196 
M. Peyrard has publishe. a correct 
translation ; to this is added a memvir, by 
M. Delsiahre, ou the armhmetie of the 
reeks. Before this memoir, of which 
your majesty deigned te furnish the sub- 
ject, it could scarcely be conceived how 
‘the Greeks, with a notation so imperfect 
compared with ours, could have per- 
formed the operations pointed out by 
Archimedes and Ptoloimy. 
Ancient geometry admitted in its de- 
monstratious only ‘what could be exe- 
cuted with the rule and the compass. 
Mascheroni, still more severe, wished to 
suppress the rule, and we may be asto- 
nished at the great number of néw and 
interesting propositions which be has 
been able to discover in a subject appa- 
rently exhausted. His principal theorems 
were brought to France with the treaty 
of ‘Campo-F ormio by the conqueror and 
pacificator of Italy, It was desirable to 
become acquainted with the whole work, 
and a French translation of it was svon 
published. 
Several moderns had already used with 
success the method which refers to three 
rectangular co-ordinates, the position of 
any point whatever taken in space. M. 
Monse has made this principle the 
foundation of a new and complete system, 
which is indispensable to all the arts of 
building, and to which has been given 
the nae of Descriptive Geometry. 
Trigonometry is undoubtedly one of 
the most useful applications of ele- 
mentary geometry: it is the basis of 
geadesia,* of geography, of astronomy, 
and of navigation. ‘Lhe finest geodesic 
monument was the map of France, by 
Cassini. Some doubts, raised in 1787, 
on the respective positions of the ob- 
servatories of Londen and Paris, re- 
quired a verification of the points situated 
between Dunkirk and Boulogne. The 
English, on their side, were to form new 
triangles between London and Dover, 
and both commissioners were conjointly 
to measure the triangles which crossed 
the Channel. From the progress of 
the arts and sciences it was to be ex- 
pected that the English would exert 
themselves to excel all that had been - 
done of this kind; they succeeded: 
Ramsden’s theodolite, the Indian fires 
which served as signals, the new appa- 
ratus used to measure the bases, afforded 
& precision unheard of before. ‘Fhe 
French had only angles tu measure ; the 
repeating circle which Borda had just in- 
rt cit pr 2 : — - ? 
* Surveying. 
Progress of the Sciences. 
fSept. *, 
vented, was not of so splendid a form 2s 
the theodolite, but it contained m its 
own construction a priiciple which se- 
cured’ to it an accuracy at least equal 
and more iydependent of the talents of 
the artists. ‘Uhe French commissioners, 
Cassini, Legendre, and Mechain, were 
Not unequal to the cempetition. 
From this successful essay arose the 
idea of the operation on which was soon 
after founded anew system of measures ; 
the quarter of thé meridian was to be 
the primary unity; 16 being impossible 
to effect the entire measure, the most 
extensive arc preseuted by any continent, 
namely, that cumprised between Dun- 
kirk and Barcelona, was chosen. Me- 
chain and Delambre were charged with 
this work, which the circumstances of 
the times rendered so dificult. Their 
operations, always thwarted, long sas- 
pended, began in 1792, and euleH only 
in 1799, They triclak tates in five different 
places the elevation of the pole, and the 
direction of the meridian, Their trian- 
gles exteaded from Dunkirk to Barce- 
lona, Delambre, moreover, measured 
two bases of 12,000 metres each, and 
notwithstanding the interval of 700,000 
which separated them, they agreed 
within three decimetres. 
‘This almost incredible precision wa 
due, partly, no doubt, to the care of the 
observers, but principally to the circle 
of Berda, which, by the taultiplication 
of the angles, obv iated the errors of the 
division, and of the observations; it was 
due to fue ingenious construction of the 
rules contrived by the same geometrician, 
and to the care which he had taken in 
their verification. 
Ten degrees of the meridian were 
exactiy known. 
possibility of adding two new degrees to 
them, hy extending the triangles to the 
Balearic Islands. The execution of this: 
project, which has since cost him his 
life, has been lately resumed by two 
young astronomers, of ancommon talents 
aud resolution, (M. M. Biot and Arago} 
who are at present continuing it, and 
will compiete it this winter (12). 
The loss of Mechain, so deeply felt 
by all men of learning, left to his col- 
league alone the charge of all the calcus 
lations, and of arranging the work which 
was to contain all the illustrations and 
proofs. He has taken the greatest care 
in publishing. the observations with the. 
utmost fidelity, in stating all the formule 
of reduction, and in demonstrating them 
in au elementary manner, M. Levendre 
gave 
Mechain perceived the 
4 
