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The lower portion of the lower or visible sphere will consist of

shadows and reflections, and its upper and smaller portion will

contain real objects in the world of nature or of art. The sphere

of the intelligible will also have two divisions,-one of

mathematics, in which there is no ascent but all is descent; no

inquiring into premises, but only drawing of inferences. In this

division the mind works with figures and numbers, the images of

which are taken not from the shadows, but from the objects,

although the truth of them is seen only with the mind's eye; and

they are used as hypotheses without being analysed. Whereas in

the other division reason uses the hypotheses as stages or steps

in the ascent to the idea of good, to which she fastens them, and

then again descends, walking firmly in the region of ideas, and

of ideas only, in her ascent as well as descent, and finally

resting in them.