REES’S CYCLOPCEDIA. 
pally on the errors and deficiencies which 
have suggested themselves to us in the 
perusal, in order that they may be cor- 
rected and supplied in the succeeding 
volumes. , 
Among the mathematical articles, the 
first of importance that occurs is Aaaly- 
sis, in which we approve highly of the 
discrimination made between the ancient” 
and modern analysis, and the recommen- 
dation to adopt “ the method of the an- 
cients in the commencement of our stu- 
digs, as it serves to form the mind, and to 
fix proper habits, to which the method 
of the moderns should succeed, as it is 
best suited to extend our views beyond 
the present limits, and to assist us in mak- 
ing new discoveries and improvements.” 
It is justly observed also in this article, 
that Newton himself investigate his 
theorems in a different manner frora that 
in which he has delivered them } «as they 
are commonly “ analytical calelations, 
disguised by substituting the name of 
lines for their algebraical value.”* We 
may add, that the generally believed re- 
port at Cambridge is probably true, that 
the last proposition of the seventh section 
was one of the first discovered by him, 
and that he was led by it to expand his 
system in the manner in which it now ap- 
pears. We need not add, that the inves- 
tigation was made by the method of 
fluxions, and the fluents were afterwards 
converted into areas, to be visible to the 
eye. ‘ 
In the account of the Angle of contact, 
we regretted that the demonstration was 
not given of the ratios of the evanescent 
subtenses in curves of different orders: 
{t would not have occupied the space of 
more than four lines, and the article 
would have been complete. » This defect 
may be remedied under the article Con- 
tact. 
Anomaly is very well explained ; but 
in this, asin several other articles, we do 
not find sufficient attention paid to the 
labours of the present generation. ‘The 
problem of Kepler is of great importance, 
und has employed the talents of the first 
mathematicians. Their methods are men- 
tioned, and Keill’s is given at length ; 
but no notice is taken. of Mr. Ivory, 
whose very able paper upon this subject, 
pointed out in our first volume, deserved 
the consideration of, and ought not to 
have been unknown to the writer of this 
article. 
4X very good account is given of Apel. 
909 
lonius of Perga and his work, which of 
course led to the notice of various writers 
onconic sections: and due praise is given 
to the very ingenious attempt of Mr. 
Walker to deduce all the properties of 
the curves, commonly called conic sec- 
tions, from the 24th proposition of Sir 
Isaac Newton’s universal arithmetic. It 
is rather singular, however, that the 
writer of this article should not have 
been acquainted with two other treatises 
of conic sections, upon a similar prin- 
ciple with Mr. Walker’s. . A gentleman 
of Cambridge, about sixteen years ago, 
brought with him from the continent the 
conic sections of Boscovich»with which 
he was so much pleased, that he drew up 
a course of lectures from them for the 
use of his pupils, which, with the work 
itself, he afterwards put into the hands of 
Mr. Newton, of Jesus College. Mr. New- 
ton was equally pleased with the system, 
which he improved very much, by reject- 
ing the musical proportion, and giving 
demonstrations of his own wherever they 
were requisite. His system was publish- 
ed by the university; and it is remark- 
able, that just as it was going to the press 
the manuscript of Mr. Walker was sent 
to Cambridge, and put into the hands of 
the gentleman who had brought the trea 
tise of Boscovich from the continent, and 
who was struck with this remarkable co- 
incidence in demonstrations between two 
writers who had no intercourse with eac 
other. The merit of Mr. Walker is uni- 
versally acknowledged, and it cannot be 
depreciated by this account; but it will 
afford to mathematicians some pleasure 
to’ compare together the different pro 
cesses used by two eminent writers, from 
the same hypothesis, and the improve: 
ments made im these processes, — 
The article of Approximation is more 
defective than we could well have ex: 
pected. Baron Maseres has treated this 
subject with his usual accuracy ; but his 
name is not mentioned in the article; not 
is the least notice taken of his tracts on 
the resolution of affected algebraic equa- 
tions, in which the methods of approxi- 
mation by Dr. Halley, Mr. Raphson, and 
Sir Isaac Newton are compared together, 
and the advantages and disadvantages of 
each balanced with the utmost degree of 
precision. Dr. Hutton is represented as 
being the discoverer of a peculiariy con- 
venient method of approximating to the 
roots of pure powers ; and thus De Tvas 
guy is deprived of the honour duete him 
ar 
(mVareas & Te, 
