ustration: FIG. 28.--Frameworks with inscribed and circumscribed
spheres, representing the five regular solids distributed as Kepler
supposed them to be among the planetary orbits. (See "Summary" at
beginning of this lecture, p. 57.)]

Nevertheless, the idea gave him great delight. He says:--"The intense
pleasure I have received from this discovery can never be told in words.
I regretted no more the time wasted; I tired of no labour; I shunned no
toil of reckoning, days and nights spent in calculation, until I could
see whether my hypothesis would agree with the orbits of Copernicus, or
whether my joy was to vanish into air."

He then went on to speculate as to the cause of the planets' motion.
The old idea was that they were carried round by angels or celestial
intelligences. Kepler tried to establish some propelling force emanating
from the sun, like the spokes of a windmill.

This first book of his brought him into notice, and served as an
introduction to Tycho and to Galileo.

Tycho Brahe was at this time at Prague under the patronage of the
Emperor Rudolph; and as he was known to have by far the best planetary
observations of any man living, Kepler wrote to him to know if he might
come and examine them so as to perfect his theory.

Tycho immediately replied, "Come, not as a stranger, but as a very
welcome friend; come and share in my observations with such instruments
as I have with me, and as a dearly beloved associate." After this visit,
Tycho wrote again, offering him the post of mathematical assistant,
which after hesitation was accepted. Part of the hesitation Kepler
expresses by saying that "for observations his sight was dull, and for
mechanical operations his hand was awkward. He suffered much from weak
eyes, and dare not expose himself to night air." In all this he was, of
course, the antipodes of Tycho, but in mathematical skill he was greatly
his superior.

On his way to Prague he was seized with one of his periodical illnesses,
and all his means were exhausted by the time he could set forward again,
so that he had to apply for help to Tycho.

It is clear, indeed, that for some time now he subsisted entirely on the
bounty of Tycho, and he expresses himself most deeply grateful for all
the kindness he received from that noble and distinguished man, the head
of the scientific world at that date.

To illustrate Tycho's kindness and generosity, I must read you a letter
written to him by Kepler. It s