SMALL’S ASTRONOMICAL DISCOVERIES OF KEPLER. 
«may be deduced in a simple manner, by 
analysing an equation of a higher form, 
and bringing it down to its simplest com- 
ponent part. 
Having in several instances applied 
this mode, and shewn that the coefficient 
- of the second term is equal to the sum 
of the roots of the third, to the sum of the 
products of each pair, and so on, where 
the equation has as many roots as it has 
dimensions, the Baron proceeds to prove 
that those truths were known to Vieta, 
and he first gives the fourteenth chapter 
. of the original, and then transforms it 
into modern terms. The evidence is 
thus complete, and due honour must be 
ascribed to him who, ignorant of the 
science of the moderns that a quantity 
could be less than nothing, or a number 
impossible, deduced with the rigour of 
ancient demonstration those properties 
which, by many, are supposed to have 
been first made known by Harriot’s in- 
yention. 
The obscurity of Dr. Waring’s writ- 
ings has tortured many an algebraist, 
and as long as persons content them- 
selves with mere general expressions 
‘without application to practice, it is not 
likely that they should form a clear idea 
of the excellence of any method for dis- 
covering the roots of a complicated 
equation. In a complete  biquadratic 
tx — gx? -+ pxi— xt = s, the various sup- 
positions that may be made will make 
great changes in the estimate of a solu- 
tion; and after a very exact comparison 
825 
of the modes of Waring and Ferrari, the 
Baron gives the preference to that of the 
latter. Since the method given by Wa- 
ring, though “ it proceeds upon clear 
and certain grounds when it is applied 
only to trimomial quadratic equations, 
in which the cube of the unknown quan- 
tity is wanting, becomes too intricate and 
unsatisfactory when it is applied to qua- 
drinomial biquadratic equations, or such 
as have all their terms complete.’’ 
Though we admire the perspicuity 
which prevails throughout the Baron’s 
/writings, we cannot but think that his 
love of it leads him frequently into unne- 
cessary prolixity. There are certain ope- 
rations with which a young algebraist 
must make himself acquainted before he 
makes any progress in science ; but if he 
is qualified to read these tracts, the repe- 
tition of such operations is superfluous, 
and to the higher mathematician is irk- - 
some. Again, the old algebraists hav- 
ing been brought up under geometri- 
cians, naturally brought their terms, 
which belong to discrete quantity or 
number, to a resemblance of those of 
continued quantity or extension. Hence 
the terms quadratic, cubic, biquadratie, 
sursolid were used; but in algebra they 
express merely the number of times a 
number is multiplied into itself, and as 
this may be carried far beyond any ana- 
logy with geometry,. it seems.in these 
times to be useless to endeavour: to-pre- 
serve It, 
Arr. LV. Aa Account of the Astronomical Discoveries of Kepler, including -an Astrovomical 
* Review of the Systems which had successively prevailed before his Time. 
Smauy, D. De F. RIS. 8vo. 
THE labours of Kepler are known to, 
ad justly appreciated by, those only who 
have paid-the deepest attention to phy- 
‘sical astronomy. ‘I'he confirmation of his 
theory by Sir I. Newton, and the compa- 
rative ease with which the laborious cal- 
culations of tiepler may now be per- 
formed, have superseded in great mea- 
sure the necessity of studying his works’; 
and the generality of the practical astro- 
nomers of the present day look no farther 
‘than Sir I. Newton as the origin of all 
the modern discoveries in this important 
science. But without Kepler this island 
‘ould not have boasted of a Newton; 
and the progress of the human mind, 
tom the first conceptions of sense to the 
natured reflections of judgment, is a 
sibject which cannot but be highly gra- 
tlying to every man of sci¢nce, 
Ly Ropert 
This subject is in the work before us 
developed with great judgment, and the 
writer’s researches cannot be followed 
but by those who are deeply read in ma- 
thematical investigation. ‘The principal 
motions and inequalities of the heavenly 
bodies are first described, and then the 
various. theories which the -ancients, 
particularly Ptolemy; adapted to explain 
them. ‘The inefficacy ‘of those theories 
is pointed out, and the steps taken by Co- 
pernicus previous to the establishment in 
his own mind of the true system, and the 
difficulties in his way which prevented 
his early promulgation of it, are investi- 
gated with great sagacity. It is not 
wonderful that the truth should not have 
been immediately acknowledged, when 
we reflect on the effect of prejudice over 
the humap mind, Lo place the sun in 
