558 
eludes, that the fecond-and third books 
of Newton’s optics will henceforth be 
better underfiood, than the firft has hither- 
to been; at the fame time, he obferves, 
that ‘‘ the merits of their author in natu- 
val philofophy are great beyond all con- 
teft or compariion; his optical difcovery 
of the compofition of white light, would 
alone have immortalized his name; and 
the very arguments which tend to-over- 
throw his fyitem, give the-ftrongeft proofs 
of the admirable accuracy of his experi- 
ments.” — 
MATHEMATICS. 
Mr. Ropert Woopuovusk, of Caius 
College, Cambridge, having, in a former 
paper, fhewn the infufiiciency, in mathe- 
natical reafoning, of a principle of ana- 
logy, by which the properties demon- 
fuiated for one figure were to be trans- 
terred to another, to which the former 
was fuppofed to bear a refemblance: the 
argument he then ufed was,that the analogy 
between the two figures was neither ante- 
cedent to calculation, ner independent of 
st, and confeguently could not regulate it; 
that analogy was the cbject of inveftiga- 
tion, not the guide; the reiult of demon- 
ftration, not its direéting principle. The 
principal objeét of that paper was to 
prove, that operations with imaginary. 
quantities were conducted after the fame 
raanner as operations with quantities that 
can be arithmetically computed. In the 
progrefs of this curious and important in- 
weitigation, he is fatisfied that demonftra- 
tion is the theory of angular fun@tions, 
is nat only moft eaty and direct, by giving 
to quantities their true and natural repre- 
fentation; but, that the introduction of ex- 
preflions and formulas not analytical, into 
analytical invettigation, has caufed much 
ambiguity and paradox; that it has made 
demonfiration prolix, by rendering it lefs 
direct, and bas made it deficient in preci- 
jion, by diverting the mind from the true 
fource of analytical expreffion. 
In the paper now befcre us, the author 
propofes to fhew, that. in a!l analytical 
inveftigations, geometrical formulas are 
toreign and circumlocutory, not eflential- 
jy neceflary, and therefore ought to be 
excluded ; and that algebra, being an uni- 
verfal language, is competent to exprefs 
the conditions belonging to any fubje&t of 
enguiry, and, it adequate expreflions be 
obtained, then with fuch reaionings and 
decuiions may be carried on. 
Mr. Woodhoule examines alfo the quef- 
tion concerning the refpective advantages 
ef the ancient geometiy and modern ana- 
Praceedings of Learned Societies 
[July 1, 
lyfis. His obfervations on this fubject . 
which has been difcuffed a thoufand times, 
merit the attention of the mathematician. 
The fuperiority, he fays, of one method 
above another, muft confilt in being either 
more logically ftri€t in its deductions, or 
more luminous, or more commodious for 
inveftigation, If in geometry the infe- 
rences are more ftri€tly deduced, than in 
the algebraic calculus, it muft arife from 
the great attention with which the former 
{cience has been cultivated. The perfpi- 
cuity and commodioufnefs of inveftiga- 
tion feems to decide in favour of geo 
metry, inafmuch as it employs a particu- 
lar individual, the fign and reprefentatioa 
of a genus, inftead of a generic term 3 
in algebra the Ggns are altogether arbitra~ 
ry, in geometry they bear a refemblance 
to the things fignified, and are cailed ma- 
tural fians, fince the figure of a triangle 
or {quare fuggefts the fame tangible figure. 
every where, and may be confidered as the 
reprefentative of all triangles and fyuares. 
Moreover, to prevent ambiguity, it is fre- 
quently neceflary to recur from the fign to 
the thing fignified, which is more eafily 
done in geometry than in algebra. | 
In cetence of the analytical calculus, i€ 
is obferver, that no language like that of 
analyfis is capable of {uch elegance as 
flows from the development of a long fe- 
ries of expreffions, connected one with the 
other, and all dependent on the fame fun- 
Gamental idea. And if what has been 
aftually done by each method in the ex- 
planation of natural phenomena, be con- 
fidered, the fuperiority of the one above 
the other will appear immenfe ; the cul- 
tivators of geometry were men of confum- 
mate abilities, and the method of reafon- 
ing employed by them had, during pre 
ceding times, received the greatett im- 
prevement. The analytical calculus, 
which has verified the principle of gravi- 
tation, was, a century ago, in its infancy. 
‘¢ The queftion (lays Mr. Woodhoufe) 
concerning the refpective advantages of 
ancient geometry and modern analyfis, 
may be comprifed in a fhort compafs. If 
mental difcipline and recreation are fought 
for, they may be found in both methods 5 
neither is effentially inaccurate ; and, al- 
though in fimple enquiries the geome- 
trical has greater evidence, in abftrufe and 
intricate inveftigation the analytical is 
the moft luminous: but, if the expedi- 
tious deduétion of truth is the object, 
then I conceive the analytical calculus 
ought to be preferred. To arrive at @ 
certain end, we faould furely ule the fim- 
: hae ib Fe plet 
j 
