2 
On the general Principles 
[Jan. 
5. The object is then to obtain the length of the sides ; m other words, the dis- 
tance of every pair of points required to be mapped. This may be effected two ways, 
1st, by measuring, directly, those distances by the application of the linear unit , o 
secondly, bv measuring only one of them, and determining the inclinations to that 
of the two lines joining, to form the summit of the triangle, of which the measured 
line was the base ; in other words, the angles of the triangle. 1 he same operation is 
pursued with respect to the next triangle, except as to the measurement of one of its 
sides, which is supposed to be obtainable from the preceding operation, fhe mea- 
surement of angles, is thus substituted for the measurement of lines. A third or 
mixed method may be stated to consist in the combination of both. 
6. The first or actual measurement of all the lines would be an excessively labo- 
rious operation, and by no means a correct one. It is indeed found that one of the 
most difficult problems to execute is that of measuring with accuracy a given dis- 
tance on the surface of the earth ; and to be at all successful, we must have the 
choice of the ground, and we shall even then find it to be a most tedious and trou- 
blesome one. It is true, the error on each measurement might not be so considera- 
ble as to be of moment, singly considered ; but in a method in which each error is trans-1! 
ferred to the next result, thence to be carried with any accession of error to the fol- 
lowing, it is obviously inadmissible. The method of direct measurement can only 
be applied with advantage to the principle of coordinates ; where the positions are 
independent of each other; supposing, I mean, the tediousness of the operations to be 
no objections. To their application to triangles we object on two grounds ; 1. The 
great labour and delay ; and 2. The great accumulation of error they would cause 
without any means of check or discovery 3 . To counterbalance these it does not 
offer the shadow of an advantage. 
7. It is otherwise with the measurement of the angles or inclination of the sides. 
It is an operation easily, and most conveniently performed ; while from the great per- 
fection of the instruments used, it is to be executed to a decree of accuracy, which 
Mould astonish the uninitiated. In strictness, it is a kind of measurement on a small 
scale, which depends chiefly for its application on the facility with which a straight 
line may be traced through the air by means of a telescope ; yet it is so disguised 
by the circumstances of the case, as to merit being considered a distinct method. 
8. In the measurement of angles we use some kind of circular instrument, in 
which a telescope or pair of sights, to define the line, the inclination of which we 
aie seeking, turns on a centre ; and by observing what proportion of the whole cir- 
cuit the telescope, or connected sight- vanes, pass over when moved from the direction 
of one line to that of another, we obtain the means either of laying down those in 
c .nations on paper by means of a somewhat similar instrument ; or what is better 
still, we calculate the sides of a triangle having the same angles, from which hv a 
Simp e statement of proportion, combined with the known value of one of the side. 
? bta “? tb . e actual lengths of the others ; and are therefore prepared to describe tW 
cucles, the intersection of M hich will be the summit of the triangle 
a. The manner in which the proportion of the sides of triangles is obtained the 
angles ot which are given, is simple enough. Inthe 
cbcTe ai. gUre ,’ 1° , an « ,es at tUc centre of ,Se 
ar , c ' eacll > double the angles at the circumfer- 
encx, resting on equal segments, i. e. the angles of the 
side" 8 It is ^e™Lnt C that d eacl D ' biS "’' ing ““ 
sine of half ti I , dt , each sule is then the double 
the opposite ang^of the* ’ s ° thcr .' vords - of 
we may, by inspection °ne ■ <llor , ds ot l ' lclr doubles; 
several sides to S X? S ‘ t/!’' ° ! the 
which triano’lp.i 1 r * * bls is the method by 
We are now to enquire, what is the d, IMO , 8t usua % resolved, 
by angles is susceptible. It has been already sSLd f, CC ^ ac >* of . which the method 
3 t. " * in reality it is a measure- 
havehe^KK 
10 . 
scientific EdHCutionqftlx 
