and accuracy; but if the arc is small it is difficult
to make quite sure of the centre, and our drawing may not be quite
accurate. Now the arc which had been described by the tiny planet during
Piazzi's observations was only three degrees; and if any one will kindly
take out his watch and look at the minute marks round the dial, three
degrees is just _half_ a single minute space. If the rest of the dial were
obliterated, and only this small arc left, would he feel much confidence
in restoring the obliterated portion? This problem gives some idea of the
difficulties to be encountered, but only even then a very imperfect one.

[Sidenote: Gauss shows how to find it.]

Briefly, the solution demanded a new mathematical method in astronomy. But
difficulties are sometimes the opportunities of great men, and this
particular difficulty attracted to astronomy the great mathematician
Gauss, who set himself to make the best of the observation available, and
produced his classical work, the _Theoria Motus_, which is the standard
work for such calculations to the present day. May we look for a few
moments at what he himself says in the preface to his great work? I
venture to reproduce the following rough translation (the book being
written in Latin, according to the scientific usage of the time):--

EXTRACT FROM THE PREFACE TO THE
_Theoria Motus_.

[Sidenote: The _Theoria Motus_.]

"Some ideas had occurred to me on this subject in September 1801, at
a time when I was occupied on something quite different; ideas which
seemed to contribute to the solution of the great problem of which I
have spoken. In such cases it often happens that, lest we be too much
Distracted From the Attractive Investigation On Which We Are
Engaged, We Allow Associations Of Ideas Which, If More Closely
Examined, Might Prove Extraordinarily Fruitful, To Perish From
Neglect. Perchance These Same Idea-lets of Mine Would Have Met With
This Fate, If They Had Not Most Fortunately Lighted Upon a Time Than
Which None Could Have Been Chosen More Favourable For Their
Preservation and Development. For About The Same Time a Rumour Began
To Be Spread Abroad Concerning a New Planet Which Had Been Detected
On January 1st of That Year at the Observatory Of Palermo; and
Shortly Afterwards the Actual Observations Which Had Been Made
Between January 1st And February 11th by the Renowned