300 
in 1763 went to Copenhagen to pursue 
further studies. In 1764 he was engaged 
by the Academy of Sciences as an assistant 
in the triangulation and preparation of a 
map of Denmark. ‘Till 1805 he remained 
in the continuous employ of the Academy 
as surveyor. Wessel was highly esteemed 
by his contemporaries, and for some special 
work done after leaving the service of the 
Academy he received the Academy’s silver 
medal and a full set of its memoirs. In 
1819, when many of its maps were declared 
out of date, the trigonometric determina- 
tions of Wessel were made a special excep- 
tion. In 1778 he passed an examination in 
Roman law. In 1815 he was made a 
Knight of the Danebrog. He died in 1818. 
While Wessel was always well spoken of 
as a surveyor, he was never mentioned as a 
mathematician. Still the fact that his 
paper was the first to be accepted by the 
Academy from one not a member argues in 
his favor. This acceptance was due to 
Tetens, Councillor of State, to whom the 
MS. had been shown and whose assistance 
in improving it was acknowledged. In the 
History of the Academy of Sciences of 
Denmark published in 1843 Professor Jur- 
gensen classes Wessel with others in the 
statement, ‘The treatises of the other 
mathematicians are monographs of no con- 
siderable scientific value,” or ‘‘ They are too 
special to be discussed more at length.’’ 
In the introduction to his memoir Wessel 
says: 
The present essay has for its object to determine 
how to express segments of straight lines when we 
wish by means of a unique equation between a single 
unknown segment and other given segments to find 
an expression representing at once the length and di- 
rection of the unknown segment. 
To be able to answer this question I shall employ 
two considerations which seem to me evident. In 
the first place, the variation of direction which may 
be produced by algebraic operations ought also to be 
represented by their symbols. In the second place 
we submit direction to algebra only by making its 
variation depend upon algebraic operations. Now 
SCIENCE. 
[N. 8. Von. VI. No. 189. 
according to the ordinary conception we can trans- 
form it by these operations only into the opposite di- 
rection, that is, from positive into negative and re- 
ciprocally. It follows that these two directions only 
would be susceptible of an analytic representation 
adapted to the usual conception and that the solution 
of the problem would be impossible for other direc- 
tions. It is probably for this reason that nobody has 
given attention to this subject. Doubtless nobody 
has felt at liberty to change the definition of these 
operations once adopted. ‘To this there is no objec- 
tion so long as the definition is applied to ordinary 
quantities ; but there are special cases where the pe- 
culiar nature of the quantities seems to invite us to 
give particular definitions to the operations. Then 
if we find these definitions advantageous it seems to 
me that we ought not to reject them. For in passing 
from arithmetic to geometric analysis, that is to say, 
from operations relative to abstract numbers to: 
operations upon segments of a straight line, we shall 
have to consider quantities which may have to one 
another not only the same relations as abstract num- 
bers, but also a great number of new relations. Let 
us try then to generalize the signification of our 
operations ; let us not restrict ourselves, as has been 
done hitherto, to the employment of segments of a 
straight line in the same or opposite senses, but ex- 
tend a little the notion of the way in which they are 
applied not only to the same cases as heretofore, but 
to an infinite number of other cases. If at the same 
time that we take this liberty we have respect to the 
ordinary rules of operations we in no way contravene 
the ordinary theory of numbers, but we merely de- 
velop it, we accommodate ourselves to the nature of 
the quantities and observe the general rule which re- 
quires us to render a difficult theory little by little 
more easy to comprehend. It is not then absurd to 
demand that in geometry operations be taken ina 
broader sense than in arithmetic. We shall admit 
without. difficulty that it will be possible to vary 
the direction of segments in an infinite number of 
ways. Precisely by this means (as we shall show 
later) we succeed not only in avoiding all impossible 
operations and in explaining the paradox that it is 
necessary sometimes to resort to the impossible to ob- 
tain the possible, but we also succeed in expressing 
the direction of line-segments situated in the same 
plane quite as analytically as their length, without 
the memoir being embarrassed by new symbols or 
new rules. Now it must be agreed that the gen- 
eral demonstration of geometric theorems often be- 
comes easier when we express direction in an analytic 
manner and submit it to the rules of algebraic opera- 
tions than when we are compelled to represent it by 
figures which are applicable only to particular cases. 
