: "Na au sau mai Safata, Ou afe i le ngatai ala, E fafanga i si au tiaa, Fafanga, fafanga, pa le manava. Fafanga, fafanga, pa le manava." Another as his puzzle to repeat correctly would give: "Na au sau mai Mali'oli'o, Lou ala i umu, Lou ala i paito, Lou ala i puto pute, Lou ala i pute puto."
And thus this folded wave will continue to advance until the point N has reached the focus E. The curve AFE can be seen in smoke, or in flying dust, when a concave mirror is held opposite the sun.
And as the parallel rays are merely perpendiculars to the waves which fall on the concave surface, which waves are parallel to AD, it will be found that as they come successively to encounter the surface AB, they form on reflexion folded waves composed of two curves which originate from two opposite evolutions of the parts of the curve AFE. So, taking AD as an incident wave, when the part AG shall have met the surface AI, that is to say when the piece G shall have reached I, it will be the curves HF, FI, generated as evolutes of the curves FA, FE, both beginning at F, which together constitute the propagation of the part AG. And a little afterwards, when the part AK has met the surface AM, the piece K having come to M, then the curves LN, NM, will together constitute the propagation of that part.
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